1. Preface

The goal of these notes is to introduce the reader to the following.

  1. Parallel computing in imperative programming languages and C++ in particular, and

  2. Real-world performance and efficiency concerns in writing parallel software and techniques for dealing with them.

For parallel programming in C++, we use a library, called PASL, that we have been developing over the past 5 years. The implementation of the library uses advanced scheduling techniques to run parallel programs efficiently on modern multicores and provides a range of utilities for understanding the behavior of parallel programs.

PASL stands for Parallel Algorithm Scheduling Library. It also sounds a bit like the French phrase "pas seul" (pronounced "pa-sole"), meaning "not alone".

We expect that the instructions in this book and PASL to allow the reader to write performant parallel programs at a relatively high level (essentially at the same level of C++ code) without having to worry too much about lower level details such as machine specific optimizations, which might otherwise be necessary.

All code that associated with this book can be found at the Github repository linked by the following URL:

This code-base includes the examples presented in the book, see file minicourse/examples.hpp.

This book does not focus on the design and analysis of parallel algorithms. The interested reader can find more details this topic in this book.

2. C++ Background

The material is entirely based on C++ and a library for writing parallel programs in C++. We use recent features of C++ such as closures or lambda expressions and templates. A deep understanding of these topics is not necessary to follow the course notes, because we explain them at a high level as we go, but such prior knowledge might be helpful; some pointers are provided below.

2.1. Template metaprogramming

Templates are C++'s way of providing for parametric polymorphism, which allows using the same code at multiple types. For example, in modern functional languages such as the ML family or Haskell, you can write a function $\lambda~x.x$ as an identity function that returns its argument for any type of $x$. You don’t have to write the function at every type that you plan to apply. Since functional languages such as ML and Haskell rely on type inference and have powerful type systems, they can infer from your code the most general type (within the constraints of the type system). For example, the function $\lambda~x.x$ can be given the type $\forall \alpha. \alpha \rightarrow \alpha$. This type says that the function works for any type $\alpha$ and given an argument of type $\alpha$, it returns a value of type $\alpha$.

C++ provides for polymorphism with templates. In its most basic form, a template is a class declaration or a function declaration, which is explicitly stated to be polymorphic, by making explicit the type variable. Since C++ does not in general perform type inference (in a rigorous sense of the word), it requires some help from the programmer.

For example, the following code below defines an array class that is parametric in the type of its elements. The declaration template <class T> says that the declaration of class array, which follows is parameterized by the identifier T. The definition of class array then uses T as a type variable. For example, the array defines a pointer to element sequences of type T, and the sub function returns an element of type T etc.

template <class T>
class array {
  public:
   array (int size) {a = new T[size];}
   T sub (int i) { a[i];}

  private:
    *T a;
}

Note that the only part of the syntax template <class T> that is changeable is the identifier T. In other words, you should think of the syntax template <class T> as a binding form that allows you to pick an identifier (in this case T). You might ask why the type identifier/variable T is a class. This is a good question. The authors find it most helpful to not think much about such questions, especially in the context of the C++ language.

Once defined a template class can be initialized with different type variables by using the < > syntax. For examples, we can define different arrays such as the following.

array<int> myFavoriteNumbers(7);
array<char*> myFavoriteNames(7);

Again, since C++ does not perform type inference for class instances, the C++ compiler expects the programmer to eliminate explicitly parametricity by specifying the argument type.

It is also possible to define polymorphic or generic functions. For example, the following declarations defines a generic identity function.

template <class T>
T identity(T x) { return x;}

Once defined, this function can be used without explicitly specializing it at various types. In contrast to templated classes, C++ does provide some type inference for calls to templated functions. So generic functions can be specialized implicitly, as shown in the examples below.

i = identity (3)
s = identity ("template programming can be ugly")

This brief summary of templates should suffice for the purposes of the material covered in this book. Templates are covered in significant detail by many books, blogs, and discussions boards. We refer the interested reader to those sources for further information.

2.2. Lambda expressions

The C++11 reference provides good documentation on lambda expressions.

3. Chapter: Fork-join parallelism

Fork-join parallelism, a fundamental model in parallel computing, dates back to 1963 and has since been widely used in parallel computing. In fork join parallelism, computations create opportunities for parallelism by branching at certain points that are specified by annotations in the program text.

Each branching point forks the control flow of the computation into two or more logical threads. When control reaches the branching point, the branches start running. When all branches complete, the control joins back to unify the flows from the branches. Results computed by the branches are typically read from memory and merged at the join point. Parallel regions can fork and join recursively in the same manner that divide and conquer programs split and join recursively. In this sense, fork join is the divide and conquer of parallel computing.

As we will see, it is often possible to extend an existing language with support for fork-join parallelism by providing libraries or compiler extensions that support a few simple primitives. Such extensions to a language make it easy to derive a sequential program from a parallel program by syntactically substituting the parallelism annotations with corresponding serial annotations. This in turn enables reasoning about the semantics or the meaning of parallel programs by essentially "ignoring" parallelism.

PASL is a C++ library that enables writing implicitly parallel programs. In PASL, fork join is expressed by application of the fork2() function. The function expects two arguments: one for each of the two branches. Each branch is specified by one C++ lambda expression.

Example 1. Fork join

In the sample code below, the first branch writes the value 1 into the cell b1 and the second 2 into b2; at the join point, the sum of the contents of b1 and b2 is written into the cell j.

long b1 = 0;
long b2 = 0;
long j  = 0;

fork2([&] {
  // first branch
  b1 = 1;
}, [&] {
  // second branch
  b2 = 2;
});
// join point
j = b1 + b2;

std::cout << "b1 = " << b1 << "; b2 = " << b2 << "; ";
std::cout << "j = " << j << ";" << std::endl;

Output:

b1 = 1; b2 = 2; j = 3;

When this code runs, the two branches of the fork join are both run to completion. The branches may or may not run in parallel (i.e., on different cores). In general, the choice of whether or not any two such branches are run in parallel is chosen by the PASL runtime system. The join point is scheduled to run by the PASL runtime only after both branches complete. Before both branches complete, the join point is effectively blocked. Later, we will explain in some more detail the scheduling algorithms that the PASL uses to handle such load balancing and synchronization duties.

In fork-join programs, a thread is a sequence of instructions that do not contain calls to fork2(). A thread is essentially a piece of sequential computation. The two branches passed to fork2() in the example above correspond, for example, to two independent threads. Moreover, the statement following the join point (i.e., the continuation) is also a thread.

Note
If the syntax in the code above is unfamiliar, it might be a good idea to review the notes on lambda expressions in C++11. In a nutshell, the two branches of fork2() are provided as lambda-expressions where all free variables are passed by reference.
Note
Fork join of arbitrary arity is readily derived by repeated application of binary fork join. As such, binary fork join is universal because it is powerful enough to generalize to fork join of arbitrary arity.

All writes performed by the branches of the binary fork join are guaranteed by the PASL runtime to commit all of the changes that they make to memory before the join statement runs. In terms of our code snippet, all writes performed by two branches of fork2 are committed to memory before the join point is scheduled. The PASL runtime guarantees this property by using a local barrier. Such barriers are efficient, because they involve just a single dynamic synchronization point between at most two processors.

Example 2. Writes and the join statement

In the example below, both writes into b1 and b2 are guaranteed to be performed before the print statement.

long b1 = 0;
long b2 = 0;

fork2([&] {
  b1 = 1;
}, [&] {
  b2 = 2;
});

std::cout << "b1 = " << b1 << "; b2 = " << b2 << std::endl;

Output:

b1 = 1; b2 = 2

PASL provides no guarantee on the visibility of writes between any two parallel branches. In the code just above, for example, writes performed by the first branch (e.g., the write to b1) may or may not be visible to the second, and vice versa.

3.1. Parallel Fibonacci

Now, we have all the tools we need to describe our first parallel code: the recursive Fibonacci function. Although useless as a program because of efficiency issues, this example is the "hello world" program of parallel computing.

Recall that the $n^{th}$ Fibonnacci number is defined by the recurrence relation

$$ \begin{array}{lcl} F(n) & = & F(n-1) + F(n-2) \end{array} $$

with base cases

$$ F(0) = 0, \, F(1) = 1 $$

Let us start by considering a sequential algorithm. Following the definition of Fibonacci numbers, we can write the code for (inefficiently) computing the $n^{th}$ Fibonnacci number as follows. This function for computing the Fibonacci numbers is inefficient because the algorithm takes exponential time, whereas there exist dynamic programming solutions that take linear time.

long fib_seq (long n) {
  long result;
  if (n < 2) {
    result = n;
  } else {
    long a, b;
    a = fib_seq(n-1);
    b = fib_seq(n-2);
    result = a + b;
  }
  return result;
}

To write a parallel version, we remark that the two recursive calls are completely independent: they do not depend on each other (neither uses a piece of data generated or written by another). We can therefore perform the recursive calls in parallel. In general, any two independent functions can be run in parallel. To indicate that two functions can be run in parallel, we use fork2().

long fib_par(long n) {
  long result;
  if (n < 2) {
    result = n;
  } else {
    long a, b;
    fork2([&] {
      a = fib_par(n-1);
    }, [&] {
      b = fib_par(n-2);
    });
    result = a + b;
  }
  return result;
}

3.2. Incrementing an array, in parallel

Suppose that we wish to map an array to another by incrementing each element by one. We can write the code for a function map_incr that performs this computation serially.

void map_incr(const long* source, long* dest, long n) {
  for (long i = 0; i < n; i++)
    dest[i] = source[i] + 1;
}
Example 3. Example: Using map_incr.

The code below illustrates an example use of map_incr.

const long n = 4;
long xs[n] = { 1, 2, 3, 4 };
long ys[n];
map_incr(xs, ys, n);
for (long i = 0; i < n; i++)
  std::cout << ys[i] << " ";
std::cout << std::endl;

Output:

2 3 4 5

This is not a good parallel algorithm but it is not difficult to give a parallel algorithm for incrementing an array. The code for such an algorithm is given below.

void map_incr_rec(const long* source, long* dest, long lo, long hi) {
  long n = hi - lo;
  if (n == 0) {
    // do nothing
  } else if (n == 1) {
    dest[lo] = source[lo] + 1;
  } else {
    long mid = (lo + hi) / 2;
    fork2([&] {
      map_incr_rec(source, dest, lo, mid);
    }, [&] {
      map_incr_rec(source, dest, mid, hi);
    });
  }
}

It is easy to see that this algorithm has O(n) work and $O(\log{n})$ span.

3.3. The sequential elision

In the Fibonacci example, we started with a sequential algorithm and derived a parallel algorithm by annotating independent functions. It is also possible to go the other way and derive a sequential algorithm from a parallel one. As you have probably guessed this direction is easier, because all we have to do is remove the calls to the fork2 function. The sequential elision of our parallel Fibonacci code can be written by replacing the call to fork2() with a statement that performs the two calls (arguments of fork2()) sequentially as follows.

long fib_par(long n) {
  long result;
  if (n < 2) {
    result = n;
  } else {
    long a, b;
    ([&] {
      a = fib_par(n-1);
    })();
    ([&] {
      b = fib_par(n-2);
    })();
    result = a + b;
  }
  return result;
}
Note
Although this code is slightly different than the sequential version that we wrote, it is not too far away, because the only the difference is the creation and application of the lambda-expressions. An optimizing compiler for C++ can easily "inline" such computations. Indeed, After an optimizing compiler applies certain optimizations, the performance of this code the same as the performance of fib_seq.

The sequential elision is often useful for debugging and for optimization. It is useful for debugging because it is usually easier to find bugs in sequential runs of parallel code than in parallel runs of the same code. It is useful in optimization because the sequentialized code helps us to isolate the purely algorithmic overheads that are introduced by parallelism. By isolating these costs, we can more effectively pinpoint inefficiencies in our code.

4. Race Conditions

A race condition is any behavior in a program that is determined by some feature of the system that cannot be controlled by the program, such as timing of the execution of instructions. In PASL, a race condition can occur whenever two parallel branches access the same location in memory and at least one of the two branches performs a write. When this situation occurs, the behavior of the program may be undefined, leading to (often) buggy behavior. We used the work "may" here because certain programs use race conditions in a careful way as a means to improve performance. Special attention to race conditions is crucial because race conditions introduce especially pernicious form error that can confound beginners and experts alike.

Race conditions can be highly problematic because, owing to their nondeterministic behavior, they are often extremely hard to debug. To make matters worse, it is also quite easy to create race conditions without even knowing it.

Example 4. Writing to the same location in parallel.

In the code below, both branches of fork2 are writing into b. What should then the output of this program be?

long b = 0;

fork2([&] {
  b = 1;
}, [&] {
  b = 2;
});

std::cout << "b = " << std::endl;

At the time of the print, the contents of b is determined by the last write. Thus depending on which of the two branches perform the write, we can see both possibilities:

Output:

b = 1

Output:

b = 2
Example 5. Fibonacci

Consider the following alternative implementation of the Fibonacci function. By "inlining" the plus operation in both branches, the programmer got rid of the addition operation after the fork2.

long fib_par_racy(long n) {
  long result = 0;
  if (n < 2) {
    result = n;
  } else {
    fork2([&] {
      result += fib_par_racy(n-1);
    },[&] {
      result += fib_par_racy(n-2);
    });
  }
  return result;
}

This code is not correct because it has a race condition.

As in the example shows, separate threads are updating the value result but it might look like this is not a race condition because the update consists of an addition operation, which reads the value and then writes to i. The race condition might be easier to see if we expand out the applications of the += operator.

long fib_par_racy(long n) {
  long result = 0;
  if (n < 2) {
    result = n;
  } else {
    fork2([&] {
      long a1 = fib_par_racy(n-1);
      long a2 = result;
      result = a1 + a2;
    },[&] {
      long b1 = fib_par_racy(n-2);
      long b2 = result;
      result = b1 + b2;
    });
  }
  return result;
}

When written in this way, it is clear that these two parallel threads are not independent: they both read result and write to result. Thus the outcome depends on the order in which these reads and writes are performed, as shown in the next example.

Example 6. Execution trace of a race condition

The following table takes us through one possible execution trace of the call fib_par_racy(2). The number to the left of each instruction describes the time at which the instruction is executed. Note that since this is a parallel program, multiple instructions can be executed at the same time. The particular execution that we have in this example gives us a bogus result: the result is 0, not 1 as it should be.

Time step Thread 1 Thread 2

1

a1 = fib_par_racy(1)

b2 = fib_par_racy(0)

2

a2 = result

b3 = result

3

result = a1 + a2

_

4

_

result = b1 + b2

The reason we get a bogus result is that both threads read the initial value of result at the same time and thus do not see each others write. In this example, the second thread "wins the race" and writes into result. The value 1 written by the first thread is effectively lost by being overwritten by the second thread.

To prevent such race conditions, we usually want to give threads mutually exclusive access to shared objects.

4.1. Synchronization Hardware

Since mutual exclusion is a common problem in computer science, many hardware systems provide specific synchronization operations that can help solve instances of the problem. These operations may allow, for example, testing the contents of a (machine) word then modifying it, perhaps by swapping it with another word. Such operations are sometimes called atomic read-modify-write or RMW, for short, operations.

A handful of different RMW operations have been proposed. They include operations such as load-link/store-conditional, fetch-and-add, and compare-and-swap. They typically take the memory location x, and a value v and replace the value of stored at x with f(x,v). For example, the fetch-and-add operation takes the location x and the increment-amount, and atomically increments the value at that location by the specified amount, i.e., f(x,v) = *x + v.

