Q1. R-tree [30 points]

We will implement a new command l for, inc(l)usion query, that prints out all the data rectangles that are contained in the closed bounding box specified by two points: low = (l₁, l₂, ..., l_n) and high = (h₁, h₂, ..., h_n). The command should ask for the low and high values by dimensions, i.e., in the order of l₁, h₁, l₂, h₂, l₃, h₃, .... You implementation should not modify the tree's data structure itself.

Please build the R-Tree Package (tar xvf; make demo). This creates the bin/DRmain program and runs it on some small datasets. Running make test1 should return an Algorithm not yet implemented message after loading the appropriate dataset.

Turn in:

1. [5 points] The list of qualifying rectangles for dataset 1 (2-d). This should run using make test1.
2. [5 points] The list of qualifying rectangles for dataset 2 (3-d). This should run using make test2.
3. [20 points] A tarball (YourAndrewID_rtree.tar) of your code emailed to the TA (dchau at cs) and a hard copy of files you changed. On the hard copy, highlight or circle the code that you modified\added.

Hint #1: you may want to modify the following files, and possible others:

• DRTree\binsrc\DRmain.c
• DRTree\libsrc\src\DR.c
• DRTree\libsrc\src\DRtree.c

Hint #2: you may want to look at the DRrectCover function in DRTree\libsrc\src\DRrect.c, which checks whether one rectangle contains another rectangle.

Q2. Measuring Mystery Data Fractal Dimension [20 points]

We will use the fdnq software package to calculate the correlation integral of a collection of 6-dimensional points. Build the fdnq package (unzip; make). This computes the fractal dimensions for several example datasets, and shows the plot for one of them.

1. [5 points] Turn in a plot of the correlation integral (fdnq -q2). Write down the command you used to generate the plot. We suggest that you use the default values for r (0.001 for minimum radius) and R (10000, for maximum radius).
2. [10 points] For every line segment in the plot that is roughly linear, write down its range and its slope.
3. [5 points] What do you think the points in the mystery dataset look like?
   For example, do they form a line, some clusters, or something else?

Hint for Cygwin users: you will need to install the "tcsh" and "gnuplot" package, and maybe others.

Q3. 2D Cantor Dust [30 points]

The 2D Cantor dust can be generated recursively, as shown in the figure above, by deleting the middle third segment on both dimensions.

1. [15 points] Write a program that generates the points (shown as black dots above) for the Cantor dust, up to 6 iterations (4⁽⁶⁺¹⁾=16,384 points total). Turn in:
   1. your code (any major language is acceptable: C/C++, Perl, Python, Java, etc.)
   2. a scatterplot of the points
2. [10 points] Using the fdnq package, or your own program (written in a major language, such as C/C++, Java, Perl, Python), turn in:
   1. the boxcounting plot (fdnq -q2)
   2. slope (fractal dimension).
3. [5 points] Using the same code, turn in:
   1. the "Hausdorff" fractal dimension (fdnq -q0)
   2. the corresponding plot.

Q4. Multifractal [20 points]

We will implement the b-model devised by Wang+ [ICDE 2002] to generate several traces (or sequences) of values, each value corresponding to, say, the number of disk accesses at a time interval. These traces follow the multifractal distribution with b:(1-b) splitting probabilities, where 0.5<=b<1. The data you generate could come in two formats:

• histogram format <timetick #accesses>. e.g.,
  1 3
  2 10
  5 30
  ...
• timestamp format <timestamp>: e.g.,
  1
  1
  1
  2
  2
  2
  ...

We prefer that you use the histogram version because it is more compact.

For example, a bias b=0.8 means that within a given time interval, 80% of the accesses happen in one half of the interval and the remaining 20% in the other half, and this splitting of accesses happens recursively for each of those halves.

Your implementation should create deterministic splits, meaning for all splits, b should be applied to the left half.

Generate three traces of values. Each trace has T=1024 timeticks (some timeticks may not have disk accesses) and the total number of disk accesses is N=10,000. The biases of the traces are 0.7, 0.9, and 0.5 respectively. Turn in:

1. [10 points] your code that generates the traces
2. [5 points] a plot for the histogram format of each trace (similar to Figure 2b of the paper); three plots total.
3. [5 points] the correlation fractal dimension plot for each trace. We recommend using the command fdnq -h -q2 <histogram-file-name> to find it. Optionally, you may use fdnq -q2 <timestamp-file-name> to verify the result.

Hints:

• You may refer to the method described on slide 42 of the lecture slides. Section 4.1 of the paper describes the naive method for generating the traces; section 4.4 describes an efficient alternative