15-451 Algorithms                  12/07/11
                        recitation notes

Note: Review session: next Wed 1:00pm Wean 7500.
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General Review
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What the course has been about:
 1. techniques for developing algorithms
 2. techniques for analysis

Break down (1) into
 (a) fast subroutines: useful tools that go *inside* an algorithm.
     E.g., data structures like B-trees, hashing, union-find.
     Sorting, selection, DFS.


 (b) problems that are important because you can reduce a lot
     of *other* problems to them: network flow, linear programming.

Notion of reduction is important in algorithms and in complexity
theory: 
 - solve a problem by reducing it to something you know how to do.
 - show a problem is hard by reducing a known hard problem to it.



Let's organize the algorithms and problems we've discussed in class
along a "running time" line.

sublinear  ---  linear --- near-linear  ---  low-degree poly  ---  general poly(n) ---  hard but probably not NP-complete  ---  NP-complete 



sublinear per operation:
 - data structures: hashing, balanced search trees, heaps, union find.

linear:
 - depth-first search, breadth-first search, topological sorting
 - selection/median-finding

near-linear:
 - greedy algs with good data structures
   -  Prim, Kruskal, Dijkstra
 - divide-and-conquer algorithms
   - sorting, FFTs

low-degree & general poly(n):
 - dynamic programming: Bellman-Ford, Floyd-Washall, etc.
 - network flow, matchings, min-cost flow
 - linear programming
 - Graph matrix algorithms

Hard-but-not-probably-not-NP-complete:
 - factoring

NP-complete:
 - 3-SAT, Vertex cover, Clique, TSP, etc.

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Algorithm tools and where they are typically used:
* Dynamic programming: typically for improving exponential time down to
	polynomial. 
* Reducing to LP, or reducing to network flow: likewise, typically for
	improving exponential time down to polynomial. 
* Divide-and-conquer: typically for getting from O(n^2) to O(n log n).
        Sometimes just for reducing exponent (like Karatsuba or Strassen).
* Data structures: ditto
* Randomization: everywhere.
* Approximation algorithms: typically for NP-complete problems, but
        sometimes also makes sense for easier problems too.

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Analysis tools:
* Amortized analysis, Potential functions, piggy banks:
    Typically for showing O(1) or O(log n) amortized cost per operation
* Reductions:  proving NP-completeness by reducing a known NP-complete
    problem to it, or giving a poly-time algorithm by reducing to a
    problem like LP or network flow. 
* Recurrences.  Esp with divide-and-conquer
* Linearity of expectation

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