The compare-and-swap operation takes the location x and takes a pair of values (a,b) as the second argument, and stores b into x if the value in x is a, i.e., f(x,(a,b)) = if *x = a then b else a; the operation returns a boolean indicating whether the operation successfully stored a new value in x. The operation "compare-and-swap" is a reasonably powerful synchronization operation: it can be used by arbitrarily many threads to agree (reach consensus) on a value. This instruction therefore is frequently provided by modern parallel architectures such as Intel’s X86.

In C$++$, the atomic class can be used to perform synchronization. Objects af this type are guarantee to be free of race conditions; and in fact, in C++, they are the only objects that are guaranteed to be free from race conditions. The contents of an atomic object can be accessed by load opeations, updated by store operation, and also updated by compare_exchange_weak and compare_exchange_strong operations, the latter of which implement the compare-and-swap operation.

Example 7. Accessing the contents of atomic memory cells

Access to the contents of any given cell is achieved by the load() and store() methods.

std::atomic<bool> flag;

flag.store(false);
std::cout << flag.load() << std::endl;
flag.store(true);
std::cout << flag.load() << std::endl;

Output:

0
1

The key operation that help with race conditions is the compare-and-exchange operation.

Definition: compare and swap

When executed with a ‘target` atomic object and an expected cell and a new value `new’ this operation performs the following steps, atomically:

  1. Read the contents of target.

  2. If the contents equals the contents of expected, then writes new into the target and returns true.

  3. Otherwise, returns false.

Example 8. Reading and writing atomic objects
std::atomic<bool> flag;

flag.store(false);
bool expected = false;
bool was_successful = flag.compare_exchange_strong(expected, true);
std::cout << "was_successful = " << was_successful << "; flag = " << flag.load() << std::endl;
bool expected2 = false;
bool was_successful2 = flag.compare_exchange_strong(expected2, true);
std::cout << "was_successful2 = " << was_successful2 << "; flag = " <<
flag.load() << std::endl;

Output:

was_successful = 1; flag = 1
was_successful2 = 0; flag = 1

As another example use of the atomic class, recall our Fibonacci example with the race condition. In that example, race condition arises because of concurrent writes to the result variable. We can eliminate this kind of race condition by using different memory locations, or by using an atomic class and using a compare_exchange_strong operation.

Example 9. Fibonacci

The following implementation of Fibonacci is not safe because the variable result is shared and updated by multiple threads.

long fib_par_racy(long n) {
  long result = 0;
  if (n < 2) {
    result = n;
  } else {
    fork2([&] {
      result += fib_par_racy(n-1);
    },[&] {
      result += fib_par_racy(n-2);
    });
  }
  return result;
}

We can solve this problem by declaring result to be an atomic type and using a standard busy-waiting protocol based on compare-and-swap.


long fib_par_atomic(long n) {
  atomic<long> result = 0;
  if (n < 2) {
    result.store(n);
  } else {
    fork2([&] {
      long r = fib_par_racy(n-1);
      // Atomically update result.
      while (true) {
        long exp = result.load();
        bool flag = result.compare_exchange_strong(exp,exp+r)
        if (flag) {break;}
      }
    },[&] {
      long r = fib_par_racy(n-2);
      // Atomically update result.
      while (true) {
        long exp = result.load();
        bool flag = result.compare_exchange_strong(exp,exp+r)
        if (flag) {break;}
      }
    });
  }
  return result;
}

The idea behind the solution is to load the current value of result and atomically update result only if it has not been modified (by another thread) since it was loaded. This guarantees that the result is always updated (read and modified) correctly without missing an update from another thread.

The example above illustrates a typical use of the compare-and-swap operation. In this particular example, we can probably prove our code is correct. But this is not always as easy due to a problem called the "ABA problem."

4.2. ABA problem

While reasonably powerful, compare-and-swap suffers from the so-called ABA problem. To see this consider the following scenario where a shared variable result is update by multiple threads in parallel: a thread, say $T$, reads the result and stores its current value, say 2, in current. In the mean time some other thread also reads result and performs some operations on it, setting it back to 2 after it is done. Now, thread $T$ takes its turn again and attempts to store a new value into result by using 2 as the old value and being successful in doing so, because the value stored in result appears to have not changed. The trouble is that the value has actually changed and has been changed back to the value that it used to be. Thus, compare-and-swap was not able to detect this change because it only relies on a simple shallow notion of equality. If for example, the value stored in result was a pointer, the fact that the pointer remains the same does not mean that values accessible from the pointer has not been modified; if for example, the pointer led to a tree structure, an update deep in the tree could leave the pointer unchanged, even though the tree has changed.

This problem is called the ABA problem, because it involves cycling the atomic memory between the three values $A$, $B$, and again $A$). The ABA problem is an important limitation of compare-and-swap: the operation itself is not atomic but is able to behave as if it is atomic if it can be ensured that the equality test of the subject memory cell suffices for correctness.

In the example below, ABA problem may happen (if the counter is incremented and decremented again in between a load and a store) but it is impossible to observe because it is harmless. If however, the compare-and-swap was on a memory object with references, the ABA problem could have had observable effects.

The ABA problem can be exploited to give seemingly correct implementations that are in fact incorrect. To reduce the changes of bugs due to the ABA problem, memory objects subject to compare-and-swap are usually tagged with an additional field that counts the number of updates. This solves the basic problem but only up to a point because the counter itself can also wrap around. The load-link/store-conditional operation solves this problem by performing the write only if the memory location has not been updated since the last read (load) but its practical implementations are hard to come by.

5. Chapter: Experimenting with PASL

We are now going to study the practical performance of our parallel algorithms written with PASL on multicore computers.

To be concrete with our instructions, we assume that our username is pasl and that our home directory is /home/pasl/. You need to replace these settings with your own where appropriate.

5.1. Obtain source files

Let’s start by downloading the PASL sources. The PASL sources that we are going to use are part of a branch that we created specifically for this course. You can access the sources either via the tarball linked by the github webpage or, if you have git, via the command below.

$ cd /home/pasl
$ git clone -b edu https://github.com/deepsea-inria/pasl.git

5.2. Software Setup

You can skip this section if you are using a computer already setup by us or you have installed an image file containing our software. To skip this part and use installed binaries, see the heading "Starting with installed binaries", below.

5.2.1. Check for software dependencies

Currently, the software associated with this course supports Linux only. Any machine that is configured with a recent version of Linux and has access to at least two processors should be fine for the purposes of this course. Before we can get started, however, the following packages need to be installed on your system.

Software dependency Version Nature of dependency

gcc

>= 4.9.0

required to build PASL binaries

php

>= 5.3.10

required by PASL makefiles to build PASL binaries

ocaml

>= 4.0.0

required to build the benchmarking tools (i.e., pbench and pview)

R

>= 2.4.1

required by benchmarking tools to generate reports in bar plot and scatter plot form

latex

recent

optional; required by benchmarking tools to generate reports in tabular form

git

recent

optional; can be used to access PASL source files

tcmalloc

>= 2.2

optional; may be useful to improve performance of PASL binaries

hwloc

recent

optional; might be useful to improve performance on large systems with NUMA (see below)

The rest of this section explains what are the optional software dependencies and how to configure PASL to use them. We are going to assume that all of these software dependencies have been installed in the folder /home/pasl/Installs/.

5.2.2. Use a custom parallel heap allocator

At the time of writing this document, the system-default implementations of malloc and free that are provided by Linux distributions do not scale well with even moderately large amounts of concurrent allocations. Fortunately, for this reason, organizations, such as Google and Facebook, have implemented their own scalable allocators that serve as drop-in replacements for malloc and free. We have observed the best results from Google’s allocator, namely, tcmalloc. Using tcmalloc for your own experiements is easy. Just add to the /home/pasl/pasl/minicourse folder a file named settings.sh with the following contents.

Example 10. Configuration to select tcmalloc

We assume that the package that contains tcmalloc, namely gperftools, has been installed already in the folder /home/pasl/Installs/gperftools-install/. The following lines need to be in the settings.sh file in the /home/pasl/pasl/minicourse folder.

USE_ALLOCATOR=tcmalloc
TCMALLOC_PATH=/home/pasl/Installs/gperftools-install/lib/

Also, the environment linkder needs to be instructed where to find tcmalloc.

export LD_PRELOAD=/home/pasl/Installs/gperftools-install/lib/libtcmalloc.so

This assignment can be issued either at the command line or in the environment loader script, e.g., ~/.bashrc.

Warning
Changes to the settings.sh file take effect only after recompiling the binaries.

5.2.3. Use hwloc

If your system has a non-uniform memory architecture (i.e., NUMA), then you may improve performance of PASL applications by using optional support for hwloc, which is a library that reports detailed information about the host system, such as NUMA layout. Currently, PASL leverages hwloc to configure the NUMA allocation policy for the program. The particular policy that works best for our applications is round-robin NUMA page allocation. Do not worry if that term is unfamiliar: all it does is disable NUMA support, anyway!

Example 11. How to know whether my machine has NUMA

Run the following command.

$ dmesg | grep -i numa

If the output that you see is something like the following, then your machine has NUMA. Otherwise, it probably does not.

[    0.000000] NUMA: Initialized distance table, cnt=8
[    0.000000] NUMA: Node 4 [0,80000000) + [100000000,280000000) -> [0,280000000)

We are going to assume that hwloc has been installed already and is located at /home/pasl/Installs/hwloc-install/. To configure PASL to use hwloc, add the following lines to the settings.sh file in the /home/pasl/pasl/minicourse folder.

Example 12. Configuration to use hwloc
USE_HWLOC=1
HWLOC_PATH=/home/pasl/Installs/hwloc-install/

5.3. Starting with installed binaries

At this point, you have either installed all the necessary software to work with PASL or these are installed for you. In either case, make sure that your PATH variable makes the software visible. For setting up your PATH variable on andrew.cmu domain, see below.

5.3.1. Specific set up for the andrew.cmu domain

We have installed much of the needed software on andrew.cmu.edu. So you need to go through a relatively minimal set up.

First set up your PATH variable to refer to the right directories. Using cshell

setenv PATH  /opt/rh/devtoolset-3/root/usr/bin:/usr/lib64/qt-3.3/bin:/usr/lib64/ccache:/usr/local/bin:/bin:/usr/bin:./

The part added to the default PATH on andrew is

/opt/rh/devtoolset-3/root/usr/bin

It is important that this is at the beginning of the PATH variable. To make interaction easier, we also added the relative path ./ to the PATH variable.

5.3.2. Fetch the benchmarking tools (pbench)

We are going to use two command-line tools to help us to run experiments and to analyze the data. These tools are part of a library that we developed, which is named pbench. The pbench sources are available via github.

$ cd /home/pasl
$ git clone https://github.com/deepsea-inria/pbench.git

The tarball of the sources can be downloaded from the github page.

5.3.3. Build the tools

The following command builds the tools, namely prun and pplot. The former handles the collection of data and the latter the human-readable output (e.g., plots, tables, etc.).

$ make -C /home/pasl/pbench/

Make sure that the build succeeded by checking the pbench directory for the files prun and pplot. If these files do not appear, then the build failed.

5.3.4. Create aliases

We recommend creating the following aliases.

$ alias prun '/home/pasl/pbench/prun'
$ alias pplot '/home/pasl/pbench/pplot'

It will be convenient for you to make these aliases persistent, so that next time you log in, the aliases will be set. Add the commands above to your shell configuration file.

5.3.5. Visualizer Tool

When we are tuning our parallel algorithms, it can be helpful to visualize their processor utilization over time, just in case there are patterns that help to assign blame to certain regions of code. Later, we are going to use the utilization visualizer that comes packaged along with PASL. To build the tool, run the following make command.

$ make -C /home/pasl/pasl/tools/pview pview

Let us create an alias for the tool.

$ alias pview '/home/pasl/pasl/tools/pview/pview'

We recommend that you make this alias persistent by putting it into your shell configuration file (as you did above for the pbench tools).

5.4. Using the Makefile

PASL comes equipped with a Makefile that can generate several different kinds of executables. These different kinds of executables and how they can be generated is described below for a benchmark program pgm.

  • baseline: build the baseline with command make pgm.baseline

  • elision: build the sequential elision with command make pgm.elision

  • optimized: build the optimized binary with command make pgm.opt

  • log: build the log binary with command make pgm.log

  • debug: build the debug binary with the command make pgm.dbg

To speed up the build process, add to the make command the option -j (e.g., make -j pgm.opt). This option enables make to parallelize the build process. Note that, if the build fails, the error messages that are printed to the terminal may be somewhat garbled. As such, it is better to use -j only if after the debugging process is complete.

5.5. Task 1: Run the baseline Fibonacci

We are going to start our experimentation with three different instances of the same program, namely bench. This program serves as a "driver" for the benchmarks that we have implemented. These implementations are good parallel codes that we expect to deliver good performance. We first build the baseline version.

$ cd /home/pasl/pasl/minicourse
$ make bench.baseline
Warning
The command-line examples that we show here assume that you have . in your $PATH. If not, you may need to prefix command-line calls to binaries with ./ (e.g., ./bench.baseline).

The file extension .baseline means that every benchmark in the binary uses the sequential-baseline version of the specified algorithm.

We can now run the baseline for one of our benchmarks, say Fibonacci by using the -bench argument to specify the benchmark and the -n argument to specify the input value for the Fibonacci function.

$ bench.baseline -bench fib -n 39

On our machine, the output of this run is the following.

exectime 0.556
utilization 1.0000
result 63245986

The three lines above provide useful information about the run.

  • The exectime indicates the wall-clock time in seconds that is taken by the benchmark. In general, this time measures only the time taken by the benchmark under consideration. It does not include the time taken to generate the input data, for example.

  • The utilization relates to the utilization of the processors available to the program. In the present case, for a single-processor run, the utilization is by definition 100%. We will return to this measure soon.

  • The result field reports a value computed by the benchmark. In this case, the value is the $39^{th}$ Fibonacci number.

5.6. Task 2: Run the sequential elision of Fibonacci

The .elision extension means that parallel algorithms (not sequential baseline algorithms) are compiled. However, all instances of fork2() are erased as described in an earlier chapter.

$ make bench.elision
$ bench.elision -bench fib -n 39

The run time of the sequential elision in this case is similar to the run time of the sequential baseline because the two are similar codes. However, for most other algorithms, the baseline will typically be at least a little faster.

exectime 0.553
utilization 1.0000
result 63245986

5.7. Task 3: Run parallel Fibonacci

The .opt extension means that the program is compiled with full support for parallel execution. Unless specified otherwise, however, the parallel binary uses just one processor.

$ make bench.opt
$ bench.opt -bench fib -n 39

The output of this program is similar to the output of the previous two programs.

exectime 0.553
utilization 1.0000
result 63245986

Because our machine has 40 processors, we can run the same application using all available processors. Before running this command, please adjust the -proc option to match the number of cores that your machine has. Note that you can use any number of cores up to the number you have available. You can use nproc or lscpu to determine the number of cores your machine has.

$ bench.opt -bench fib -n 39 -proc 40

We see from the output of the 40-processor run that our program ran faster than the sequential runs. Moreover, the utilization field tells us that approximately 86% of the total time spent by the 40 processors was spent performing useful work, not idling.

exectime 0.019
utilization 0.8659
result 63245986
Warning
PASL allows the user to select the number of processors by the -proc key. The maximum value for this key is the number of processors that are available on the machine. PASL raises an error if the programmer asks for more processors than are available.

5.8. Measuring performance with "speedup"

We may ask at this point: What is the improvement that we just observed from the parallel run of our program? One common way to answer this question is to measure the "speedup".

Definition: $P$-processor speedup

The speedup on $P$ processors is the ratio $T_B/T_P$, where the term $T_B$ represents the run time of the sequential baseline program and the term $T_P$ the time measured for the $P$-processor run.

