puzzlef/pagerank-barrierfree-openmp-static-vs-dynamic: Performance of static vs dynamic barrier-free iterations in OpenMP-based ordered PageRank algorithm for link analysis

Performance of static vs dynamic barrier-free iterations in OpenMP-based ordered PageRank algorithm for link analysis.

Unordered PageRank is the standard approach of PageRank computation (as described in the original paper by Larry Page et al. (1)), where two different rank vectors are maintained; one representing the current ranks of vertices, and the other representing the previous ranks. On the other hand, ordered PageRank uses a single rank vector, representing the current ranks of vertices (2). This is similar to barrierless non-blocking implementations of the PageRank algorithm by Hemalatha Eedi et al. (3). As ranks are updated in the same vector (with each iteration), the order in which vertices are processed affects the final result (hence the adjective ordered). However, as PageRank is an iteratively converging algorithm, results obtained with either approach are mostly the same. Barrier-free PageRank is an ordered PageRank where each thread processes a subset of vertices in the graph independently, without waiting (with a barrier) for other threads to complete an iteration. This minimizes unnecessary waits and allows each thread to be on a different iteration number (which may or may not be beneficial for convergence) (3).

Dynamic graphs, which change with time, have many applications. Computing ranks of vertices from scratch on every update (static PageRank) may not be good enough for an interactive system. In such cases, we only want to process ranks of vertices which are likely to have changed. To handle any new vertices added/removed, we first adjust the previous ranks (before the graph update/batch) with a scaled 1/N-fill approach (4). Then, with naive dynamic approach we simply run the PageRank algorithm with the initial ranks set to the adjusted ranks. Alternatively, with the (fully) dynamic approach we first obtain a subset of vertices in the graph which are likely to be affected by the update (using BFS/DFS from changed vertices), and then perform PageRank computation on only this subset of vertices.

In this experiment, we compare the performance of static, naive dynamic, and (fully) barrier-free iterations in dynamic OpenMP-based ordered PageRank (along with similar unordered/ordered OpenMP-based approaches). We take temporal graphs as input, and add edges to our in-memory graph in batches of size 10^2 to 10^6. However we do not perform this on every point on the temporal graph, but only on 5 time samples of the graph (5 samples are good enough to obtain an average). At each time sample we load B edges (where B is the batch size), and perform static, naive dynamic, and dynamic PageRank. At each time sample, each approach is performed 5 times to obtain an average time for that sample. A schedule of dynamic, 2048 is used for OpenMP-based PageRank as obtained in (5). We use the follwing PageRank parameters: damping factor α = 0.85, tolerance τ = 10^-6, and limit the maximum number of iterations to L = 500. The error between the current and the previous iteration is obtained with L1-norm, and is used to detect convergence. Dead ends in the graph are handled by adding self-loops to all vertices in the graph (loopall approach (6)). Error in ranks obtained for each approach is measured relative to the sequential static approach using L1-norm.

From the results, we observe that OpenMP-based unordered PageRank is faster than ordered approach, and Barrier-free PageRank with partial error measurement is faster than the one with full error measurement. Partial barrier-free dynamic PageRank is the fastest approach for batch sizes below or equal to 10^3, while OpenMP-based unordered dynamic PageRank is the fastest for larger batch sizes. We also note that OpenMP-based dynamic and naive dynamic have almost identical performance, while barrier-free dynamic approaches are faster what naive dynamic approaches. In terms of the number of iterations, we observe that parital barrier-free dynamic/naive dynamic approaches requires the fewest iterations to converge for all batch sizes. We also note that all dynamic and naive dynamic approaches require similar number of iterations to converge.

All outputs are saved in a gist and a small part of the output is listed here. Some charts are also included below, generated from sheets. The input data used for this experiment is available from the SuiteSparse Matrix Collection. This experiment was done with guidance from Prof. Kishore Kothapalli, Prof. Dip Sankar Banerjee, and Prof. Sathya Peri.

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$ g++ -std=c++17 -O3 -fopenmp main.cxx
$ ./a.out ~/data/email-Eu-core-temporal.txt
$ ./a.out ~/data/CollegeMsg.txt
$ ...

