Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing

Bernstein, Aaron; Gutenberg, Maximilian Probst; Saranurak, Thatchaphol

doi:10.1109/focs46700.2020.00108

Cited by 27 publications

(46 citation statements)

References 40 publications

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“…The simpler problem of maintain Single-Source Reachability in a decremental digraph was further solved to near-otimality [BPW19]. For Decremental Single-Source Reachability and SSSP, deterministic algorithms that improve over the classic ES-tree were given by Bernstein et al [BPS20]. For the incremental setting, Probst Gutenberg et al [PVW20] recently obtained a deterministic (1 + ǫ)-approximate algorithm with total update time Õ(n 2 ).…”

Section: Running Timementioning

confidence: 99%

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

Evald,

Fredslund-Hansen,

Gutenberg

et al. 2020

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Given a directed graph G = (V, E), undergoing an online sequence of edge deletions with m edges in the initial version of G and n = |V |, we consider the problem of maintaining all-pairs shortest paths (APSP) in G.Whilst this problem has been studied in a long line of research [ACM'81, FOCS'99, FOCS'01, STOC'02, STOC'03, SWAT'04, STOC'13] and the problem of (1+ǫ)-approximate, weighted APSP was solved to near-optimal update time Õ(mn) by Bernstein [STOC'13], the problem has mainly been studied in the context of oblivious adversaries, which assumes that the adversary fixes the update sequence before the algorithm is started.In this paper, we make significant progress on the problem in the setting were the adversary is adaptive, i.e. can base the update sequence on the output of the data structure queries. We present three new data structures that fit different settings:• We first present a deterministic data structure that maintains the exact distances with total update time Õ(n 3 ) 1 .• We also present a deterministic data structure that maintains (1 + ǫ)-approximate distance estimates with total update time Õ( √ mn 2 /ǫ) which for sparse graphs is Õ(n 2+1/2 /ǫ).

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Section: Running Timementioning

confidence: 99%

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

Evald,

Fredslund-Hansen,

Gutenberg

et al. 2020

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“…Step (I): Design an efficient dynamic algorithm that maintains an approximately maximum fractional matching 2 w : E → [0, 1] in the input graph G = (V, E). All the known algorithms for this first step are deterministic [9,12,13,14,15,16,23].…”

Section: Introductionmentioning

confidence: 99%

“…Next, very recently Bernstein et al [9] showed how to maintain a (1+ δ)-approximate maximum fractional matching in a bipartite graph with O(log 3 n) amortized update time in the decremental setting, where the input graph only undergoes edge-deletions. Applying our dynamic rounding framework on top of their result, we get the first deterministic algorithm for maximum (integral) matching in an analogous decremental setting, with the same approximation ratio and similar update time.…”

Section: Introductionmentioning

confidence: 99%

Deterministic Rounding of Dynamic Fractional Matchings

Bhattacharya¹,

Kiss²

2021

Preprint

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We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a (2 − δ)-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a (1 + δ)approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time.(3) The first deterministic algorithm for maintaining a (2 + δ)-approximate maximum matching in a fully dynamic general graph, in small polylogarithmic (specifically, O(log 4 n)) update time. These results are respectively obtained by applying our framework on top of the fractional matching algorithms of Bhattacharya et al. [STOC'16], Bernstein et al. [FOCS'20], and Bhattacharya and Kulkarni [SODA'19].Prior to our work, there were two known general-purpose rounding schemes for dynamic fractional matchings. Both these schemes, by Arar et al. [ICALP'18] and Wajc [STOC'20], were randomized.Our rounding scheme works by maintaining a good matching-sparsifier with bounded arboricity, and then applying the algorithm of Peleg and Solomon [SODA'16] to maintain a near-optimal matching in this low arboricity graph. To the best of our knowledge, this is the first dynamic matching algorithm that works on general graphs by using an algorithm for low-arboricity graphs as a black-box subroutine. This feature of our rounding scheme might be of independent interest.

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“…We believe that the LCD data structure is of independent interest and will be useful in future adaptive-update dynamic algorithms. Indeed, a near-optimal short-path oracle on decremental expanders (from Section 3.2), which is one of the technical ingredients of our LCD data structure, has already found further applications in other algorithms for dynamic problems [BGS20].…”

Section: Introductionmentioning

confidence: 99%

“…We believe that our general approach of "rooting a tree at an expander" instead of "rooting a tree at a random location" will be a key technique for future adaptive-update algorithms. This idea was already exploited in a different way in a recent subsequent work [BGS20].…”

Section: Introductionmentioning

confidence: 99%

Deterministic Algorithms for Decremental Shortest Paths via Layered Core Decomposition

Chuzhoy¹,

Saranurak²

2020

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In the decremental single-source shortest paths (SSSP) problem, the input is an undirected graph G = (V, E) with n vertices and m edges undergoing edge deletions, together with a fixed source vertex s ∈ V . The goal is to maintain a data structure that supports shortest-path queries: given a vertex v ∈ V , quickly return an (approximate) shortest path from s to v. The decremental all-pairs shortest paths (APSP) problem is defined similarly, but now the shortest-path queries are allowed between any pair of vertices of V .Both problems have been studied extensively since the 80's, and algorithms with near-optimal total update time and query time have been discovered for them. Unfortunately, all these algorithms are randomized and, more importantly, they need to assume an oblivious adversary -a drawback that prevents them from being used as subroutines in several known algorithms for classical static problems. In this paper, we provide new deterministic algorithms for both problems, which, by definition, can handle an adaptive adversary.Our first result is a deterministic algorithm for the decremental SSSP problem on weighted graphs with O(n 2+o(1) ) total update time, that supports (1 + ǫ)-approximate shortest-path queries, with query time O(|P | • n o(1) ), where P is the returned path. This is the first (1 + ǫ)-approximation adaptive-update algorithm supporting shortest-path queries in time below O(n), that breaks the O(mn) total update time bound of the classical algorithm of Even and Shiloah from 1981. Previously, Bernstein and Chechik [STOC'16, ICALP'17] provided a Õ(n 2 )-time deterministic algorithm that supports approximate distance queries, but unfortunately the algorithm cannot return the approximate shortest paths. Chuzhoy and Khanna [STOC'19] showed an O(n 2+o(1) )-time randomized algorithm for SSSP that supports approximate shortest-path queries in the adaptive adversary regime, but their algorithm only works in the restricted setting where only vertex deletions, and not edge deletions are allowed, and it requires Ω(n) time to respond to shortest-path queries.Our second result is a deterministic algorithm for the decremental APSP problem on unweighted graphs that achieves total update time O(n 2.5+δ ), for any constant δ > 0, supports approximate distance queries in O(log log n) time, and supports approximate shortest-path queries in time), where P is the returned path; the algorithm achieves an O(1)-multiplicative and n o(1) -additive approximation on the path length. All previous algorithms for APSP either assume an oblivious adversary or have an Ω(n 3 ) total update time when m = Ω(n 2 ), even if an o(n)-multiplicative approximation is allowed.To obtain both our results, we improve and generalize the layered core decomposition data structure introduced by Chuzhoy and Khanna to be nearly optimal in terms of various parameters, and introduce a new generic approach of rooting Even-Shiloach trees at expander sub-graphs of the given graph. We believe both these technical tools to be interesting in their...

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Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing

Cited by 27 publications

References 40 publications

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

Deterministic Rounding of Dynamic Fractional Matchings

Deterministic Algorithms for Decremental Shortest Paths via Layered Core Decomposition

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