Important
The importance of selecting a good baseline

Note that speedup is defined with respect to a baseline program. How exactly should this baseline program be chosen? One option is to take the sequential elision as a baseline. The speedup curve with such a baseline can be helpful in determining the scalability of a parallel algorithm but it can also be misleading, especially if speedups are taken as a indicator of good performance, which they are not because they are only relative to a specific baseline. For speedups to be a valid indication of good performance, they must be calculated against an optimized implementation of the best serial algorithm (for the same problem.)

The speedup at a given number of processors is a good starting point on the way to evaluating the scalability of the implementation of a parallel algorithm. The next step typically involves considering speedups taken from varying numbers of processors available to the program. The data collected from such a speedup experiment yields a speedup curve, which is a curve that plots the trend of the speedup as the number of processors increases. The shape of the speedup curve provides valuable clues for performance and possibly for tuning: a flattening curve suggests lack of parallelism; a curve that arcs up and then downward suggests that processors may be wasting time by accessing a shared resource in an inefficient manner (e.g., false sharing); a speedup curve with a constant slope indicates at least some scaling.

Example 13. Speedup for our run of Fibonacci on 40 processors

The speedup $T_B/T_{40}$ equals $0.556/0.019 = 29.26$x. Although not linear (i.e., 40x), this speedup is decent considering factors such as: the capabilities of our machine; the overheads relating to parallelism; and the small size of the problem compared to the computing power that our machine offers.

5.8.1. Generate a speedup plot

Let us see what a speedup curve can tell us about our parallel Fibonacci program. We need to first get some data. The following command performs a sequence of runs of the Fibonacci program for varying numbers of processors. You can now run the command yourself.

$ prun speedup -baseline "bench.baseline" -parallel "bench.opt -proc 1,10,20,30,40" -bench fib -n 39

Here is another example on a 24-core machine.

$ prun speedup -baseline "bench.baseline" -parallel "bench.opt -proc 1,4,8,16,24" -bench fib -n 39

Run the following command to generate the speedup plot.

$ pplot speedup

If successful, the command generates a file named plots.pdf. The output should look something like the plot in speedup plot below.

Starting to generate 1 charts.
Produced file plots.pdf.
Speedup curve for the computation of the 39th Fibonacci number.
Figure 1. Speedup curve for the computation of the 39th Fibonacci number.

The plot shows that our Fibonacci application scales well, up to about twenty processors. As expected, at twenty processors, the curve dips downward somewhat. We know that the problem size is the primary factor leading to this dip. How much does the problem size matter? The speedup plot in the Figure below shows clearly the trend. As our problem size grows, so does the speedup improve, until at the calculation of the $45^{th}$ Fibonacci number, the speedup curve is close to being linear.

Speedup plot showing speedup curves at different problem sizes.
Figure 2. Speedup plot showing speedup curves at different problem sizes.
Note
The prun and pplot tools have many more features than those demonstrated here. For details, see the documentation provided with the tools in the file named README.md.
Warning
Noise in experiments

The run time that a given parallel program takes to solve the same problem can vary noticeably because of certain effects that are not under our control, such as OS scheduling, cache effects, paging, etc. We can consider such noise in our experiments random noise. Noise can be a problem for us because noise can lead us to make incorrect conclusions when, say, comparing the performance of two algorithms that perform roughly the same. To deal with randomness, we can perform multiple runs for each data point that we want to measure and consider the mean over these runs. The prun tool enables taking multiple runs via the -runs argument. Moreover, the pplot tool by default shows mean values for any given set of runs and optionally shows error bars. The documentation for these tools gives more detail on how to use the statistics-related features.

5.8.2. Superlinear speedup

Suppose that, on our 40-processor machine, the speedup that we observe is larger than 40x. It might sound improbable or even impossible. But it can happen. Ordinary circumstances should preclude such a superlinear speedup, because, after all, we have only forty processors helping to speed up the computation. Superlinear speedups often indicate that the sequential baseline program is suboptimal. This situation is easy to check: just compare its run time with that of the sequential elision. If the sequential elision is faster, then the baseline is suboptimal. Other factors can cause superlinear speedup: sometimes parallel programs running on multiple processors with private caches benefit from the larger cache capacity. These issues are, however, outside the scope of this course. As a rule of thumb, superlinear speedups should be regarded with suspicion and the cause should be investigated.

5.9. Visualize processor utilization

The 29x speedup that we just calculated for our Fibonacci benchmark was a little dissapointing, and the 86% processor utilization of the run left 14% utilization for improvement. We should be suspicious that, although seemingly large, the problem size that we chose, that is, $n = 39$, was probably a little too small to yield enough work to keep all the processors well fed. To put this hunch to the test, let us examine the utilization of the processors in our system. We need to first build a binary that collects and outputs logging data.

$ make bench.log

We run the program with the new binary in the same fashion as before.

$ bench.log -bench fib -proc 40 -n 39

The output looks something like the following.

exectime 0.019
launch_duration 0.019
utilization     0.8639
thread_send     205
thread_exec     4258
thread_alloc    2838
utilization 0.8639
result 63245986

We need to explain what the new fields mean.

  • The thread_send field tells us that 233 threads were exchaged between processors for the purpose of load balancing;

  • the thread_exec field that 5179 threads were executed by the scheduler;

  • the thread_alloc field that 3452 threads were freshly allocated.

Each of these fields can be useful for tracking down inefficiencies. The number of freshly allocated threads can be a strong indicator because in C++ thread allocation costs can sometimes add up to a significant cost. In the present case, however, none of the new values shown above are highly suspicious, considering that there are all at most in the thousands.

Since we have not yet found the problem, let us look at the visualization of the processor utilization using our pview tool. To get the necessary logging data, we need to run our program again, this time passing the argument --pview.

$ bench.log -bench fib -n 39 -proc 40 --pview

When the run completes, a binary log file named LOG_BIN should be generated in the current directory. Every time we run with --pview this binary file is overwritten. To see the visualization of the log data, we call the visualizer tool from the same directory.

$ pview

The output we see on our 40-processor machine is shown in the Figure below. The window shows one bar per processor. Time goes from left to right. Idle time is represented by red and time spent busy with work by grey. You can zoom in any part of the plot by clicking on the region with the mouse. To reset to the original plot, press the space bar. From the visualization, we can see that most of the time, particularly in the middle, all of the processors keep busy. However, there is a lot of idle time in the beginning and end of the run. This pattern suggests that there just is not enough parallelism in the early and late stages of our Fibonacci computation.

Utilization plot for computation of 39th Fibonacci number.
Figure 3. Utilization plot for computation of 39th Fibonacci number.

5.10. Strong versus weak scaling

We are pretty sure that or Fibonacci program is not scaling as well is it could. But poor scaling on one particular input for $n$ does not necessarily mean there is a problem with the scalability our parallel Fibonacci program in general. What is important is to know more precisely what it is that we want our Fibonacci program to achieve. To this end, let us consider a distinction that is important in high-performance computing: the distinction between strong and weak scaling. So far, we have been studying the strong-scaling profile of the computation of the $39^{th}$ Fibonacci number. In general, strong scaling concerns how the run time varies with the number of processors for a fixed problem size. Sometimes strong scaling is either too ambitious, owing to hardware limitations, or not necessary, because the programmer is happy to live with a looser notion of scaling, namely weak scaling. In weak scaling, the programmer considers a fixed-size problem per processor. We are going to consider something similar to weak scaling. In the Figure below, we have a plot showing how processor utilization varies with the input size. The situation dramatically improves from 12% idle time for the $39^{th}$ Fibonacci number down to 5% idle time for the $41^{st}$ and finally to 1% for the $45^{th}$. At just 1% idle time, the utilization is excellent.

How processor utilization of Fibonacci computation varies with input size.
Figure 4. How processor utilization of Fibonacci computation varies with input size.

The scenario that we just observed is typical of multicore systems. For computations that perform relatively little work, such as the computation of the $39^{th}$ Fibonacci number, properties that are specific to the hardware, OS, and PASL load-balancing algorithm can noticeably limit processor utilization. For computations that perform lots of highly parallel work, such limitations are barely noticeable, because processors spend most of their time performing useful work. Let us return to the largest Fibonacci instance that we considered, namely the computation of the $45^{th}$ Fibonacci number, and consider its utilization plot.

$ bench.log -bench fib -n 45 -proc 40 --pview
$ pview

The utilization plot is shown in the Figure below. Compared the to utilization plot we saw in the Figure above for n=39, the red regions are much less prominent overall and the idle regions at the beginning and end are barely noticeable.

Utilization plot for computation of 45th Fibonacci number.
Figure 5. Utilization plot for computation of 45th Fibonacci number.

5.11. Chapter Summary

We have seen in this lab how to build, run, and evaluate our parallel programs. Concepts that we have seen, such as speedup curves, are going to be useful for evaluating the scalability of our future solutions. Strong scaling is the gold standard for a parallel implementation. But as we have seen, weak scaling is a more realistic target in most cases.

6. Chapter: Work efficiency

In many cases, a parallel algorithm which solves a given problem performs more work than the fastest sequential algorithm that solves the same problem. This extra work deserves careful consideration for several reasons. First, since it performs additional work with respect to the serial algorithm, a parallel algorithm will generally require more resources such as time and energy. By using more processors, it may be possible to reduce the time penalty, but only by using more hardware resources.

For example, if an algorithm performs $O(\log{n})$-factor more work than the serial algorithm, then, assuming that the constant factor hidden by the asymptotic notation is $1$, when $n = 2^{20}$, it will perform $20$-times more actual work, consuming $20$ times more energy consumption than the serial algorithm. Assuming perfect scaling, we can reduce the time penalty by using more processors. For example, with $40$ processors, the algorithm may require half the time of the serial algorithm.

Sometimes, a parallel algorithm has the same asymptotic complexity of the best serial algorithm for the problem but it has larger constant factors. This is generally true because scheduling friction, especially the cost of creating threads, can be significant. In addition to friction, parallel algorithms can incur more communication overhead than serial algorithms because data and processors may be placed far away in hardware. For example, it is not unusual for a parallel algorithm to incur a $10-100\times$ overhead over a similar serial algorithm because of scheduling friction and communication.

These considerations motivate considering "work efficiency" of parallel algorithm. Work efficiency is a measure of the extra work performed by the parallel algorithm with respect to the serial algorithm. We define two types of work efficiency: asymptotic work efficiency and observed work efficiency. The former relates to the asymptotic performance of a parallel algorithm relative to the fastest sequential algorithm. The latter relates to running time of a parallel algorithm relative to that of the fastest sequential algorithm.

Definition: asymptotic work efficiency

An algorithm is asymptotically work efficient if the work of the algorithm is the same as the work of the best known serial algorithm.

Example 14. Asymptotic work efficiency
  • A parallel algorithm that comparison-sorts $n$ keys in span $O(\log^3 n)$ and work $O(n \log n)$ is asymptotically work efficient because the work cost is as fast as the best known sequential comparison-based sorting algorithm. However, a parallel sorting algorithm that takes $O(n\log^2{n})$ work is not asymptotically work efficient.

  • The parallel array increment algorithm that we consider in an earlier Chapter is asymptotically work efficient, because it performs linear work, which is optimal (any sequential algorithm must perform at least linear work).

To assess the practical effectiveness of a parallel algorithm, we define observed work efficiency, parameterized a value $r$.

Definition: observed work efficiency

A parallel algorithm that runs in time $T_1$ on a single processor has observed work efficient factor of $r$ if $r = \frac{T_1}{T_{seq}}$, where $T_{seq}$ is the time taken by the fastest known sequential algorithm.

Example 15. Observed work efficiency
  • A parallel algorithm that runs $10\times$ slower on a single processor that the fastest sequential algorithm has an observed work efficiency factor of $10$. We consider such algorithms unacceptable, as they are too slow and wasteful.

  • A parallel algorithm that runs $1.1\times-1.2\times$. slower on a single processor that the fastest sequential algorithm has observed work efficiency factor of $1.2$. We consider such algorithms to be acceptable.

Example 16. Observed work efficiency of parallel increment

To obtain this measure, we first run the baseline version of our parallel-increment algorithm.

$ bench.baseline -bench map_incr -n 100000000
exectime 0.884
utilization 1.0000
result 2

We then run the parallel algorithm, which is the same exact code as map_incr_rec. We build this code by using the special optfp "force parallel" file extension. This special file extension forces parallelism to be exposed all the way down to the base cases. Later, we will see how to use this special binary mode for other purposes.

$ make bench.optfp
$ bench.optfp -bench map_incr -n 100000000
exectime 45.967
utilization 1.0000
result 2

Our algorithm has an observed work efficiency factor of $60\times$. Such poor observed work efficiency suggests that the parallel algorithm would require more than an order of magnitude more energy and that it would not run faster than the serial algorithm even when using less than $60$ processors.

In practice, observed work efficiency is a major concern. First, the whole effort of parallel computing is wasted if parallel algorithms consistently require more work than the best sequential algorithms. In other words, in parallel computing, both asymptotic complexity and constant factors matter.

Based on these discussions, we define a good parallel algorithm as follows.

Definition: good parallel algorithm

We say that a parallel algorithm is good if it has the following three characteristics:

  1. it is asymptotically work efficient;

  2. it is observably work efficient;

  3. it has low span.

6.1. Improving work efficiency with granularity control

It is common for a parallel algorithm to be asymptotically and/or observably work inefficient but it is often possible to improve work efficiency by observing that work efficiency increases with parallelism and can thus be controlled by limiting it.

For example, a parallel algorithm that performs linear work and has logarithmic span leads to average parallelism in the orders of thousands with the small input size of one million. For such a small problem size, we usually would not need to employ thousands of processors. It would be sufficient to limit the parallelism so as to feed tens of processors and as a result reduce impact of excess parallelism on work efficiency.

In many parallel algorithms such as the algorithms based on divide-and-conquer, there is a simple way to achieve this goal: switch from parallel to sequential algorithm when the problem size falls below a certain threshold. This technique is sometimes called coarsening or granularity control.

But which code should we switch to: one idea is to simply switch to the sequential elision, which we always have available in PASL. If, however, the parallel algorithm is asymptotically work inefficient, this would be ineffective. In such cases, we can specify a separate sequential algorithm for small instances.

Optimizing the practical efficiency of a parallel algorithm by controlling its parallelism is sometimes called optimization, sometimes it is called performance engineering, and sometimes performance tuning or simply tuning. In the rest of this document, we use the term "tuning."

Example 17. Tuning the parallel array-increment function

We can apply coarsening to map_inc_rec code by switching to the sequential algorithm when the input falls below an established threshold.

long threshold = Some Number;

void map_incr_rec(const long* source, long* dest, long lo, long hi) {
  long n = hi - lo;
  if (n <= threshold) {
    for (long i = lo; i < hi; i++)
      dest[i] = source[i] + 1;
  } else {
    long mid = (lo + hi) / 2;
    fork2([&] {
      map_incr_rec(source, dest, lo, mid);
    }, [&] {
      map_incr_rec(source, dest, mid, hi);
    });
  }
}
Note
Even in sequential algorithms, it is not uncommon to revert to a different algorithm for small instances of the problem. For example, it is well known that insertion sort is faster than other sorting algorithms for very small inputs containing 30 keys or less. Many optimize sorting algorithm therefore revert to insertion sort when the input size falls within that range.
Example 18. Observed work efficiency of tuned array increment

As can be seen below, after some tuning, map_incr_rec program becomes highly work efficient. In fact, there is barely a difference between the serial and the parallel runs. The tuning is actually done automatically here by using an automatic-granularity-control technique described in the section.

$ bench.baseline -bench map_incr -n 100000000
exectime 0.884
utilization 1.0000
result 2
$ bench.opt -bench map_incr -n 100000000
exectime 0.895
utilization 1.0000
result 2

In this case, we have $r = \frac{T_1}{T_{seq}} = \frac{0.895}{0.884} = 1.012$ observed work efficiency. Our parallel program on a single processor is one percent slower than the sequential baseline. Such work efficiency is excellent.

