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Register a new userFully Dynamic Four-Vertex Subgraph Counting
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Kathrin Hanauer
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper presents a comprehensive study of algorithms for maintaining the number of all connected four-vertex subgraphs in a dynamic graph. Specifically, our algorithms maintain the number of any four-vertex subgraph, which is not a clique, in deterministic amortized update time $mathcal{O}(m^{1/2})$, resp., $mathcal{O}(m^{2/3})$. Queries can be answered in constant time. For length-3 paths, paws, 4-cycles, and diamonds these bounds match or are not far from (conditional) lower bounds: Based on the OMv conjecture we show that any dynamic algorithm that detects the existence of length-3 paths, 4-cycles, diamonds, or 4-cliques takes amortized update time $Omega(m^{1/2-delta})$. Additionally, for 4-cliques and all connected induced subgraphs, we show a lower bound of $Omega(m^{1-delta})$ for any small constant $delta > 0$ for the amortized update time, assuming the static combinatorial 4-clique conjecture holds. This shows that the $mathcal{O}(m)$ algorithm by Eppstein et al. [9] for these subgraphs cannot be improved by a polynomial factor.
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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.
We present a space- and time-efficient fully dynamic implementation de Bruijn graphs, which can also support fixed-length jumbled pattern matching.
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.
We present a practically efficient algorithm for maintaining a global minimum cut in large dynamic graphs under both edge insertions and deletions. While there has been theoretical work on this problem, our algorithm is the first implementation of a fully-dynamic algorithm. The algorithm uses the theoretical foundation and combines it with efficient and finely-tuned implementations to give an algorithm that can maintain the global minimum cut of a graph with rapid update times. We show that our algorithm gives up to multiple orders of magnitude speedup compared to static approaches both on edge insertions and deletions.
The fully dynamic transitive closure problem asks to maintain reachability information in a directed graph between arbitrary pairs of vertices, while the graph undergoes a sequence of edge insertions and deletions. The problem has been thoroughly investigated in theory and many specialized algorithms for solving it have been proposed in the last decades. In two large studies [Frigioni ea, 2001; Krommidas and Zaroliagis, 2008], a number of these algorithms have been evaluated experimentally against simple static algorithms for graph traversal, showing the competitiveness and even superiority of the simple algorithms in practice, except for very dense random graphs or very high ratios of queries. A major drawback of those studies is that only small and mostly randomly generated graphs are considered. In this paper, we engineer new algorithms to maintain all-pairs reachability information which are simple and space-efficient. Moreover, we perform an extensive experimental evaluation on both generated and real-world instances that are several orders of magnitude larger than those in the previous studies. Our results indicate that our new algorithms outperform all state-of-the-art algorithms on all types of input considerably in practice.
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