Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate.
Given a graph G = (V, E) with n vertices, m edges and T triangles, our first result is an algorithm that outputs a (1+ε)-approximation to T, with asymptotically optimal round and total space complexity provided any S ≥ max{(√ m, n²/m)} space per machine and assuming T = Ω(√{m/n}). Our result gives a quadratic improvement on the bound on T over previous works. We also provide a simple extension of our result to counting any subgraph of k size for constant k ≥ 1. Our second result is an O_δ(log log n)-round algorithm for exactly counting the number of triangles, whose total space usage is parametrized by the arboricity α of the input graph. We extend this result to exactly counting k-cliques for any constant k. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most 5 can be implemented in the MPC model in Õ_δ(√{log n}) rounds, O(n^δ) space per machine and O(mα³) total space.
In addition to our theoretical results, we simulate our triangle counting algorithms in real-world graphs obtained from the Stanford Network Analysis Project (SNAP) database. Our results show that both our approximate and exact counting algorithms exhibit improvements in terms of round complexity and approximation ratio, respectively, compared to two previous widely used algorithms for these problems.
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Theoretically-Efficient and Practical Parallel In-Place Radix Sorting
Parallel radix sorting has been extensively studied in the literature
for decades. However, the most efficient implementations require
auxiliary memory proportional to the input size, which can be
prohibitive for large inputs. The standard serial in-place radix
sorting algorithm is based on swapping each key to its correct
place in the array on each level of recursion. However, it is not
straightforward to parallelize this algorithm due to dependencies among
the swaps.
This paper presents Regions Sort, a new parallel in-place radix sorting algorithm
that is efficient both in theory and in practice. Our algorithm uses a
graph structure to model dependencies across elements that need to be
swapped, and generates independent tasks from this graph that can be
executed in parallel. For sorting $n$ integers from a range $r$, and
a parameter $K$, Regions Sort requires only $O(K\log r\log n)$
auxiliary memory. Our algorithm requires $O(n\log r)$ work and
$O((n/K+\log K)\log r)$ span, making it work-efficient and highly
parallel. In addition, we present several optimizations that
significantly improve the empirical performance of our algorithm. We
compare the performance of Regions Sort to existing parallel in-place
and out-of-place sorting algorithms on a variety of input
distributions and show that Regions Sort is faster than optimized
out-of-place radix sorting and comparison sorting algorithms, and is
almost always faster than the fastest publicly-available in-place
sorting algorithm.
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- Award ID(s):
- 1845763
- NSF-PAR ID:
- 10137104
- Date Published:
- Journal Name:
- ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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