6.2. Determining the threshold

The basic idea behind coarsening or granularity control is to revert to a fast serial algorithm when the input size falls below a certain threshold. To determine the optimal threshold, we can simply perform a search by running the code with different threshold settings.

While this approach can help find the right threshold on the particular machine that we performed the search, there is no guarantee that the same threshold would work on another machine. In fact, there are examples in the literature that show that such optimizations are not portable, i.e., a piece of code optimized for a particular architecture may behave poorly on another.

In the general case, determining the right threshold is even more difficult. To see the difficulty consider a generic (polymorphic), higher-order function such as map that takes a sequence and a function and applies the function to the sequence. The problem is that the threshold depends both on the type of the elements of the sequence and the function to be mapped over the sequence. For example, if each element itself is a sequence (the sequence is nested), the threshold can be relatively small. If, however, the elements are integers, then the threshold will likely need to be relatively large. This makes it difficult to determine the threshold because it depends on arguments that are unknown at compile time. Essentially the same argument applies to the function being mapped over the sequence: if the function is expensive, then the threshold can be relatively small, but otherwise it will need to be relatively large.

As we describe in this chapter, it is sometimes possible to determine the threshold completely automatically.

7. Chapter: Automatic granularity control

There has been significant research into determining the right threshold for a particular algorithm. This problem, known as the granularity-control problem, turns out to be a rather difficult one, especially if we wish to find a technique that can ensure close-to-optimal performance across different architectures. In this section, we present a technique for automatically controlling granularity by using asymptotic cost functions.

7.1. Complexity functions

Our automatic granularity-control technique requires assistance from the application programmer: for each parallel region of the program, the programmer must annotate the region with a complexity function, which is simply a C++ function that returns the asymptotic work cost associated with running the associated region of code.

Example 19. Complexity function of map_incr_rec

An application of our map_incr_rec function on a given range $[lo, hi)$ of an array has work cost $cn = c (hi - lo)$ for some constant $c$. As such, the following lambda expression is one valid complexity function for our parallel map_incr_rec program.

auto map_incr_rec_complexity_fct = [&] (long lo, long hi) {
  return hi - lo;
};

In general, the value returned by the complexity function need only be precise with respect to the asymptotic complexity class of the associated computation. The following lambda expression is another valid complexity function for function map_incr_rec. The complexity function above is preferable, however, because it is simpler.

const long k = 123;
auto map_incr_rec_complexity_fct2 = [&] (long lo, long hi) {
  return k * (hi - lo);
};

More generally, suppose that we know that a given algorithm has work cost of $W = n + \log n$. Although it would be fine to assign to this algorithm exactly $W$, we could just as well assign to the algorithm the cost $n$, because the second term is dominated by the first. In other words, when expressing work costs, we only need to be precise up to the asymptotic complexity class of the work.

7.2. Controlled statements

In PASL, a controlled statement, or cstmt, is an annotation in the program text that activates automatic granularity control for a specified region of code. In particular, a controlled statement behaves as a C++ statement that has the special ability to choose on the fly whether or not the computation rooted at the body of the statement spawns parallel threads. To support such automatic granularity control PASL uses a prediction algorithm to map the asymptotic work cost (as returned by the complexity function) to actual processor cycles. When the predicted processor cycles of a particular instance of the controlled statement falls below a threshold (determined automatically for the specific machine), then that instance is sequentialized, by turning off the ability to spawn parallel threads for the execution of that instance. If the predicted processor cycle count is higher than the threshold, then the statement instance is executed in parallel.

In other words, the reader can think of a controlled statement as a statement that executes in parallel when the benefits of parallel execution far outweigh its cost and that executes sequentially in a way similar to the sequential elision of the body of the controlled statement would if the cost of parallelism exceeds its benefits. We note that while the sequential exection is similar to a sequential elision, it is not exactly the same, because every call to fork2 must check whether it should create parallel threads or run sequentially. Thus the execution may differ from the sequential elision in terms of performance but not in terms of behavior or semantics.

Example 20. Array-increment function with automatic granularity control

The code below uses a controlled statement to automatically select, at run time, the threshold size for our parallel array-increment function.

controller_type map_incr_rec_contr("map_incr_rec");

void map_incr_rec(const long* source, long* dest, long lo, long hi) {
  long n = hi - lo;
  cstmt(map_incr_rec_contr, [&] { return n; }, [&] {
    if (n == 0) {
      // do nothing
    } else if (n == 1) {
      dest[lo] = source[lo] + 1;
    } else {
      long mid = (lo + hi) / 2;
      fork2([&] {
        map_incr_rec(source, dest, lo, mid);
      }, [&] {
        map_incr_rec(source, dest, mid, hi);
      });
    }
  });
}

The controlled statement takes three arguments, whose requirements are specified below, and returns nothing (i.e., void). The effectiveness of the granularity controller may be compromised if any of the requirements are not met.

  • The first argument is a reference to the controller object. The controller object is used by the controlled statement to collect profiling data from the program as the program runs. Every controller object is initialized with a string label (in the code above "map_incr_rec"). The label must be unique to the particular controller. Moreover, the controller must be declared as a global variable.

  • The second argument is the complexity function. The type of the return value should be long.

  • The third argument is the body of the controlled statement. The return type of the controlled statement should be void.

When the controlled statement chooses sequential evaluation for its body the effect is similar to the effect where in the code above the input size falls below the threshold size: the body and the recursion tree rooted there is sequentialized. When the controlled statement chooses parallel evaluation, the calls to fork2() create parallel threads.

7.2.1. Granularity control with alternative sequential bodies

It is not unusual for a divide-and-conquer algorithm to switch to a different algorithm at the leaves of its recursion tree. For example, sorting algorithms, such as quicksort, may switch to insertion sort at small problem sizes. In the same way, it is not unusual for parallel algorithms to switch to different sequential algorithms for handling small problem sizes. Such switching can be beneficial especially when the parallel algorithm is not asymptotically work efficient.

To provide such algorithmic switching, PASL provides an alternative form of controlled statement that accepts a fourth argument: the alternative sequential body. This alternative form of controlled statement behaves essentially the same way as the original described above, with the exception that when PASL run time decides to sequentialize a particular instance of the controlled statement, it falls through to the provided alternative sequential body instead of the "sequential elision."

Example 21. Array-increment function with automatic granularity control and sequential body
controller_type map_incr_rec_contr("map_incr_rec");

void map_incr_rec(const long* source, long* dest, long lo, long hi) {
  long n = hi - lo;
  cstmt(map_incr_rec_contr, [&] { return n; }, [&] {
    if (n == 0) {
      // do nothing
    } else if (n == 1) {
      dest[lo] = source[lo] + 1;
    } else {
      long mid = (lo + hi) / 2;
      fork2([&] {
        map_incr_rec(source, dest, lo, mid);
      }, [&] {
        map_incr_rec(source, dest, mid, hi);
      });
    }
  }, [&] {
    for (long i = lo; i < hi; i++)
      dest[i] = source[i] + 1;
  });
}

Even though the parallel and sequential array-increment algorithms are algorithmically identical, except for the calls to fork2(), there is still an advantage to using the alternative sequential body: the sequential code does not pay for the parallelism overheads due to fork2(). Even when eliding fork2(), the run-time-system has to perform a conditional branch to check whether or not the context of the fork2() call is parallel or sequential. Because the cost of these conditional branches adds up, the version with the sequential body is going to be more work efficient. Another reason for why a sequential body may be more efficient is that it can be written more simply, as for example using a for-loop instead of recursion, which will be faster in practice.

Note
Recommended style for programming with controlled statements

In general, we recommend that the code of the parallel body be written so as to be completely self contained, at least in the sense that the parallel body code contains the logic that is necessary to handle recursion all the way down to the base cases. The code for map_incr_rec honors this style by the fact that the parallel body handles the cases where n is zero or one (base cases) or is greater than one (recursive case). Put differently, it should be the case that, if the parallelism-specific annotations (including the alternative sequential body) are erased, the resulting program is a correct program.

We recommend this style because such parallel codes can be debugged, verified, and tuned, in isolation, without relying on alternative sequential codes.

7.3. Controlled parallel-for loops

Let us add one more component to our granularity-control toolkit: the parallel-for from. By using this loop construct, we can avoid having to explicitly express recursion-trees over and over again. For example, the following function performs the same computation as the example function we defined in the first lecture. Only, this function is much more compact and readable. Moreover, this code takes advantage of our automatic granularity control, also by replacing the parallel-for with a serial-for.

loop_controller_type map_incr_contr("map_incr");

void map_incr(const long* source, long* dest, long n) {
  parallel_for(map_incr_contr, (long)0, n, [&] (long i) {
    dest[i] = source[i] + 1;
  });
}

Underneath, the parallel-for loop uses a divide-and-conquer routine whose structure is similar to the structure of the divide-and-conquer routine of our map_incr_rec. Because the parallel-for loop generates the log-height recursion tree, the map_incr routine just above has the same span as the map_incr routine that we defined earlier: $\log n$, where $n$ is the size of the input array.

Notice that the code above specifies no complexity function. The reason is that this particular instance of the parallel-for loop implicitly defines a complexity function. The implicit complexity function reports a linear-time cost for any given range of the iteration space of the loop. In other words, the implicit complexity function assumes that per iteration the body of the loop performs a constant amount of work. Of course, this assumption does not hold in general. If we want to specify explicitly the complexity function, we can use the form shown in the example below. The complexity function is passed to the parallel-for loop as the fourth argument. The complexity function takes as argument the range [lo, hi). In this case, the complexity is linear in the number of iterations. The function simply returns the number of iterations as the complexity.

loop_controller_type map_incr_contr("map_incr");

void map_incr(const long* source, long* dest, long n) {
  auto linear_complexity_fct = [] (long lo, long hi) {
    return hi-lo;
  };
  parallel_for(map_incr_contr, linear_complexity_fct, (long)0, n, [&] (long i) {
    dest[i] = source[i] + 1;
  });
}

The following code snippet shows a more interesting case for the complexity function. In this case, we are performing a multiplication of a dense matrix by a dense vector. The outer loop iterates over the rows of the matrix. The complexity function in this case gives to each of these row-wise iterations a cost in proportion to the number of scalars in each column.

loop_controller_type dmdvmult_contr("dmdvmult");

// mtx: nxn dense matrix, vec: length n dense vector
// dest: length n dense vector
void dmdvmult(double* mtx, double* vec, double* dest, long n) {
  auto compl_fct = [&] (long lo, long hi) {
    return (hi-lo)*n;
  };
  parallel_for(dmdvmult_contr, compl_fct, (long)0, n, [&] (long i) {
    ddotprod(mtx, v, dest, i);
  });
  return dest;
}
Example 22. Speedup for matrix multiply

Matrix multiplication has been widely used as an example for parallel computing since the early days of the field. There are good reasons for this. First, matrix multiplication is a key operation that can be used to solve many interesting problems. Second, it is an expansive computation that is nearly cubic in the size of the input---it can thus can become very expensive even with modest inputs.

Fortunately, matrix multiplication can be parallelized relatively easily as shown above. The figure below shows the speedup for a sample run of this code. Observe that the speedup is rather good, achieving nearly excellent utilization.

Speedup plot for matrix multiplication for latexmath:[$25000 	imes 25000$] matrices.
Figure 6. Speedup plot for matrix multiplication for $25000 \times 25000$ matrices.

While parallel matrix multiplication delivers excellent speedups, this is not common for many other algorithms on modern multicore machines where many computations can quickly become limited by the availability of bandwidth. Matrix multiplication does not suffer as much from the memory-bandwidth limitations because it performs significant work per memory operation: it touches $\Theta(n^2)$ memory cells, while performing nearly $\Theta(n^3)$ work.

8. Simple Parallel Arrays

Arrays are a fundamental data structure in sequential and parallel computing. When computing sequentially, arrays can sometimes be replaced by linked lists, especially because linked lists are more flexible. Unfortunately, linked lists are deadly for parallelism, because they require serial traversals to find elements; this makes arrays all the more important in parallel computing.

Unfortunately, it is difficult to find a good treatment of parallel arrays in C++: the various array implementations provided by C++ have been designed primarily for sequential computing. Each one has various pitfalls for parallel use.

Example 23. C++ arrays

By default, C++ arrays that are created by the new[] operator are initialized sequentially. Therefore, the work and span cost of the call new[n] is n. But we can initialize an array in logarithmic span in the number of items.

Example 24. STL vectors

The "vector" data structure that is provided by the Standard Template Library (STL) has similar issues. The STL vector implements a dynamically resizable array that provides push, pop, and indexing operations. The push and pop operations take amortized constant time and the indexing operation constant time. As with C++ arrays, initialization of vectors can require linear work and span. The STL vector also provides the method resize(n) which changes the size of the array to be n. The resize operation takes, in the worst case, linear work and span in proportion to the new size, n. In other words, the resize function uses a sequential algorithm to fill the cells in the vector. The resize operation is therefore not arrays.

Such sequential computations that exist behind the wall of abstraction of a language or library can harm parallelism by introducing implicit sequential dependencies. Finding the source of such sequential bottlenecks can be time consuming, because they are hidden behind the abstraction boundary of the native array abstraction that is provided by the programming language.

We can avoid such pitfalls by carefully designing our own array data structure. Because array implementations are quite subtle, we consider our own implementation of parallel arrays, which makes explicit the cost of array operation, allowing us to control them quite carefully. Specifically, we carefully control initialization and disallow implicit copy operations on arrays, because copy operations can harm observable work efficiency (their asymptotic work cost is linear).

8.1. Interface and cost model

The key components of our array data structure, sparray, are shown by the code snippet below. An sparray can store 64-bit words only; in particular, they are monomorphic and fixed to values of type long. Generalizing sparray to store values of arbitrary types can be achieved by use of C++ templates. We stick to monomorphic arrays here to simplify the presentation.

using value_type = long;

class sparray {
public:

  // n: size to give to the array; by default 0
  sparray(unsigned long n = 0);

  // constructor from list
  sparray(std::initializer_list<value_type> xs);

  // indexing operator
  value_type& operator[](unsigned long i);

  // size of the array
  unsigned long size() const;

};

The class sparray provides two constructors. The first one takes in the size of the array (set to 0 by default) and allocates an unitialized array of the specified size (nullptr if size is 0). The second constructor takes in a list specified by curly braces and allocates an array with the same size. Since the argument to this constructor must be specified explicitly in the program, its size is constant by definition.

The cost model guaranteed by our implementation of parallel array is as follows:

  • Constructors/Allocation: The work and span of simply allocating an array on the heap, without initialization, is constant. The second constructor performs initialization, based on constant-size lists, and thus also has constant work and span.

  • Array indexing: Each array-indexing operation, that is the operation which accesses an individual cell, requires constant work and constant span.

  • Size operation: The work and the span of accessing the size of the array is constant.

  • Destructors/Deallocation: Not shown, the class includes a destructor that frees the array. Combined with the "move assignment operator" that C++ allows us to define, destructors can be used to deallocate arrays when they are out of scope. The destructor takes constant time because the contents of the array are just bits that do not need to be destructed individually.

  • Move assignment operator: Not shown, the class includes a move-assignment operator that gets fired when an array is assigned to a variable. This operator moves the contents of the right-hand side of the assigned array into that of the left-hand side. This operation takes constant time.

  • Copy constructor: The copy constructor of sparray is disabled. This prohibits copying an array unintentionally, for example, by passing the array by value to a function.

