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Register a new userParallel Algorithms for Small Subgraph Counting
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Added by
Quanquan C. Liu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Subgraph counting is a fundamental problem in analyzing massive graphs, often studied in the context of social and complex networks. There is a rich literature on designing efficient, accurate, and scalable algorithms for this problem. In this work, we tackle this challenge and design several new algorithms for subgraph counting in the Massively Parallel Computation (MPC) model: Given a graph $G$ over $n$ vertices, $m$ edges and $T$ triangles, our first main result is an algorithm that, with high probability, outputs a $(1+varepsilon)$-approximation to $T$, with optimal round and space complexity provided any $S geq max{(sqrt m, n^2/m)}$ space per machine, assuming $T=Omega(sqrt{m/n})$. Our second main result is an $tilde{O}_{delta}(log log n)$-rounds algorithm for exactly counting the number of triangles, parametrized by the arboricity $alpha$ of the input graph. The space per machine is $O(n^{delta})$ for any constant $delta$, and the total space is $O(malpha)$, which matches the time complexity of (combinatorial) triangle counting in the sequential model. We also prove that this result can be extended to exactly counting $k$-cliques for any constant $k$, with the same round complexity and total space $O(malpha^{k-2})$. Alternatively, allowing $O(alpha^2)$ space per machine, the total space requirement reduces to $O(nalpha^2)$. 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 $tilde{O}_{delta}(sqrt{log n})$ rounds, $O(n^{delta})$ space per machine and $O(malpha^3)$ total space. Therefore, this result also exhibits the phenomenon that a time bound in the sequential model translates to a space bound in the MPC model.
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For over a decade now we have been witnessing the success of {em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the {em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in $O(log{n})$ rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. showed that if each machine has $n^{1+Omega(1)}$ memory, this problem can also be solved $2$-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly $O(log{n})$ rounds once we enter the near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that perplexing possibility. That is, we break the above $O(log n)$ round complexity bound even in the case of {em slightly sublinear} memory per machine. In fact, our improvement here is {em almost exponential}: we are able to deliver a $(2+epsilon)$-approximation to maximum matching, for any fixed constant $epsilon>0$, in $O((log log n)^2)$ rounds.
The problem of finding dense components of a graph is a widely explored area in data analysis, with diverse applications in fields and branches of study including community mining, spam detection, computer security and bioinformatics. This research project explores previously available algorithms in order to study them and identify potential modifications that could result in an improved version with considerable performance and efficiency leap. Furthermore, efforts were also steered towards devising a novel algorithm for the problem of densest subgraph discovery. This paper presents an improved implementation of a widely used densest subgraph discovery algorithm and a novel parallel algorithm which produces better results than a 2-approximation.
In this paper, we study new batch-dynamic algorithms for the $k$-clique counting problem, which are dynamic algorithms where the updates are batches of edge insertions and deletions. We study this problem in the parallel setting, where the goal is to obtain algorithms with low (polylogarithmic) depth. Our first result is a new parallel batch-dynamic triangle counting algorithm with $O(Deltasqrt{Delta+m})$ amortized work and $O(log^* (Delta+m))$ depth with high probability, and $O(Delta+m)$ space for a batch of $Delta$ edge insertions or deletions. Our second result is an algebraic algorithm based on parallel fast matrix multiplication. Assuming that a parallel fast matrix multiplication algorithm exists with parallel matrix multiplication constant $omega_p$, the same algorithm solves dynamic $k$-clique counting with $Oleft(minleft(Delta m^{frac{(2k - 1)omega_p}{3(omega_p + 1)}}, (Delta+m)^{frac{2(k + 1)omega_p}{3(omega_p + 1)}}right)right)$ amortized work and $O(log (Delta+m))$ depth with high probability, and $Oleft((Delta+m)^{frac{2(k + 1)omega_p}{3(omega_p + 1)}}right)$ space. Using a recently developed parallel $k$-clique counting algorithm, we also obtain a simple batch-dynamic algorithm for $k$-clique counting on graphs with arboricity $alpha$ running in $O(Delta(m+Delta)alpha^{k-4})$ expected work and $O(log^{k-2} n)$ depth with high probability, and $O(m + Delta)$ space. Finally, we present a multicore CPU implementation of our parallel batch-dynamic triangle counting algorithm. On a 72-core machine with two-way hyper-threading, our implementation achieves 36.54--74.73x parallel speedup, and in certain cases achieves significant speedups over existing parallel algorithms for the problem, which are not theoretically-efficient.
Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms -- usually sublogarithmic-time and often $poly(loglog n)$-time, or even faster -- for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present $poly(log k) in poly(loglog n)$ round MPC algorithms for computing $O(k^{1+{o(1)}})$-spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmic-time MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: -For the MPC setting, we get an $O(log^2log n)$-round algorithm for $O(log^{1+o(1)} n)$ approximation of all pairs shortest paths (APSP) in the near-linear regime of local memory. To the best of our knowledge, this is the first sublogarithmic-time MPC algorithm for distance approximations. -Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sub-logarithmic algorithm for approximating APSP in weighted graphs in the Congested Clique model.
In this paper, we study the single-source shortest-path (SSSP) problem with positive edge weights, which is a notoriously hard problem in the parallel context. In practice, the $Delta$-stepping algorithm proposed by Meyer and Sanders has been widely adopted. However, $Delta$-stepping has no known worst-case bounds for general graphs. The performance of $Delta$-stepping also highly relies on the parameter $Delta$. There have also been lots of algorithms with theoretical bounds, such as Radius-stepping, but they either have no implementations available or are much slower than $Delta$-stepping in practice. We propose a stepping algorithm framework that generalizes existing algorithms such as $Delta$-stepping and Radius-stepping. The framework allows for similar analysis and implementations of all stepping algorithms. We also propose a new ADT, lazy-batched priority queue (LaB-PQ), that abstracts the semantics of the priority queue needed by the stepping algorithms. We provide two data structures for LaB-PQ, focusing on theoretical and practical efficiency, respectively. Based on the new framework and LaB-PQ, we show two new stepping algorithms, $rho$-stepping and $Delta^*$-stepping, that are simple, with non-trivial worst-case bounds, and fast in practice. The stepping algorithm framework also provides almost identical implementations for three algorithms: Bellman-Ford, $Delta^*$-stepping, and $rho$-stepping. We compare our code with four state-of-the-art implementations. On five social and web graphs, $rho$-stepping is 1.3--2.5x faster than all the existing implementations. On two road graphs, our $Delta^*$-stepping is at least 14% faster than existing implementations, while $rho$-stepping is also competitive. The almost identical implementations for stepping algorithms also allow for in-depth analyses and comparisons among the stepping algorithms in practice.
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