Decremental Single-Source Reachability and Strongly Connected Components in Õ(m√n) Total Update Time

Chechik, Shiri; Hansen, Thomas Dueholm; Italiano, Giuseppe F.; Łącki, Jakub; Parotsidis, Nikos

doi:10.1109/focs.2016.42

Cited by 16 publications

(50 citation statements)

References 15 publications

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“…Together, our results improve on the running time of [7] for all graph densities. For instance, when m = Θ(n 2 ), we improve their bound fromÕ(n 2+3/4+o (1) log W ) toÕ(n 2+1/2 ) for unweighted graphs and toÕ(n 2+2/3 log W ) for weighted graphs; our bound for unweighted graphs in fact matches (up to logarithmic factors) the bound for decremental reachability in [4]. When m = Θ(n), we get an improvement from theÕ(n 1+5/6+o (1) log W ) bound of [7] toÕ(n 1+3/4 log W ).…”

Section: Our Resultsmentioning

confidence: 62%

Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Gutenberg¹,

Wulff-Nilsen²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Given a dynamic digraph G = (V, E) undergoing edge deletions and given s ∈ V and constant ǫ with 0 < ǫ ≤ 1, we consider the problem of maintaining (1 + ǫ)-approximate shortest path distances from s to all vertices in G over the sequence of deletions. Even and Shiloach (J. ACM'81) give a deterministic data structure for the exact version of the problem in unweighted graphs with total update time O(mn). Henzinger et al. (STOC'14, ICALP'15) give a Monte Carlo data structure for the approximate version with an improved total update time bound of O(mn 0.9+o(1) log W ) with better bounds for sufficiently dense and sufficiently sparse graphs; here W is the ratio between the largest and smallest edge weight. A drawback of their data structure and in fact of all previous randomized data structures is that they only work against an oblivious adversary, meaning that the sequence of deletions needs to be fixed in advance. This severely limits its application as a black box inside algorithms. We present the following (1 + ǫ)-approximate data structures:1. the first data structure is Las Vegas and works against an adaptive adversary; it has total expected update timeÕ(m 2/3 n 4/3 ) 1 for unweighted graphs andÕ(m 3/4 n 5/4 log W ) for weighted graphs, 2. the second data structure is Las Vegas and assumes an oblivious adversary; it has total expected update timeÕ( √ mn 3/2 ) for unweighted graphs andÕ(m 2/3 n 4/3 log W ) for weighted graphs, 3. the third data structure is Monte Carlo and is correct w.h.p. against an oblivious adversary; it has total expected update timeÕ((mn) 7/8 log W ) =Õ(mn 3/4 log W ).Each of our data structures can report the length of a (1+ǫ)-approximate shortest path from s to any query vertex in constant time at any point during the sequence of updates; if the adversary is oblivious, a query can be extended to also report such a path in time proportional to its length. Our update times are faster than those of Henzinger et al. for all graph densities. For instance, when m = Θ(n 2 ), our second result improves their bound fromÕ(n 2+3/4+o(1) log W ) toÕ(n 2+1/2 ) in the unweighted setting and toÕ(n 2+2/3 log W ) in the weighted setting. When m = Θ(n), our third result gives an improvement fromÕ(n 1+5/6+o(1) log W ) toÕ(n 1+3/4 log W ). Furthermore, our first data structure is the first to improve on the O(mn) bound of Even and Shiloach for all but the sparsest graphs while still working against an adaptive adversary and works even in weighted graphs; this answers an open problem by Henzinger et al.