Note
The constructors of our array class do not perform initializations that involve non-constant work. If desired, the programmer can write an initializer that performs linear work and logarithmic span (if the values used for initialization have non-constant time cost, these bounds may need to be scaled accordingly).
Example 25. Simple use of arrays

This program below shows a basic use sparray's. The first line allocates and initializes the contents of the array to be three numbers. The second uses the familiar indexing operator to access the item at the second position in the array. The third line extracts the size of the array. The fourth line assigns to the second cell the value 5. The fifth prints the contents of the cell.

sparray xs = { 1, 2, 3 };
std::cout << "xs[1] = " << xs[1] << std::endl;
std::cout << "xs.size() = " << xs.size() << std::endl;
xs[2] = 5;
std::cout << "xs[2] = " << xs[2] << std::endl;

Output:

xs[1] = 2
xs.size() = 3
xs[2] = 5

8.2. Allocation and deallocation

Arrays can be allocated by specifying the size of the array.

Example 26. Allocation and deallocation
sparray zero_length = sparray();
sparray another_zero_length = sparray(0);
sparray yet_another_zero_length;
sparray length_five = sparray(5);    // contents uninitialized
std::cout << "|zero_length| = " << zero_length.size() << std::endl;
std::cout << "|another_zero_length| = " << another_zero_length.size() << std::endl;
std::cout << "|yet_another_zero_length| = " << yet_another_zero_length.size() << std::endl;
std::cout << "|length_five| = " << length_five.size() << std::endl;

Output:

|zero_length| = 0
|another_zero_length| = 0
|yet_another_zero_length| = 0
|length_five| = 5

Just after creation, the array contents consist of uninitialized bits. We use this convention because the programmer needs flexibility to decide the parallelization strategy to initialize the contents. Internally, the sparray class consists of a size field and a pointer to the first item in the array. The contents of the array are heap allocated (automatically) by constructor of the sparray class. Deallocation occurs when the array’s destructor is called. The destructor can be called by the programmer or by run-time system (of C++) if an object storing the array is destructed. Since C++ destructs (stack allocated) variables that go out of scope when a function returns, we can combine the stack discipline with heap-allocated arrays to manage the deallocation of arrays mostly automatically. We give several examples of this automatic deallocation scheme below.

Example 27. Automatic deallocation of arrays upon return

In the function below, the sparray object that is allocated on the frame of foo is deallocated just before foo returns, because the variable xs containing it goes out of scope.

void foo() {
  sparray xs = sparray(10);
  // array deallocated just before foo() returns
}
Example 28. Dangling pointers in arrays

Care must be taken when managing arrays, because nothing prevents the programmer from returning a dangling pointer.

value_type* foo() {
  sparray xs = sparray(10);
  ...
  // array deallocated just before foo() returns
  return &xs[0]
}

...

std::cout << "contents of deallocated memory: " << *foo() << std::endl;

Output:

contents of deallocated memory: .... (undefined)

It is safe to take a pointer to a cell in the array, when the array itself is still in scope. For example, in the code below, the contents of the array are used strictly when the array is in scope.

void foo() {
  sparray xs = sparray(10);
  xs[0] = 34;
  bar(&xs[0]);
  ...
  // array deallocated just before foo() returns
}

void bar(value_type* p) {
  std::cout << "xs[0] = " << *p << std::endl;
}

Output:

xs[0] = 34

We are going to see that we can rely on cleaner conventions for passing to functions references on arrays.

8.3. Passing to and returning from functions

If you are familiar with C++ container libraries, such as STL, this aspect of our array implementation, namely the calling conventions, may be unfamiliar: our arrays cannot be passed by value. We forbid passing by value because passing by value implies creating a fresh copy for each array being passed to or returned by a function. Of course, sometimes we really need to copy an array. In this case, we choose to copy the array explicitly, so that it is obvious where in our code we are paying a linear-time cost for copying out the contents. We will return to the issue of copying later.

Example 29. Incorrect use of copy constructor

What then happens if the program tries to pass an array to a function? The program will be rejected by the compiler. The code below does not compile, because we have disabled the copy constructor of our sparray class.

value_type foo(sparray xs) {
  return xs[0];
}

void bar() {
  sparray xs = { 1, 2 };
  foo(xs);
}
Example 30. Correctly passing an array by reference

The following code does compile, because in this case we pass the array xs to foo by reference.

value_type foo(const sparray& xs) {
  return xs[0];
}

void bar() {
  sparray xs = { 1, 2 };
  foo(xs);
}

Returning an array is straightforward: we take advantage of a feature of modern C++11 which automatically detects when it is safe to move a structure by a constant-time pointer swap. Code of the following form is perfectly legal, even though we disabled the copy constructor of sparray, because the compiler is able to transfer ownership of the array to the caller of the function. Moreover, the transfer is guaranteed to be constant work — not linear like a copy would take. The return is fast, because internally all that happens is that a couple words are being exchanged. Such "move on return" is achieved by the "move-assignment operator" of sparray class.

Example 31. Create and initialize an array (sequentially)
sparray fill_seq(long n, value_type x) {
  sparray tmp = sparray(n);
  for (long i = 0; i < n; i++)
    tmp[i] = x;
  return tmp;
}

void bar() {
  sparray xs = fill_seq(4, 1234);
  std::cout << "xs = " << xs << std::endl;
}

Output after calling bar():

xs = { 1234, 1234, 1234, 1234 }

Although it is perfectly fine to assign to an array variable the contents of a given array, what happens may be surprising to those who know the usual conventions of C++11 container libraries. Consider the following program.

Example 32. Move constructor
sparray xs = fill_seq(4, 1234);
sparray ys = fill_seq(3, 333);
ys = std::move(xs);
std::cout << "xs = " << xs << std::endl;
std::cout << "ys = " << ys << std::endl;

The assignment from xs to ys simultaneously destroys the contents of ys (by calling its destructor, which nulls it out), namely the array { 333, 333, 333 }, moves the contents of xs to ys, and empties out the contents of xs. This behavior is defined as part of the move operator of sparray. The result is the following.

xs = { }
ys = { 1234, 1234, 1234, 1234 }

The reason we use this semantics for assignment is that the assignment takes constant time. Later, we are going to see that we can efficiently copy items out of an array. But for reasons we already discussed, the copy operation is going to be explicit.

Exercise: duplicating items in parallel

The aim of this exercise is to combine our knowledge of parallelism and arrays. To this end, the exercise is to implement two functions. The first, namely duplicate, is to return a new array in which each item appearing in the given array xs appears twice.

sparray duplicate(const sparray& xs) {
  // fill in
}

For example:

sparray xs = { 1, 2, 3 };
std::cout << "xs = " << duplicate(xs) << std::endl;

Expected output:

xs = { 1, 1, 2, 2, 3, 3 }

The second function is a generalization of the first: the value returned by ktimes should be an array in which each item x that is in the given array xs is replaced by k duplicate items.

sparray ktimes(const sparray& xs, long k) {
  // fill in
}

For example:

sparray xs = { 5, 7 };
std::cout << "xs = " << ktimes(xs, 3) << std::endl;

Expected output:

xs = { 5, 5, 5, 7, 7, 7 }

Notice that the k parameter of ktimes is not bounded. Your solution to this problem should be highly parallel not only in the number of items in the input array, xs, but also in the duplication-degree parameter, k.

  1. What is the work and span complexity of your solution?

  2. Does your solution expose ample parallelism? How much, precisely?

  3. What is the speedup do you observe in practice on various input sizes?

8.4. Tabulation

A tabulation is a parallel operation which creates a new array of a given size and initializes the contents according to a given "generator function". The call tabulate(g, n) allocates an array of length n and assigns to each valid index in the array i the value returned by g(i).

template <class Generator>
sparray tabulate(Generator g, long n);

Tabulations can be used to generate sequences according to a specified formula.

Example 33. Sequences of even numbers
sparray evens = tabulate([&] (long i) { return 2*i; }, 5);
std::cout << "evens = " << evens << std::endl;

Output:

evens = { 0, 2, 4, 6, 8 }

Copying an array can be expressed as a tabulation.

Example 34. Parallel array copy function using tabulation
sparray mycopy(const sparray& xs) {
  return tabulate([&] (long i) { return xs[i]; }, xs.size());
}
Exercise

Solve the duplicate and ktimes problems that were given in homework, this time using tabulations.

Solutions appear below.

Example 35. Solution to duplicate and ktimes exercises
sparray ktimes(const sparray& xs, long k) {
  long m = xs.size() * k;
  return tabulate([&] (long i) { return xs[i/k]; }, m);
}

sparray duplicate(const sparray& xs) {
  return ktimes(xs, 2);
}

The implementation of tabulate is a straightforward application of the parallel-for loop.

loop_controller_type tabulate_contr("tabulate");

template <class Generator>
sparray tabulate(Generator g, long n) {
  sparray tmp = sparray(n);
  parallel_for(tabulate_contr, (long)0, n, [&] (long i) {
    tmp[i] = g(i);
  });
  return tmp;
}

Note that the work and span of the generator function depends on the generator function passed as an argument to the tabulation. Let us first analyze for the simple case, where the generator function takes constant work (and hence, constant span). In this case, it should be clear that a tabulation should take work linear in the size of the array. The reason is that the only work performed by the body of the loop is performed by the constant-time generator function. Since the loop itself performs as many iterations as positions in the array, the work cost is indeed linear in the size of the array. The span cost of the tabulation is the sum of two quantitles: the span taken by the loop and the maximum value of the spans taken by the applications of the generator function. Recall that we saw before that the span cost of a parallel-for loop with $n$ iterations is $\log n$. The maximum of the spans of the generator applications is a constant. Therefore, we can conclude that, in this case, the span cost is logarithmic in the size of the array.

The story is only a little more complicated when we generalize to consider non-constant time generator functions. Let $W(\mathtt{g}(i))$ denote the work performed by an application of the generator function to a specified value $i$. Similarly, let $S(\mathtt{g}(i))$ denote the span. Then the tabulation takes work

$$ \sum_{i=0}^{n} W(\mathtt{g}(i)) $$

and span

$$ \log n + \max_{i=0}^{n} S(\mathtt{g}(i))$$

8.5. Higher-order granularity controllers

We just observed that each application of our tabulate operation can have different work and span cost depending on the selection of the generator function. Pause for a moment and consider how this observation could impact our granularity-control scheme. Consider, in particular, the way that the tabulate function uses its granularity-controller object, tabulate_contr. This one controller object is shared by every call site of tabulate().

The problem is that all of the profiling data that the granularity controller collects at run time is lumped together, even though each generator function that is passed to the tabulate function can have completely different performance characteristics. The threshold that is best for one generator function is not necessarily good for another generator function. For this reason, there must be one distinct granularity-control object for each generator function that is passed to tabulate. For this reason, we refine our solution from the one above to the one below, which relies on C++ template programming to effectively key accesses to the granularity controller object by the type of the generator function.

template <class Generator>
class tabulate_controller {
public:
  static loop_controller_type contr;
};

template <class Generator>
loop_controller_type
  tabulate_controller<Generator>::contr("tabulate"+std::string(typeid(Generator).name()));

template <class Generator>
sparray tabulate(Generator g, long n) {
  sparray tmp = sparray(n);
  parallel_for(tabulate_controller<Generator>::contr, (long)0, n, [&] (long i) {
    tmp[i] = g(i);
  });
  return tmp;
}

To be more precise, the use of the template parameter in the class tabulate_controller ensures that each generator function in the program that is passed to tabulate() gets its own unique instance of the controller object. The rules of the template system regarding static class members that appear in templated classes ensure this behavior. Although it is not essential for our purposes to have a precise understanding of the template system, it is useful to know that the template system provides us with the exact mechanism that we need to properly separate granularity controllers of distinct instances of higher-order functions, such as tabulation.

Note
Above, we still assume constant-work generator functions.

8.6. Reduction

A reduction is an operation which combines a given set of values according to a specified identity element and a specified associative combining operator. Let $S$ denote a set. Recall from algebra that an associative combining operator is any binary operator $\oplus$ such that, for any three items $x,y,z \in S$, the following holds.

$$ x \oplus (y \oplus z) = (x \oplus y) \oplus z $$

An element $\mathbf{I} \in S$ is an identity element if for any $x \in S$ the following holds.

$$ (x \oplus \mathbf{I}) \, = \, (\mathbf{I} \oplus x) \, = \, x $$

This algebraic structure consisting of $(S, \oplus, \mathbf{I})$ is called a monoid and is particularly worth knowing because this structure is a common pattern in parallel computing.

Example 36. Addition monoid
  • $S$ = the set of all 64-bit unsigned integers; $\oplus$ = addition modulo $2^{64}$; $\mathbf{I}$ = 0

Example 37. Multiplication monoid
  • $S$ = the set of all 64-bit unsigned integers; $\oplus$ = multiplication modulo $2^{64}$; $\mathbf{I}$ = 1

Example 38. Max monoid
  • $S$ = the set of all 64-bit unsigned integers; $\oplus$ = max function; $\mathbf{I}$ = 0

The identity element is important because we are working with sequences: having a base element is essential for dealing with empty sequences. For example, what should the sum of the empty sequence? More interestingly, what should be the maximum (or minimum) element of an empty sequence? The identity element specifies this behavior.

What about the associativity of $\oplus$? Why does associativity matter? Suppose we are given the sequence $ [ a_0, a_1, \ldots, a_n ] $. The serial reduction of this sequence always computes the expression $(a_0 \oplus a_1 \oplus \ldots \oplus a_n)$. However, when the reduction is performed in parallel, the expression computed by the reduction could be $( (a_0 \oplus a_1 \oplus a_2 \oplus a_3) \oplus (a_4 \oplus a_5) \oplus \ldots \oplus (a_{n-1} \oplus a_n) )$ or $( (a_0 \oplus a_1 \oplus a_2) \oplus (a_3 \oplus a_4 \oplus a_5) \oplus \ldots \oplus (a_{n-1} \oplus a_n) )$. In general, the exact placement of the parentheses in the parallel computation depends on the way that the parallel algorithm decomposes the problem. Associativity gives the parallel algorithm the flexibility to choose an efficient order of evaluation and still get the same result in the end. The flexibility to choose the decomposition of the problem is exploited by efficient parallel algorithms, for reasons that should be clear by now. In summary, associativity is a key building block to the solution of many problems in parallel algorithms.

Now that we have monoids for describing a generic method for combining two items, we can consider a generic method for combining many items in parallel. Once we have this ability, we will see that we can solve the remaining problems from last homework by simply plugging the appropriate monoids into our generic operator, reduce. The interface of this operator in our framework is specified below. The first parameter corresponds to $\oplus$, the second to the identity element, and the third to the sequence to be processed.

template <class Assoc_binop>
value_type reduce(Assoc_binop b, value_type id, const sparray& xs);

We can solve our first problem by plugging integer plus as $\oplus$ and 0 as $\mathbf{I}$.

Example 39. Summing elements of array
auto plus_fct = [&] (value_type x, value_type y) {
  return x+y;
};

sparray xs = { 1, 2, 3 };
std::cout << "sum_xs = " << reduce(plus_fct, 0, xs) << std::endl;

Output:

reduce(plus_fct, 0, xs) = 6

We can solve our second problem in a similar fashion. Note that in this case, since we know that the input sequence is nonempty, we can pass the first item of the sequence as the identity element. What could we do if we instead wanted a solution that can deal with zero-length sequences? What identity element might make sense in that case? Why?

Example 40. Taking max of elements of array

Let us start by solving a special case: the one where the input sequence is nonempty.

auto max_fct = [&] (value_type x, value_type y) {
  return std::max(x, y);
};

sparray xs = { -3, 1, 634, 2, 3 };
std::cout << "reduce(max_fct, xs[0], xs) = " << reduce(max_fct, xs[0], xs) << std::endl;

Output:

reduce(max_fct, xs[0], xs) = 634

Observe that in order to seed the reduction we selected the provisional maximum value to be the item at the first position of the input sequence. Now let us handle the general case by seeding with the smallest possible value of type long.

long max(const sparray& xs) {
  return reduce(max_fct, LONG_MIN, xs);
}

The value of LONG_MIN is defined by <limits.h>.

