Matrix Walker

Inspired by ps5, we implement cache oblivious NxN matrix storage, as well as a 3-D tensor generalization. Our solution involves storing elements in order of so-called “pseudo-Hilbert curves” (finite approximations of Hilbert’s space-filling curve), with padding for matrices of size not a power of two. We evaluate the cache performance of these by experimentation, comparing the Hilbert ordering to naive row-major ordering as well as Z-ordering.

The deliverables of our project include:

• code implementing a cache-oblivious $N\times N$ matrix using Hilbert and Z-indexing and their generalizations to higher dimensional tensors
• analysis on asymptotic behavior (runtime and cache misses) of a generalized $k$ dimensional tensor version of the data structure
• visualizing the cache and block accesses for matrix walkers in a web app (2-D only)
• benchmarking our implementation by comparing cache misses for various tests (see Running below)

Usage: Benchmarking

Compiling

We compile C++ files using make. To compile file.cpp, run make file from the cpp folder. The most relevant files are benchmark_matrix.cpp and benchmark_tensor.cpp respectively. To run a debug version append the argument DEBUG=1, which will include any bits of DEBUG code written. If make claims that binaries are up to date, try running make clean to remove them.

Running

Each benchmarking script takes command line arguments, specifying the type of matrix_walker or tensor_walker to be benchmarked, and the test to apply. The options are as follows:

• walker type: naive, z, hilbert
• matrix tests: move_up, move_down, move_left, move_right, rand_move, row_order, col_order
• tensor tests: move_in, move_out, move_up, move_down, move_left, move_right, rand_move, ijk_order, col_order For example, running the ijk_order test on a hilbert_tensor_walker from the command line looks like: ./benchmark_tensor hilbert ijk_order Simply running the scripts doesn't provide info on cache misses; to do this, we need the cachegrind tool from valgrind, which simulates your machine's cache while running a given program.

Profiling & Benchmarking

• Most pertinent: To view cache behavior, run valgrind --tool=cachegrind --cachegrind-out-file=OUT_FILE BINARY_NAME
  • For example:valgrind --tool=cachegrind --cachegrind-out-file=results/z_ijk.out ./benchmark_tensor z ijk_order
• To profile the code, run perf record BINARY_NAME followed by perf report. This gives some interesting statistics regarding what percent of execution time is spent in each chunk of the code.
• For more stats, run perf stat BINARY_NAME. More command line arguments / flags can be used to specify what stats to display.
• To check for memory leaks, simply run valgrind BINARY_NAME.

Viewing Benchmarking results

Cachegrind writes its detailed output to a file, by default named cachegrind.out.{pid}. This output file can be renamed using the ---cachegrind-out-file=OUT_FILE option, as seen above. To view the results in a nicely tabulated format (function-by-function summary), run cg_annotate OUT_FILE. The relevant statistics are D1mr (misses on data reads) and Dr (total data reads). If you can't find a function in the results, chances are it didn't have enough cache misses and was thresholded out of the display; use the option --threshold=0 in your call to cg_annotate to see every single function called.

Our Results

The below plots summarize our results on each test for each matrix/tensor ordering method.

The raw results are also available here in CSV format.

Visualization

We visualized the cache for three different representations of matrices: row major indexing, z indexing, and hilbert indexing. On both sides of the page, the entries of a 32 by 32 matrix are displayed in row major order. The black square is the current entry being accessed in the array. The grayed squares are the entries that reside in a cache with width 8 and height 8 using an LRU policy. Row-major indexing stores the entries of the array behind the scenes in a row major order, z indexing does so using a z-order curve, and Hilbert indexing uses a pseudo Hilbert curve. Finally, below each array, some basic statistics regarding cache hits and misses are shown.

Usage

• Use the dropdowns on the two sides of the page to select the type of indexing
• Use the teleport UI in the center to teleport the pointer of all matrices to any entry of the array
• Use WASD to move the pointer through the array

Deployment

The application is deployed at https://6851-2021.github.io/matrix-walker.

Codebase

The key components for the visualization, all in the vis directory were

• App.js the main page for the app
• MatrixWalker.js the implementation of each of the three matrix walkers and the cache simulator
• Help.js the help modal for the app
• Selector.js the component to help select matrices

Misc

Originally, we had intended to do a visualization in python, which turned out to be harder than expected. We did, however, implement the three matrix walkers in python, which are located in the py directory.