Scalable matrix computations on large scale-free graphs using 2D graph partitioning

Abstract

Scalable parallel computing is essential for processing large scale-free (power-law) graphs. The distribution of data across processes becomes important on distributed-memory computers with thousands of cores. It has been shown that twodimensional layouts (edge partitioning) can have significant advantages over traditional one-dimensional layouts. However, simple 2D block distribution does not use the structure of the graph, and more advanced 2D partitioning methods are too expensive for large graphs. We propose a new two-dimensional partitioning algorithm that combines graph partitioning with 2D block distribution. The computational cost of the algorithm is essentially the same as 1D graph partitioning. We study the performance of sparse matrix-vector multiplication (SpMV) for scale-free graphs from the web and social networks using several different partitioners and both 1D and 2D data layouts. We show that SpMV run time is reduced by exploiting the graph's structure. Contrary to popular belief, we observe that current graph and hypergraph partitioners often yield relatively good partitions on scale-free graphs. We demonstrate that our new 2D partitioning method consistently outperforms the other methods considered, for both SpMV and an eigensolver, on matrices with up to 1.6 billion nonzeros using up to 16,384 cores.

Reducing latency cost in 2D sparse matrix partitioning models

“…Some works on 2D models do not take the communication volume into account, however they provide an upper bound on the number of messages communicated [40][41][42][43][44] . On the other hand, there are 2D models that aim at reducing volume, with or without providing a bound on the maximum number of messages [33,[45][46][47][48][49][50] . 2D partitioning models in the literature can further be categorized into three classes: checkerboard partitioning [47,49,50] (also known as coarse-grain partitioning), jagged partitioning [45,49] and fine-grain partitioning [46,4 8,4 9] .…”

“…On the other hand, there are 2D models that aim at reducing volume, with or without providing a bound on the maximum number of messages [33,[45][46][47][48][49][50] . 2D partitioning models in the literature can further be categorized into three classes: checkerboard partitioning [47,49,50] (also known as coarse-grain partitioning), jagged partitioning [45,49] and fine-grain partitioning [46,4 8,4 9] . Notably, a recent work [50] proposes a fast 2D partitioning for scale-free graphs via a two-phase approach.…”

“…2D partitioning models in the literature can further be categorized into three classes: checkerboard partitioning [47,49,50] (also known as coarse-grain partitioning), jagged partitioning [45,49] and fine-grain partitioning [46,4 8,4 9] . Notably, a recent work [50] proposes a fast 2D partitioning for scale-free graphs via a two-phase approach. This method uses 1D partitioning to reduce volume in the first phase and an efficient heuristic in the second phase to obtain a bound on the maximum number of messages.…”

An exact algorithm for sparse matrix bipartitioning

“…Boman, Devine, and Rajamanickam [6] present a 2D method based on combining 1D graph/hypergraph partitioning with a 2D block distribution that also limits the total number of messages, besides trying to minimize the communication volume. A different approach to minimize other metrics as well is taken by the authors of the UMPa package [13], where the total and maximum volume per processor, and the total and maximum number of messages per processor can be chosen as primary or secondary objectives, and the secondary objective is used to break ties.…”

GraSP: distributed streaming graph partitioning

“…Boman, et. al show how conventional graph partitioning can be used to optimize distributed SpmV [5]. However, recent approaches to scale conventional multi-level partitioners to billion-node graphs can still take hours [25].…”