Like the tabulate function, reduce is a higher-order function. Just like any other higher-order function, the work and span costs have to account for the cost of the client-supplied function, which is in this case, the associative combining operator.

8.7. Scan

A scan is an iterated reduction that is typically expressed in one of two forms: inclusive and exclusive. The inclusive form maps a given sequence $ [ x_0, x_1, x_2, \ldots, x_{n-1} ] $ to $ [ x_0, x_0 \oplus x_1, x_0 \oplus x_1 \oplus x_2, \ldots, x_0 \oplus x_1 \oplus \ldots \oplus x_{n-1} ] $.

template <class Assoc_binop>
sparray scan_incl(Assoc_binop b, value_type id, const sparray& xs);
Example 41. Inclusive scan
  scan_incl(b, 0, sparray({ 2, 1, 8, 3 }))
= { reduce(b, id, { 2 }),       reduce(b, id, { 2, 1 }),
    reduce(b, id, { 2, 1, 8 }), reduce(b, id, { 2, 1, 8, 3}) }
= { 0+2, 0+2+1, 0+2+1+8, 0+2+1+8+3 }
= { 2, 3, 11, 14 }

The exclusive form maps a given sequence $ [ x_0, x_1, x_2, \ldots, x_{n-1} ] $ to $ [ \mathbf{I}, x_0, x_0 \oplus x_1, x_0 \oplus x_1 \oplus x_2, \ldots, x_0 \oplus x_1 \oplus \ldots \oplus x_{n-2} ] $. For convenience, we extend the result of the exclusive form with the total $ x_0 \oplus \ldots \oplus x_{n-1} $.

class scan_excl_result {
public:
  sparray partials;
  value_type total;
};

template <class Assoc_binop>
scan_excl_result scan_excl(Assoc_binop b, value_type id, const sparray& xs);
Example 42. Exclusive scan

The example below represents the logical behavior of scan, but actually says nothing about the way scan is implemented.

  scan_excl(b, 0, { 2, 1, 8, 3 }).partials
= { reduce(b, 0, { }),      reduce(b, 0, { 2 }),
    reduce(b, 0, { 2, 1 }), reduce(b, 0, { 2, 1, 8}) }
= { 0, 0+2, 0+2+1, 0+2+1+8 }
= { 0, 2, 3, 11 }

  scan_excl(b, 0, { 2, 1, 8, 3 }).total
= reduce(b, 0, { 2, 1, 8, 3})
= { 0+2+1+8+3 }
= 14

Scan has applications in many parallel algorithms. To name just a few, scan has been used to implement radix sort, search for regular expressions, dynamically allocate processors, evaluate polynomials, etc. Suffice to say, scan is important and worth knowing about because scan is a key component of so many efficient parallel algorithms. In this course, we are going to study a few more applications not in this list.

The expansions shown above suggest the following sequential algorithm.

template <class Assoc_binop>
scan_excl_result scan_excl_seq(Assoc_binop b, value_type id, const sparray& xs) {
  long n = xs.size();
  sparray r = array(n);
  value_type x = id;
  for (long i = 0; i < n; i++) {
    r[i] = x;
    x = b(x, xs[i]);
  }
  return make_scan_result(r, x);
}

If we just blindly follow the specification above, we might be tempted to try the solution below.

loop_controller_type scan_contr("scan");

template <class Assoc_binop>
sparray scan_excl(Assoc_binop b, value_type id, const sparray& xs) {
  long n = xs.size();
  sparray result = array(n);
  result[0] = id;
  parallel_for(scan_contr, 1l, n, [&] (long i) {
    result[i] = reduce(b, id, slice(xs, 0, i-1));
  });
  return result;
}
Question

Although it is highly parallel, this solution has a major problem. What is it?

Consider that our sequential algorithm takes linear time in the size of the input array. As such, finding a work-efficient parallel solution means finding a solution that also takes linear work in the size of the input array. The problem is that our parallel algorithm takes quadratic work: it is not even asymptotically work efficient! Even worse, the algorithm performs a lot of redundant work.

Can we do better? Yes, in fact, there exist solutions that take, in the size of the input, both linear time and logarithmic span, assuming that the given associative operator takes constant time. It might be worth pausing for a moment to consider this fact, because the specification of scan may at first look like it would resist a solution that is both highly parallel and work efficient.

8.8. Derived operations

The remaining operations that we are going to consider are useful for writing more succinct code and for expressing special cases where certain optimizations are possible. All of the the operations that are presented in this section are derived forms of tabulate, reduce, and scan.

8.8.1. Map

The map(f, xs) operation applies f to each item in xs returning the array of results. It is straightforward to implement as a kind of tabulation, as we have at our disposal efficient indexing.

template <class Func>
sparray map(Func f, sparray xs) {
  return tabulate([&] (long i) { return f(xs[i]); }, xs.size());
}

The array-increment operation that we defined on the first day of lecture is simple to express via map.

Example 43. Incrementing via map
sparray map_incr(sparray xs) {
  return map([&] (value_type x) { return x+1; }, xs);
}

The work and span costs of map are similar to those of tabulate. Granularity control is handled similarly as well. However, that the granularity controller object corresponding to map is instantiated properly is not obvious. It turns out that, for no extra effort, the behavior that we want is indeed preserved: each distinct function that is passed to map is assigned a distinct granularity controller. Although it is outside the scope of this course, the reason that this scheme works in our current design owes to specifics of the C++ template system.

8.8.2. Fill

The call fill(v, n) creates an array that is initialized with a specified number of items of the same value. Although just another special case for tabulation, this function is worth having around because internally the fill operation can take advantage of special hardware optimizations, such as SIMD instructions, that increase parallelism.

sparray fill(value_type v, long n);
Example 44. Creating an array of all 3s
sparray threes = fill(3, 5);
std::cout << "threes = " << threes << std::endl;

Output:

threes = { 3, 3, 3, 3, 3 }

8.8.3. Copy

Just like fill, the copy operation can take advantage of special hardware optimizations that accellerate memory traffic. For the same reason, the copy operation is a good choice when a full copy is needed.

sparray copy(const sparray& xs);
Example 45. Copying an array
sparray xs = { 3, 2, 1 };
sparray ys = copy(xs);
std::cout << "xs = " << xs << std::endl;
std::cout << "ys = " << ys << std::endl;

Output:

xs = { 3, 2, 1 }
ys = { 3, 2, 1 }

8.8.4. Slice

We now consider a slight generalization on the copy operator: with the slice operation we can copy out a range of positions from a given array rather than the entire array.

sparray slice(const sparray& xs, long lo, long hi);

The slice operation takes a source array and a range to copy out and returns a fresh array that contains copies of the items in the given range.

Example 46. Slicing an array
sparray xs = { 1, 2, 3, 4, 5 };
std::cout << "slice(xs, 1, 3) = " << slice(xs, 1, 3) << std::endl;
std::cout << "slice(xs, 0, 4) = " << slice(xs, 0, 4) << std::endl;

Output:

{ 2, 3 }
{ 1, 2, 3, 4 }

8.8.5. Concat

In contrast to slice, the concat operation lets us "copy in" to a fresh array.

sparray concat(const sparray& xs, const sparray& ys);
Example 47. Concatenating two arrays
sparray xs = { 1, 2, 3 };
sparray ys = { 4, 5 };
std::cout << "concat(xs, ys) = " << concat(xs, ys) << std::endl;

Output:

{ 1, 2, 3, 4, 5 }

8.8.6. Prefix sums

The prefix sums problem is a special case of the scan problem. We have defined two solutions for two variants of the problem: one for the exclusive prefix sums and one for the inclusive case.

sparray prefix_sums_incl(const sparray& xs);
scan_excl_result prefix_sums_excl(const sparray& xs);
Example 48. Inclusive and exclusive prefix sums
sparray xs = { 2, 1, 8, 3 };
sparray incl = prefix_sums_incl(xs);
scan_excl_result excl = prefix_sums_excl(xs);
std::cout << "incl = " << incl << std::endl;
std::cout << "excl.partials = " << excl.partials << "; excl.total = " << excl.total << std::endl;

Output:

incl = { 2, 3, 11, 14 }
excl.partials = { 0, 2, 3, 11 }; excl.total = 147

8.8.7. Filter

The last data-parallel operation that we are going to consider is the operation that copies out items from a given array based on a given predicate function.

template <class Predicate>
sparray filter(Predicate pred, const sparray& xs);

For our purposes, a predicate function is any function that takes a value of type long (i.e., value_type) and returns a value of type bool.

Example 49. Extracting even numbers

The following function copies out the even numbers it receives in the array of its argument.

bool is_even(value_type x) {
  return (x%2) == 0;
}

sparray extract_evens(const sparray& xs) {
  return filter([&] (value_type x) { return is_even(x); }, xs);
}

sparray xs = { 3, 5, 8, 12, 2, 13, 0 };
std::cout << "extract_evens(xs) = " << extract_evens(xs) << std::endl;

Output:

extract_evens(xs) = { 8, 12, 2, 0 }
Example 50. Solution to the sequential-filter problem

The particular instance of the filter problem that we are considering is a little tricky because we are working with fixed-size arrays. In particular, what requires care is the method that we use to copy the selected items out of the input array to the output array. We need to first run a pass over the input array, applying the predicate function to the items, to determine which items are to be written to the output array. Furthermore, we need to track how many items are to be written so that we know how much space to allocate for the output array.

template <class Predicate>
sparray filter(Predicate pred, const sparray& xs) {
  long n = xs.size();
  long m = 0;
  sparray flags = array(n);
  for (long i = 0; i < n; i++)
    if (pred(xs[i])) {
      flags[i] = true;
      m++;
    }
  sparray result = array(m);
  long k = 0;
  for (long i = 0; i < n; i++)
    if (flags[i])
      result[k++] = xs[i];
  return result;
}
Question

In the sequential solution above, it appears that there are two particular obstacles to parallelization. What are they?

Hint: the obstacles relate to the use of variables m and k.

Question

Under one particular assumption regarding the predicate, this sequential solution takes linear time in the size of the input, using two passes. What is the assumption?

8.8.8. Parallel-filter problem

The starting point for our solution is the following code.

template <class Predicate>
sparray filter(Predicate p, const sparray& xs) {
  sparray flags = map(p, xs);
  return pack(flags, xs);
}

The challenge of this exercise is to solve the following problem: given two arrays of the same size, the first consisting of boolean valued fields and the second containing the values, return the array that contains (in the same relative order as the items from the input) the values selected by the flags. Your solution should take linear work and logarithmic span in the size of the input.

sparray pack(const sparray& flags, const sparray& xs);
Example 51. The allocation problem
sparray flags = { true, false, false, true, false, true, true };
sparray xs    = {   34,    13,     5,    1,    41,   11,   10 };
std::cout << "pack(flags, xs) = " << pack(flags, xs) << std::endl;

Output:

pack(flags, xs) = { 34, 1, 11, 10 }
Tip
You can use scans to implement pack.
Note
Even though our arrays can store only 64-bit values of type long, we can nevertheless store values of type bool, as we have done just above with the flags array. The compiler automatically promotes boolean values to long values without causing us any problems, at least with respect to the correctness of our solutions. However, if we want to be more space efficient, we need to use arrays that are capable of packing values of type bool more efficiently, e.g., into single- or eight-bit fields. It should easy to convince yourself that achieving such specialized arrays is not difficult, especially given that the template system makes it easy to write polymorphic containers.

8.9. Summary of operations

8.9.1. Tabulate

template <class Generator>
sparray tabulate(Generator g, long n);

The call tabulate(g, n) returns the length-n array where the i th element is given by g(i).

Let $W(\mathtt{g}(i))$ denote the work performed by an application of the generator function to a specified value $i$. Similarly, let $S(\mathtt{g}(i))$ denote the span. Then the tabulation takes work

$$ \sum_{i=0}^{n} W(\mathtt{g}(i)) $$

and span

$$ \log \mathtt{n} + \max_{i=0}^{n} S(\mathtt{g}(i))$$

8.9.2. Reduce

template <class Assoc_binop>
value_type reduce(Assoc_binop b, value_type id, const sparray& xs);

The call reduce(b, id, xs) is logically equal to id if xs.size() == 0, xs[0] if xs.size() == 1, and

b(reduce(b, id, slice(xs, 0, n/2)),
  reduce(b, id, slice(xs, n/2, n)))

otherwise where n == xs.size().

The work and span cost are $O(n)$ and $O(\log n)$ respectively, where $n$ denotes the size of the input sequence xs. This cost assumes that the work and span of b are constant. If it’s not the case, then refer directly to the implementation of reduce.

8.9.3. Scan

template <class Assoc_binop>
sparray scan_incl(Assoc_binop b, value_type id, const sparray& xs);

For an associative function b and corresponding identity id, the return result of the call scan_incl(b, id, xs) is logically equivalent to

tabulate([&] (long i) { return reduce(b, id, slice(xs, 0, i+1)); }, xs.size())
class scan_excl_result {
public:
  sparray partials;
  value_type total;
};

template <class Assoc_binop>
scan_excl_result scan_excl(Assoc_binop b, value_type id, const sparray& xs);

For an associative function b and corresponding identity id, the call scan_excl(b, id, xs) returns the object res, such that res.partials is logically equivalent to

tabulate([&] (long i) { return reduce(b, id, slice(xs, 0, i)); }, xs.size())

and res.total is logically equivalent to

reduce(b, id, xs)

The work and span cost are $O(n)$ and $O(\log n)$ respectively, where $n$ denotes the size of the input sequence xs. This cost assumes that the work and span of b are constant. If it’s not the case, then refer directly to the implementation of scan_incl and scan_excl.

8.9.4. Map

template <class Func>
sparray map(Func f, sparray xs) {
  return tabulate([&] (long i) { return f(xs[i]); }, xs.size());
}

Let $W(\mathtt{f}(x))$ denote the work performed by an application of the function f to a specified value $x$. Similarly, let $S(\mathtt{f}(x))$ denote the span. Then the map takes work

$$ \sum_{x \in xs} W(\mathtt{f}(x)) $$

and span

$$ \log \mathtt{xs.size()} + \max_{x \in xs} S(\mathtt{f}(x))$$

8.9.5. Fill

sparray fill(value_type v, long n);

Returns a length-n array with all cells initialized to v.

Work and span are linear and logarithmic, respectively.

8.9.6. Copy

sparray copy(const sparray& xs);

Returns a fresh copy of xs.

Work and span are linear and logarithmic, respectively.

8.9.7. Slice

sparray slice(const sparray& xs, long lo, long hi);

The call slice(xs, lo, hi) returns the array {xs[lo], xs[lo+1], ..., xs[hi-1]}.

Work and span are linear and logarithmic, respectively.

8.9.8. Concat

sparray concat(const sparray& xs, const sparray& ys);

Concatenate the two sequences.

Work and span are linear and logarithmic, respectively.

8.9.9. Prefix sums

sparray prefix_sums_incl(const sparray& xs);

The result of the call prefix_sums_incl(xs) is logically equivalent to the result of the call

scan_incl([&] (value_type x, value_type y) { return x+y; }, 0, xs)
scan_excl_result prefix_sums_excl(const sparray& xs);

The result of the call prefix_sums_excl(xs) is logically equivalent to the result of the call

scan_excl([&] (value_type x, value_type y) { return x+y; }, 0, xs)

Work and span are linear and logarithmic, respectively.