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Section: Our Resultsmentioning

confidence: 62%

Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Gutenberg¹,

Wulff-Nilsen²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Łącki [23] showed that for graph classes admitting balanced separators of size O(n α ) (α > 0), the decremental stronglyconnected components problem can be solved in O(n α m) time. This property of Łącki's data structure was instrumental in obtaining the best known algorithm for decremental single-source reachability [5]. By wellknown reductions [13], from [23] it also follows that the decremental single-source reachability for O(n α )-separable graph classes can be solved in O(n α m) time.…”

Section: Introductionmentioning

confidence: 96%

“…Only recently, Henzinger et al [12] presented a randomized decremental single-source reachability algorithm with total update time O(mn 0.984+o (1) ), which they later improved to O(mn 0.9+o (1) ) [13]. Very recently, Chechik et al [5] improved the total update time to O(m √ n ) 1 . For the closely related decremental transitive closure problem, where the algorithm has to answer pointto-point reachability queries when the graph undergoes edge deletions, algorithms with total update time O(mn) and O(1) query time are known [23,29].…”

Section: Introductionmentioning

confidence: 99%

“…First we note that the decremental (1 + )-approximate APSP algorithm of Bernstein [3] can be simplified and adapted to solve decremental transitive closure in O(nm) time. This algorithm, although randomized and very non-trivial, is conceptually quite simple and its main advantage is that its efficiency relies only a single bottleneck -the efficiency of the ubiquitous Even-Shiloach tree 5 . Specifically, in the algorithm of [3], 4 There exist trivial incremental transitive closure algorithms running in O(nm) time, whereas the decremental transitive closure algorithms [23,29] are highly non-trivial.…”

Section: Introductionmentioning

confidence: 99%

“…Specifically, in the algorithm of [3], 4 There exist trivial incremental transitive closure algorithms running in O(nm) time, whereas the decremental transitive closure algorithms [23,29] are highly non-trivial. 5 Other decremental transitive closure algorithms [29,23] we are asked to solve O(log n) instances of the following subproblem. For p, d, such that p·d = Ω(n), and graphs G 1 , .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Decrementai Transitive Closure and Shortest Paths for Planar Digraphs and Beyond

Karczmarz¹

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we show that the tools used to obtain the best state-of-the-art decremental algorithms for reachability and approximate shortest paths in directed graphs can be successfully combined with the existence of small separators in certain graph classes.In particular, for graph classes admitting balanced separators of size O( √ n), such as planar, bounded-genus and minor-free graphs, we show that for both transitive closure and (1 + )-approximate all pairs shortest paths (where is constant), there exist decremental algorithms with O(n 3/2 ) total update time and O( √ n) worst-case query time. Additionally, for the case of planar graphs, we show that for any t ∈ [1, n], there exists a decremental transitive closure algorithm with O(n 2 /t) total update time and O( √ t) worst-case query time. In particular, for t = n 2/3 , if all the edges are eventually deleted, we obtain O(n 1/3 ) amortized update and query times.Most of the algorithms we obtain are correct with high probability against an oblivious adversary.

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An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model

2018

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In this paper we study the problem of maintaining the strongly connected components of a graph in the presence of failures. In particular, we show that given a directed graph G = (V, E) with n = |V | and m = |E|, and an integer value k ≥ 1, there is an algorithm that computes in O(2 k n log 2 n) time for any set F of size at most k the strongly connected components of the graph G \ F . The running time of our algorithm is almost optimal since the time for outputting the SCCs of G \ F is at least Ω(n). The algorithm uses a data structure that is computed in a preprocessing phase in polynomial time and is of size O(2 k n 2 ).Our result is obtained using a new observation on the relation between strongly connected components (SCCs) and reachability. More specifically, one of the main building blocks in our result is a restricted variant of the problem in which we only compute strongly connected components that intersect a certain path. Restricting our attention to a path allows us to implicitly compute reachability between the path vertices and the rest of the graph in time that depends logarithmically rather than linearly in the size of the path. This new observation alone, however, is not enough, since we need to find an efficient way to represent the strongly connected components using paths. For this purpose we use a mixture of old and classical techniques such as the heavy path decomposition of Sleator and Tarjan [29] and the classical Depth-First-Search algorithm. Although, these are by now standard techniques, we are not aware of any usage of them in the context of dynamic maintenance of SCCs. Therefore, we expect that our new insights and mixture of new and old techniques will be of independent interest.

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Decremental Single-Source Reachability and Strongly Connected Components in Õ(m√n) Total Update Time

Cited by 16 publications

References 15 publications

Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Decrementai Transitive Closure and Shortest Paths for Planar Digraphs and Beyond

An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model

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