# Using graph /home/subhajit/data/email-Eu-core-temporal.txt ...
# OMP_NUM_THREADS=12
# Temporal edges: 332335
#
# # Batch size 1e+02
# [751 order; 7703 size; 00000.619 ms; 063 iters.] [0.0000e+00 err.] pagerankOmpUnorderedStatic
# [751 order; 7703 size; 00000.859 ms; 075 iters.] [5.6016e-07 err.] pagerankOmpOrderedStatic
# [751 order; 7703 size; 00020.857 ms; 089 iters.] [6.1278e-07 err.] pagerankBarrierfreeFullOmpStatic
# [751 order; 7703 size; 00006.136 ms; 067 iters.] [2.1334e-06 err.] pagerankBarrierfreePartOmpStatic
# [751 order; 7703 size; 00000.620 ms; 057 iters.] [9.0126e-07 err.] pagerankOmpUnorderedNaiveDynamic
# [751 order; 7703 size; 00000.675 ms; 057 iters.] [8.4747e-07 err.] pagerankOmpOrderedNaiveDynamic
# [751 order; 7703 size; 00006.907 ms; 062 iters.] [8.4526e-07 err.] pagerankBarrierfreeFullOmpNaiveDynamic
# [751 order; 7703 size; 00003.605 ms; 037 iters.] [2.2513e-06 err.] pagerankBarrierfreePartOmpNaiveDynamic
# [751 order; 7703 size; 00000.612 ms; 057 iters.] [9.0126e-07 err.] pagerankOmpUnorderedDynamic
# [751 order; 7703 size; 00000.659 ms; 057 iters.] [8.4745e-07 err.] pagerankOmpOrderedDynamic
# [751 order; 7703 size; 00005.945 ms; 062 iters.] [8.3778e-07 err.] pagerankBarrierfreeFullOmpDynamic
# [751 order; 7703 size; 00003.136 ms; 037 iters.] [2.2675e-06 err.] pagerankBarrierfreePartOmpDynamic
# ...
# [986 order; 25915 size; 00001.915 ms; 064 iters.] [0.0000e+00 err.] pagerankOmpUnorderedStatic
# [986 order; 25915 size; 00002.375 ms; 072 iters.] [5.7090e-07 err.] pagerankOmpOrderedStatic
# [986 order; 25915 size; 00027.283 ms; 080 iters.] [1.0225e-06 err.] pagerankBarrierfreeFullOmpStatic
# [986 order; 25915 size; 00020.180 ms; 064 iters.] [2.0462e-06 err.] pagerankBarrierfreePartOmpStatic
# [986 order; 25915 size; 00000.861 ms; 026 iters.] [1.3071e-06 err.] pagerankOmpUnorderedNaiveDynamic
# [986 order; 25915 size; 00000.957 ms; 026 iters.] [1.2276e-06 err.] pagerankOmpOrderedNaiveDynamic
# [986 order; 25915 size; 00010.578 ms; 027 iters.] [1.2453e-06 err.] pagerankBarrierfreeFullOmpNaiveDynamic
# [986 order; 25915 size; 00004.964 ms; 017 iters.] [1.8024e-06 err.] pagerankBarrierfreePartOmpNaiveDynamic
# [986 order; 25915 size; 00000.856 ms; 026 iters.] [1.3071e-06 err.] pagerankOmpUnorderedDynamic
# [986 order; 25915 size; 00000.955 ms; 026 iters.] [1.2276e-06 err.] pagerankOmpOrderedDynamic
# [986 order; 25915 size; 00007.328 ms; 028 iters.] [1.2456e-06 err.] pagerankBarrierfreeFullOmpDynamic
# [986 order; 25915 size; 00003.954 ms; 017 iters.] [1.8240e-06 err.] pagerankBarrierfreePartOmpDynamic
#
# # Batch size 1e+03
# [751 order; 7703 size; 00000.623 ms; 063 iters.] [0.0000e+00 err.] pagerankOmpUnorderedStatic
# [751 order; 7703 size; 00000.815 ms; 075 iters.] [5.6016e-07 err.] pagerankOmpOrderedStatic
# [751 order; 7703 size; 00012.586 ms; 091 iters.] [6.4177e-07 err.] pagerankBarrierfreeFullOmpStatic
# [751 order; 7703 size; 00008.811 ms; 067 iters.] [2.1239e-06 err.] pagerankBarrierfreePartOmpStatic
# [751 order; 7703 size; 00000.654 ms; 060 iters.] [1.0349e-06 err.] pagerankOmpUnorderedNaiveDynamic
# [751 order; 7703 size; 00000.707 ms; 060 iters.] [9.6145e-07 err.] pagerankOmpOrderedNaiveDynamic
# [751 order; 7703 size; 00011.240 ms; 073 iters.] [8.0149e-07 err.] pagerankBarrierfreeFullOmpNaiveDynamic
# [751 order; 7703 size; 00006.967 ms; 050 iters.] [1.9816e-06 err.] pagerankBarrierfreePartOmpNaiveDynamic
# [751 order; 7703 size; 00000.662 ms; 060 iters.] [1.0360e-06 err.] pagerankOmpUnorderedDynamic
# [751 order; 7703 size; 00000.715 ms; 060 iters.] [9.6246e-07 err.] pagerankOmpOrderedDynamic
# [751 order; 7703 size; 00010.556 ms; 068 iters.] [8.4651e-07 err.] pagerankBarrierfreeFullOmpDynamic
# [751 order; 7703 size; 00006.814 ms; 050 iters.] [2.0133e-06 err.] pagerankBarrierfreePartOmpDynamic
# ...

<br> <br>

• An Efficient Practical Non-Blocking PageRank Algorithm for Large Scale Graphs; Hemalatha Eedi et al. (2021)
• PageRank Algorithm, Mining massive Datasets (CS246), Stanford University
• The PageRank Citation Ranking: Bringing Order to the Web; Larry Page et al. (1998)
• The University of Florida Sparse Matrix Collection; Timothy A. Davis et al. (2011)
• What's the difference between "static" and "dynamic" schedule in OpenMP?
• OpenMP Dynamic vs Guided Scheduling

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