8.9.10. Filter

template <class Predicate>
sparray filter(Predicate p, const sparray& xs);

The call filter(p, xs) returns the subsequence of xs which contains each xs[i] for which p(xs[i]) returns true.

Warning
Depending on the implementation, the predicate function p may be called multiple times by the filter function.

Work and span are linear and logarithmic, respectively.

9. Chapter: Parallel Sorting

In this chapter, we are going to study parallel implementations of quicksort and mergesort.

9.1. Quicksort

The quicksort algorithm for sorting an array (sequence) of elements is known to be a very efficient sequential sorting algorithm. A natural question thus is whether quicksort is similarly effective as a parallel algorithm?

Let us first convince ourselves, at least informally, that quicksort is actually a good parallel algorithm. But first, what do we mean by "parallel quicksort." Chances are that you think of quicksort as an algorithm that, given an array, starts by reorganizing the elements in the array around a randomly selected pivot by using an in-place partitioning algorithm, and then sorts the two parts of the array to the left and the right of the array recursively.

While this implementation of the quicksort algorithm is not immediately parallel, it can be parallelized. Note that the recursive calls are naturally independent. So we really ought to focus on the partitioning algorithm. There is a rather simple way to do such a partition in parallel by performing three filter calls on the input array, one for picking the elements less that the pivot, one for picking the elements equal to the pivot, and another one for picking the elements greater than the pivot. This algorithm can be described as follows.

Algorithm: parallel quicksort
  1. Pick from the input sequence a pivot item.

  2. Based on the pivot item, create a three-way partition of the input sequence:

    1. the sequence of items that are less than the pivot item,

    2. those that are equal to the pivot item, and

    3. those that are greater than the pivot item.

  3. Recursively sort the "less-than" and "greater-than" parts

  4. Concatenate the sorted arrays.

Now that we have a parallel algorithm, we can check whether it is a good algorithm or not. Recall that a good parallel algorithm is one that has the following three characteristics

  1. It is asymptotically work efficient

  2. It is observably work efficient

  3. It is highly parallel, i.e., has low span.

Let us first convince ourselves that Quicksort is a highly parallel algorithm. Observe that

  1. the dividing process is highly parallel because no dependencies exist among the intermediate steps involved in creating the three-way partition,

  2. two recursive calls are parallel, and

  3. concatenations are themselves highly parallel.

9.1.1. Asymptotic Work Efficiency and Parallelism

Let us now turn our attention to asymptotic and observed work efficiency. Recall first that quicksort can exhibit a quadratic-work worst-case behavior on certain inputs if we select the pivot deterministically. To avoid this, we can pick a random element as a pivot by using a random-number generator, but then we need a parallel random number generator. Here, we are going to side-step this issue by assuming that the input is randomly permuted in advance. Under this assumption, we can simply pick the pivot to be the first item of the sequence. With this assumption, our algorithm performs asymptotically the same work as sequential quicksort implementations that perform $\Theta(n\log{n})$ in expectation.

For the analysis, let’s assume a version of quicksort that compares the pivot $p$ to each key in the input once (instead of 3 times. The figure below illustrates the structure of an execution of quicksort by using a tree. Each node corresponds to a call to the quicksort function and is labeled with the key at that call. Note that the tree is a binary search tree.

Quicksort call tree
Figure 7. Quicksort call tree.

Let’s observe some properties of quicksort.

  1. In quicksort, a comparison always involves a pivot and another key. Since, the pivot is never sent to a recursive call, a key is selected as a pivot exactly once, and is not involved in further comparisons (after is becomes a pivot). Before a key is selected as a pivot, it may be compared to other pivots, once per pivot, and thus two keys are never compared more than once.

  2. When the algorithm selects a key $y$ as a pivot and if $y$ is between two other keys $x, z$ such that $x < y < z$, it sends the two keys latexmath:[$y, z$ to two separate subtrees. The two keys $x$ and $z$ separated in this way are never compared again.

  3. We can sum up the two observations: a key is compared with all its ancestors in the call tree and all its descendants in the call tree, and with no other keys.

Since a pair of key are never compared more than once, the total number of comparisons performed by quicksort can be expressed as the sum over all pairs of keys.

Let $X_n$ be the random variable denoting the total number of comparisons performed in an execution of quicksort with a randomly permuted input of size latexmath:[$n$.

We want to bound the expectation of $X_n$, $E \lbrack X_n \rbrack$.

For the analysis, let’s consider the final sorted order of the keys $T$, which corresponds to the output. Consider two positions $i, j \in \{1, \dots, n\}$ in the sequence $T$. We define following random variable:

$$ \begin{array}{lll} A_{ij} & = &\left\{ \begin{array}{ll} 1 & \mbox{if}~T_i~\mbox{and}~T_j \mbox{are compared}\\ 0 & \mbox{otherwise} \end{array} \right. \end{array} $$

We can write $X_n$ by summing over all $A_{ij}$'s:

$$ X_n \leq \sum_{i=1}^n \sum_{j=i+1}^n A_{ij} $$

By linearity of expectation, we have

$$ E \lbrack X_n \rbrack \leq \sum_{i=1}^n \sum_{j=i+1}^n E \lbrack A_{ij} \rbrack $$

Furthermore, since each $A_{ij}$ is an indicator random variable, $E \lbrack A_{ij} \rbrack = P(A_{ij}) = 1$. Our task therefore comes down to computing the probability that $T_i$ and $T_j$ are compared, i.e., $P(A_{ij} = 1)$, and working out the sum.

Relationship between the pivot and other keys.}
Figure 8. Relationship between the pivot and other keys.

To compute this probability, note that each call takes the pivot $p$ and splits the sequence into two parts, one with keys larger than $p$ and the other with keys smaller than $p$. For any one call to quicksort there are three possibilities as illustrated in the figure above.

  1. The pivot is (equal to) either $T_i$ or $T_j$, in which case $T_i$ and $T_j$ are compared and $A_{ij} = 1$. Since there are $j-i+1$ keys in the interval $T_i \ldots T_j$ and since each one is equally likely to be the first in the randomly permuted input, the $P(A_{ij} = 1) = \frac{2}{j-i+1}$.

  2. The pivot is a key between $T_i$ and $T_j$, and $T_i$ and $T_j$ will never be compared; thus $A_{ij} = 0$.

  3. The pivot is less than $T_i$ or greater than $T_j$. Then latexmath:[$T_i$ and $T_j$ are sent to the same recursive call. Whether $T_i$ and $T_j$ are compared will be determined in some later call.

$$ \begin{array}{ll} E \lbrack X_n \rbrack & \leq \sum_{i=1}^{n-1} \sum_{j=i+1}^n E \lbrack A_{ij} \rbrack \\ & = \sum_{i=1}^{n-1} \sum_{j=i+1}^n \frac{2}{j-i+1} \\ & = \sum_{i=1}^{n-1} n \sum_{k=2}^{n-i+1} \frac{2}{k} \\ & \leq 2\sum_{i=1}^{n-1} H_n \\ & = 2nH_n \in O(n\log n) \end{array} $$

The last step follows by the fact that $H_n = \ln{n} + O(1)$.

Having completed work, let’s analyze the span of quicksort. Recall that each call to quicksort partitions the input sequence of length $n$ into three subsequences $L$, $E$, and $R$, consisting of the elements less than, equal to, and greater than the pivot.

Let’s first bound the size of the left sequence $L$.

$$ \begin{array}{ll} E \lbrack |L| \rbrack & = \sum_{i=1}^{n-1}{\frac{i-1}{n}} \\ & \le \frac{n}{2}. \end{array} $$

By symmetry, $E \lbrack |R| \rbrack \le \frac{n}{2}$. This reasoning applies at any level of the quicksort call tree. In other words, the size of the input decreases by $1/2$ in expectation at each level in the call tree.

Since pivot choice at each call is independent of the other calls. The expected size of the input at level $i$ is $E \lbrack Y_i \rbrack = \frac{n}{2^i}$.

Let’s calculate the expected size of the input at depth $i = 5\lg{n}$. By basic arithmetic, we obtain

$$ E \lbrack Y_{5\lg{n}} \rbrack = n\, \frac{1}{2^{5\lg{n}}} = n\, n^{-5\lg{2}} = n^{-4}. $$

Since $Y_i$'s are always non-negative, we can use Markov’s inequality, to turn expectations into probabilities as follows.

$$ P(Y_{5\lg{n}} \ge 1) \le \frac{E \lbrack Y_{5\lg{n}}\rbrack}{1} = n^{-4}. $$

In other words, the probability that a given path in the quicksort call tree has a depth that exceeds $5 \lg{n} $ is tiny.

From this bound, we can calculute a bound on the depth of the whole tree. Note first that the tree has exactly $n+1$ leaves because it has $n$ internal nodes. Thus we have at most $n+1$ to consider. The probability that a given path expeeds a depth of $5\lg{n}$ is $n^{-4}$. Thus, by union bound the probability that any one of the paths exceed the depeth of $5\lg{n}$ is $(n+1) \cdot n^{-4} \le n^{-2}$. Thus the probability that the depth of the tree is greater than $5\lg{n}$ is $n^{-2}$.

By using Total Expectation Theorem or the Law of total expectation, we can now calculate expected span by dividing the sample space into mutually exclusive and exhaustive space as follows.

$$ \begin{array}{ll} E \lbrack S \rbrack & = E \lbrack S ~|~\mbox{Depth is no greater than}~5\lg{n} \rbrack \cdot P(\mbox{Depth is no greater than}~5\lg{n}) + E \lbrack S ~|~\mbox{Depth is greater than}~5\lg{n} \rbrack \cdot P(\mbox{Depth is greater than}~5\lg{n}) \\ & \le \lg^2{n} \cdot (1 - 1/n^{2}) + n^2 \cdot 1/n^{2} \\ & = O(lg^2{n}) \end{array} $$

In thes bound, we used the fact that each call to quicksort has a span of $O(\lg{n})$ because this is the span of filter.

Here is an alternative analysis.

Let $M_n = \max\{|L|, |R|\}$, which is the size of larger subsequence. The span of quicksort is determined by the sizes of these larger subsequences. For ease of analysis, we will assume that $|E| = 0$, as more equal elements will only decrease the span. As the partition step uses filter we have the following recurrence for span:

$$ S(n) = S(M_n) + O(\log n) $$

To develop some intuition for the span analysis, let’s consider the probability that we split the input sequence more or less evenly. If we select a pivot that is greater than $T_{n/4}$ and less than $T_{3n/4}$] then $M_n$ is at most $3n/4$. Since all keys are equally likely to be selected as a pivot this probability is $\frac{3n/4 - n/4}{n} = 1/2$. The figure below illustrates this.

Quicksort span intuition.
Figure 9. Quicksort span intuition.

This observations implies that at each level of the call tree (every time a new pivot is selected), the size of the input to both calls decrease by a constant fraction (of $4/3$). At every two levels, the probability that the input size decreases by latexmath:[$4/3$ is the probability that it decreases at either step, which is at least $3/4$, etc. Thus at a small costant number of steps, the probability that we observe a $4/3$ factor decrease in the size of the input approches $1$ quickly. This suggest that at some after $c\log{n}$ levels quicksort should complete. We now make this intuition more precise.

For the analysis, we use the conditioning technique for computing expectations as suggested by the total expectation theorem. Let $X$ be a random variable and let $A_i$ be disjoint events that form a a partition of the sample space such that $P(A_i) > 0$. The Total Expectation Theorem or the Law of total expectation states that

$$ E[X] = \sum_{i=1}^{n}{P(A_i) \cdot E \lbrack X | A_i \rbrack}. $$

Note first that $P(X_n \leq 3n/4) = 1/2$, since half of the randomly chosen pivots results in the larger partition to be at most $3n/4$ elements: any pivot in the range $T_{n/4}$ to $T_{3n/4}$ will do, where $T$ is the sorted input sequence.

By conditioning $S_n$ on the random variable $M_n$, we write,

$$ E \lbrack S_n \rbrack = \sum_{m=n/2}^{n}{P(M_n=m) \cdot E \lbrack S_n | (M_n=m) \rbrack}. $$

We can re-write this

$$ E \lbrack S_n \rbrack = \sum_{m=n/2}^{n}{P(M_n=m) \cdot E \lbrack S_m \rbrack} $$

The rest is algebra

$$ \begin{array}{ll} E \lbrack S_n \rbrack & = \sum_{m=n/2}^{n}{P(M_n=m) \cdot E \lbrack S_m \rbrack} \\ & \leq P(M_n \leq \frac{3n}{4}) \cdot E \lbrack S_{\frac{3n}{4}} \rbrack + P(M_n > \frac{3n}{4}) \cdot E \lbrack S_n \rbrack + c\cdot \log n \\ & \leq \frac{1}{2} E \lbrack S_{\frac{3n}{4}} \rbrack + \frac{1}{2} E \lbrack S_n \rbrack \\ & \implies E \lbrack S_n \rbrack \leq E \lbrack S_{\frac{3n}{4}} \rbrack + 2c \log n. \\ \end{array} $$

This is a recursion in $E \lbrack S(\cdot) \rbrack $ and solves easily to $E \lbrack S(n) \rbrack = O(\log^2 n)$.

9.1.2. Observable Work Efficency and Scalability

For an implementation to be observably work efficient, we know that we must control granularity by switching to a fast sequential sorting algorithm when the input is small. This is easy to achieve using our granularity control technique by using seqsort(), a fast sequential algorithm provided in the code base; seqsort() is really a call to STL’s sort function. Of course, we have to assess observable work efficiency experimentally after specifying the implementation.

The code for quicksort is shown below. Note that we use our array class sparray to store the input and output. To partition the input, we use our parallel filter function from the previous lecture to parallelize the partitioning phase. Similarly, we use our parallel concatenation function to constructed the sorted output.

controller_type quicksort_contr("quicksort");

sparray quicksort(const sparray& xs) {
  long n = xs.size();
  sparray result = { };
  cstmt(quicksort_contr, [&] { return n * std::log2(n); }, [&] {
    if (n == 0) {
      result = { };
    } else if (n == 1) {
      result = { xs[0] };
    } else {
      value_type p = xs[0];
      sparray less = filter([&] (value_type x) { return x < p; }, xs);
      sparray equal = filter([&] (value_type x) { return x == p; }, xs);
      sparray greater = filter([&] (value_type x) { return x > p; }, xs);
      sparray left = { };
      sparray right = { };
      fork2([&] {
        left = quicksort(less);
      }, [&] {
        right = quicksort(greater);
      });
      result = concat(left, equal, right);
    }
  }, [&] {
    result = seqsort(xs);
  });
  return result;
}

By using randomized-analysis techniques, it is possible to analyze the work and span of this algorithm. The techniques needed to do so are beyond the scope of this book. The interested reader can find more details in another book.

Fact

The randomized quicksort algorithm above has expected work of $O(n \log n)$ and expected span of $O(\log^2 n)$, where $n$ is the number of items in the input sequence.

One consequence of the work and span bounds that we have stated above is that our quicksort algorithm is highly parallel: its average parallelism is $\frac{O(n \log n)}{O(\log^2 n)} = \frac{n}{\log n}$. When the input is large, there should be ample parallelism to keep many processors well fed with work. For instance, when $n = 100$ million items, the average parallelism is $\frac{10^8}{\log 10^8} \approx \frac{10^8}{40} \approx 3.7$ million. Since 3.7 million is much larger than the number of processors in our machine, that is, forty, we have a ample parallelism.

Unfortunately, the code that we wrote leaves much to be desired in terms of observable work efficiency. Consider the following benchmarking runs that we performed on our 40-processor machine.

$ prun speedup -baseline "bench.baseline" -parallel "bench.opt -proc 1,10,20,30,40" -bench quicksort -n 100000000

The first two runs show that, on a single processor, our parallel algorithm is roughly 6x slower than the sequential algorithm that we are using as baseline! In other words, our quicksort appears to have "6-observed work efficiency". That means we need at least six processors working on the problem to see even a small improvement compared to a good sequential baseline.

[1/6]
bench.baseline -bench quicksort -n 100000000
exectime 12.518
[2/6]
bench.opt -bench quicksort -n 100000000 -proc 1
exectime 78.960

The rest of the results confirm that it takes about ten processors to see a little improvement and forty processors to see approximately a 2.5x speedup. This plot shows the speedup plot for this program. Clearly, it does not look good.

[3/6]
bench.opt -bench quicksort -n 100000000 -proc 10
exectime 9.807
[4/6]
bench.opt -bench quicksort -n 100000000 -proc 20
exectime 6.546
[5/6]
bench.opt -bench quicksort -n 100000000 -proc 30
exectime 5.531
[6/6]
bench.opt -bench quicksort -n 100000000 -proc 40
exectime 4.761
Speedup plot for quicksort with latexmath:[$100000000$] elements.
Figure 10. Speedup plot for quicksort with $100000000$ elements.

Our analysis suggests that we have a good parallel algorithm for quicksort, yet our observations suggest that, at least on our test machine, our implementation is rather slow relative to our baseline program. In particular, we noticed that our parallel quicksort started out being 6x slower than the baseline algorithm. What could be to blame?

In fact, there are many implementation details that could be to blame. The problem we face is that identifying those causes experimentally could take a lot of time and effort. Fortunately, our quicksort code contains a few clues that will guide us in a good direction.

It should be clear that our quicksort is copying a lot of data and, moreover, that much of the copying could be avoided. The copying operations that could be avoided, in particular, are the array copies that are performed by each of the the three calls to filter and the one call to concat. Each of these operations has to touch each item in the input array.

Let us now consider a (mostly) in-place version of quicksort. This code is mostly in place because the algorithm copies out the input array in the beginning, but otherwise sorts in place on the result array. The code for this algorithm appears just below.

sparray in_place_quicksort(const sparray& xs) {
  sparray result = copy(xs);
  long n = xs.size();
  if (n == 0) {
    return result;
  }
  in_place_quicksort_rec(&result[0], n);
  return result;
}
controller_type in_place_quicksort_contr("in_place_quicksort");

void in_place_quicksort_rec(value_type* A, long n) {
  if (n < 2) {
    return;
  }
  cstmt(in_place_quicksort_contr, [&] { return nlogn(n); }, [&] {
    value_type p = A[0];
    value_type* L = A;   // below L are less than pivot
    value_type* M = A;   // between L and M are equal to pivot
    value_type* R = A+n-1; // above R are greater than pivot
    while (true) {
      while (! (p < *M)) {
        if (*M < p) std::swap(*M,*(L++));
        if (M >= R) break;
        M++;
      }
      while (p < *R) R--;
      if (M >= R) break;
      std::swap(*M,*R--);
      if (*M < p) std::swap(*M,*(L++));
      M++;
    }
    fork2([&] {
      in_place_quicksort_rec(A, L-A);
    }, [&] {
      in_place_quicksort_rec(M, A+n-M); // Exclude all elts that equal pivot
    });
  }, [&] {
    std::sort(A, A+n);
  });
}

We have good reason to believe that this code is, at least, going to be more work efficient than our original solution. First, it avoids the allocation and copying of intermediate arrays. And, second, it performs the partitioning phase in a single pass. There is a catch, however: in order to work mostly in place, our second quicksort code sacrified on parallelism. In specific, observe that the partitioning phase is now sequential. The span of this second quicksort is therefore linear in the size of the input and its average parallelism is therefore logarithmic in the size of the input.

Exercise

Verify that the span of our second quicksort has linear span and that the average parallelism is logarithmic.

So, we expect that the second quicksort is more work efficient but should scale poorly. To test the first hypothesis, let us run the second quicksort on a single processor.

$ bench.opt -bench in_place_quicksort -n 100000000 -proc 1

Indeed, the running time of this code is essentially same as what we observed for our baseline program.

exectime 12.500
total_idle_time 0.000
utilization 1.0000
result 1048575

Now, let us see how well the second quicksort scales by performing another speedup experiment.

$ prun speedup -baseline "bench.baseline" -parallel "bench.opt -proc 1,20,30,40" -bench quicksort,in_place_quicksort -n 100000000
[1/10]
bench.baseline -bench quicksort -n 100000000
exectime 12.031
[2/10]
bench.opt -bench quicksort -n 100000000 -proc 1
exectime 68.998
[3/10]
bench.opt -bench quicksort -n 100000000 -proc 20
exectime 5.968
[4/10]
bench.opt -bench quicksort -n 100000000 -proc 30
exectime 5.115
[5/10]
bench.opt -bench quicksort -n 100000000 -proc 40
exectime 4.871
[6/10]
bench.baseline -bench in_place_quicksort -n 100000000
exectime 12.028
[7/10]
bench.opt -bench in_place_quicksort -n 100000000 -proc 1
exectime 12.578
[8/10]
bench.opt -bench in_place_quicksort -n 100000000 -proc 20
exectime 1.731
[9/10]
bench.opt -bench in_place_quicksort -n 100000000 -proc 30
exectime 1.697
[10/10]
bench.opt -bench in_place_quicksort -n 100000000 -proc 40
exectime 1.661
Benchmark successful.
Results written to results.txt.
$ pplot speedup -series bench

The plot below shows one speedup curve for each of our two quicksort implementations. The in-place quicksort is always faster. However, the in-place quicksort starts slowing down a lot at 20 cores and stops after 30 cores. So, we have one solution that is observably not work efficient and one that is, and another that is the opposite. The question now is whether we can find a happy middle ground.

Speedup plot for quicksort with latexmath:[$100000000$] elements.
Figure 11. Speedup plot showing our quicksort and the in-place quicksort side by side. As before, we used $100000000$ elements.
Question

What can we do to write a better quicksort?

Tip
Eliminate unecessary copying and array allocations.
Tip
Eliminate redundant work by building the partition in one pass instead of three.
Tip
Find a solution that has the same span as our first quicksort code.

We encourage students to look for improvements to quicksort independently. For now, we are going to consider parallel mergesort. This time, we are going to focus more on achieving better speedups.

9.2. Mergesort

As a divide-and-conquer algorithm, the mergesort algorithm, is a good candidate for parallelization, because the two recursive calls for sorting the two halves of the input can be independent. The final merge operation, however, is typically performed sequentially. It turns out to be not too difficult to parallelize the merge operation to obtain good work and span bounds for parallel mergesort. The resulting algorithm turns out to be a good parallel algorithm, delivering asymptotic, and observably work efficiency, as well as low span.

Mergesort algorithm
  1. Divide the (unsorted) items in the input array into two equally sized subrange.

  2. Recursively and in parallel sort each subrange.

  3. Merge the sorted subranges.

This process requires a "merge" routine which merges the contents of two specified subranges of a given array. The merge routine assumes that the two given subarrays are in ascending order. The result is the combined contents of the items of the subranges, in ascending order.

The precise signature of the merge routine appears below and its description follows. In mergesort, every pair of ranges that are merged are adjacent in memory. This observation enables us to write the following function. The function merges two ranges of source array xs: [lo, mid) and [mid, hi). A temporary array tmp is used as scratch space by the merge operation. The function writes the result from the temporary array back into the original range of the source array: [lo, hi).

void merge(sparray& xs, sparray& tmp, long lo, long mid, long hi);
Example 52. Use of merge function
sparray xs = {
  // first range: [0, 4)
  5, 10, 13, 14,
  // second range: [4, 9)
  1, 8,  10, 100, 101 };

merge(xs, sparray(xs.size()), (long)0, 4, 9);

std::cout << "xs = " << xs << std::endl;

Output:

xs = { 1, 5, 8, 10, 10, 13, 14, 100, 101 }

To see why sequential merging does not work, let us implement the merge function by using one provided by STL: std::merge(). This merge implementation performs linear work and span in the number of items being merged (i.e., hi - lo). In our code, we use this STL implementation underneath the merge() interface that we described just above.

Now, we can assess our parallel mergesort with a sequential merge, as implemented by the code below. The code uses the traditional divide-and-conquer approach that we have seen several times already.

Question

Is the implementation asymptotically work efficient?

The code is asymptotically work efficient, because nothing significant has changed between this parallel code and the serial code: just erase the parallel annotations and we have a textbook sequential mergesort!

sparray mergesort(const sparray& xs) {
  long n = xs.size();
  sparray result = copy(xs);
  mergesort_rec(result, sparray(n), (long)0, n);
  return result;
}

controller_type mergesort_contr("mergesort");

void mergesort_rec(sparray& xs, sparray& tmp, long lo, long hi) {
  long n = hi - lo;
  cstmt(mergesort_contr, [&] { return n * std::log2(n); }, [&] {
    if (n == 0) {
      // nothing to do
    } else if (n == 1) {
      tmp[lo] = xs[lo];
    } else {
      long mid = (lo + hi) / 2;
      fork2([&] {
        mergesort_rec(xs, tmp, lo, mid);
      }, [&] {
        mergesort_rec(xs, tmp, mid, hi);
      });
      merge(xs, tmp, lo, mid, hi);
    }
  }, [&] {
    if (hi-lo < 2)
      return;
    std::sort(&xs[lo], &xs[hi-1]+1);
  });
}
Question

How well does our "parallel" mergesort scale to multiple processors, i.e., does it have a low span?

Unfortunately, this implementation has a large span: it is linear, owing to the sequential merge operations after each pair of parallel calls. More precisely, we can write the work and span of this implementation as follows:

Analyzing work and span of mergesort
$$ W(n) = \left\{ \begin{array}{ll} 1 & \mbox{if}~n \le 1 \\ W(n/2) + W(n/2) + n \end{array} \right. $$ $$ S(n) = \left\{ \begin{array}{ll} 1 & \mbox{if}~n \le 1 \\ \max (W(n/2),W(n/2)) + n \end{array} \right. $$

It is not difficult to show that these recursive equation solve to $W(n) = \Theta (n\log{n})$ and $S(n) = \Theta (n)$.

With these work and span costs, the average parallelism of our solution is $\frac{cn \log n}{2cn} = \frac{\log n}{2}$. Consider the implication: if $n = 2^{30}$, then the average parallelism is $\frac{\log 2^{30}}{2} = 15$. That is terrible, because it means that the greatest speedup we can ever hope to achieve is 15x!

The analysis above suggests that, with sequential merging, our parallel mergesort does not expose ample parallelism. Let us put that prediction to the test. The following experiment considers this algorithm on our 40-processor test machine. We are going to sort a random sequence of 100 million items. The baseline sorting algorithm is the same sequential sorting algorithm that we used for our quicksort experiments: std::sort().

$ prun speedup -baseline "bench.baseline" -parallel "bench.opt -proc 1,10,20,30,40" -bench mergesort_seqmerge -n 100000000

The first two runs suggest that our mergesort has better observable work efficiency than our quicksort. The single-processor run of parallel mergesort is roughly 50% slower than that of the sequential baseline algorithm. Compare that to the 6x-slower running time for single-processor parallel quicksort! We have a good start.

[1/6]
bench.baseline -bench mergesort_seqmerge -n 100000000
exectime 12.483
[2/6]
bench.opt -bench mergesort_seqmerge -n 100000000 -proc 1
exectime 19.407

The parallel runs are encouraging: we get 5x speedup with 40 processors.

[3/6]
bench.opt -bench mergesort_seqmerge -n 100000000 -proc 10
exectime 3.627
[4/6]
bench.opt -bench mergesort_seqmerge -n 100000000 -proc 20
exectime 2.840
[5/6]
bench.opt -bench mergesort_seqmerge -n 100000000 -proc 30
exectime 2.587
[6/6]
bench.opt -bench mergesort_seqmerge -n 100000000 -proc 40
exectime 2.436

But we can do better by using a parallel merge instead of a sequential one: the speedup plot in [mergesort-speedups] shows three speedup curves, one for each of three mergesort algorithms. The mergesort() algorithm is the same mergesort routine that we have seen here, except that we have replaced the sequential merge step by our own parallel merge algorithm. The cilksort() algorithm is the carefully optimized algorithm taken from the Cilk benchmark suite. What this plot shows is, first, that the parallel merge significantly improves performance, by at least a factor of two. The second thing we can see is that the optimized Cilk algorithm is just a little faster than the one we presented here. That’s pretty good, considering the simplicity of the code that we had to write.

Speedup plot for three different implementations of mergesort using 100 million 32-bit items.
Figure 12. Speedup plot for three different implementations of mergesort using 100 million items.

It turns out that we can do better by simply changing some of the variables in our experiment. The plot shown in [better-speedups] shows the speedup plot that we get when we change two variables: the input size and the sizes of the items. In particular, we are selecting a larger number of items, namely 250 million instead of 100 million, in order to increase the amount of parallelism. And, we are selecting a smaller type for the items, namely 32 bits instead of 64 bits per item. The speedups in this new plot get closer to linear, topping out at approximately 20x.

Practically speaking, the mergesort algorithm is memory bound because the amount of memory used by mergesort and the amount of work performed by mergesort are both approximately roughly linear. It is an unfortunate reality of current multicore machines that the main limiting factor for memory-bound algorithms is amount of parallelism that can be achieved by the memory bus. The memory bus in our test machine simply lacks the parallelism needed to match the parallelism of the cores. The effect is clear after just a little experimentation with mergesort. You can see this effect yourself, if you are interested to change in the source code the type aliased by value_type. For a sufficiently large input array, you should observe a significant performance improvement by changing just the representation of value_type from 64 to 32 bits, owing to the fact that with 32-bit items is a greater amount of computation relative to the number of memory transfers.

Speedup plot for three different implementations of mergesort using 250 million 32-bit items.
Figure 13. Speedup plot for three different implementations of mergesort using 250 million items.
Question: Stable Mergesort

An important property of the sequential merge-sort algorithm is that it is stable: it can be written in such a way that it preserves the relative order of equal elements in the input. Is the parallel merge-sort algorithm that you designed stable? If not, then can you find a way to make it stable?

10. Chapter: Graphs

In just the past few years, a great deal of interest has grown for frameworks that can process very large graphs. Interest comes from a diverse collection of fields. To name a few: physicists use graph frameworks to simulate emergent properties from large networks of particles; companies such as Google mine the web for the purpose of web search; social scientists test theories regarding the origins of social trends.

In response, many graph-processing frameworks have been implemented both in academia and in the industry. Such frameworks offer to client programs a particular application programming interface. The purpose of the interface is to give the client programmer a high-level view of the basic operations of graph processing. Internally, at a lower level of abstraction, the framework provides key algorithms to perform basic functions, such as one or more functions that "drive" the traversal of a given graph.

The exact interface and the underlying algorithms vary from one graph-processing framework to another. One commonality among the frameworks is that it is crucial to harness parallelism, because interesting graphs are often huge, making it practically infeasible to perform sequentially interesting computations.

10.1. Graph representation

We will use an adjacency lists representation based on compressed arrays to represent directed graphs. In this representation, a graph is stored as a compact array containing the neighbors of each vertex. Each vertex in the graph $G = (V, E)$ is assigned an integer identifier $v \in \{ 0, \ldots, n-1 \}$, where $n = |V|$ is the number of vertices in the graph. The representation then consists of two array.

  1. The edge array contains the adjacency lists of all vertices ordered by the vertex ids.

  2. The vertex array stores an index for each vertex that indicates the starting position of the adjacency list for that vertex in the edge array. This array implements the

A graph (top) and its compressed-array representation (bottom)

consisting of the vertex and the edge arrays. The sentinel value "-1" is used to indicate a non-vertex id.

An example di-graph.