Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

Brand, Jan van den; Nanongkai, Danupon

doi:10.1109/focs.2019.00035

Cited by 31 publications

(14 citation statements)

References 60 publications

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“…The fastest fully dynamic exact all-pairs-shortest path algorithm takes time O(n 2 (log n + log 2 (m/n))) per edge update operation and constant time per query [10,48]. For single-source shortest paths the fastest exact fully dynamic algorithm is still O(m); in graphs with real edge weights in [1, W ] there exists a (1 + ǫ)-approximation algorithm in time O(n 1.823 /ǫ 2 ) per update for any small ǫ > 0 using fast matrix multiplication and O(n 2.621 ) preprocessing time [49]. There are also algorithms that realize interesting tradeoffs between approximation ratio and query time [15], e.g., guaranteeing a O(log 4 n)-approximation with O(m 1/2+o (1) ) time per operation, or a n o(1) -approximation with O(n o(1) ) update time.…”

Section: Related Workmentioning

confidence: 99%

On the Complexity of Weight-Dynamic Network Algorithms

Henzinger¹,

Paz²,

Schmid³

2021

Preprint

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While operating communication networks adaptively may improve utilization and performance, frequent adjustments also introduce an algorithmic challenge: the re-optimization of traffic engineering solutions is time-consuming and may limit the granularity at which a network can be adjusted. This paper is motivated by question whether the reactivity of a network can be improved by re-optimizing solutions dynamically rather than from scratch, especially if inputs such as link weights do not change significantly.This paper explores to what extent dynamic algorithms can be used to speed up fundamental tasks in network operations. We specifically investigate optimizations related to traffic engineering (namely shortest paths and maximum flow computations), but also consider spanning tree and matching applications. While prior work on dynamic graph algorithms focuses on link insertions and deletions, we are interested in the practical problem of link weight changes.We revisit existing upper bounds in the weight-dynamic model, and present several novel lower bounds on the amortized runtime for recomputing solutions. In general, we find that the potential performance gains depend on the application, and there are also strict limitations on what can be achieved, even if link weights change only slightly.

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Section: Related Workmentioning

confidence: 99%

On the Complexity of Weight-Dynamic Network Algorithms

Henzinger¹,

Paz²,

Schmid³

2021

Preprint

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“…There are many different applications and areas where such matrix data structures are used. Applications include graph theoretic problems like reachability [San04,BS19], shortest paths [San05,BN19], matchings [San07,MS06], but also in convex optimization, e.g. linear programming [Kar84, Vai89, LS15, CLS19, LSZ19, Bra20, JSWZ20, BLSS20], cutting plane algorithms [LSW15,JLSW20], and ℓ p -norm regression [AKPS19].…”

Section: Introductionmentioning

confidence: 99%

Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms

Brand¹

2021

Symposium on Simplicity in Algorithms (SOSA)

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Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex hull and volume computation. Given the wide range of applications, the exact property these data structures must maintain varies from one application to another, forcing algorithm designers to invent them from scratch or modify existing ones. Thus it is not surprising that these data structures and their proofs are usually tailor-made for their specific application and that maintaining more complicated properties results in more complicated proofs.In this paper we present a unifying framework that captures a wide range of these data structures. The simplicity of this framework allows us to give short proofs for many existing data structures regardless of how complicated the to be maintained property is. We also show how the framework can be used to speed up existing iterative algorithms, such as the simplex algorithm.More formally, consider any rational function f (A 1 , ..., A d ) with input matrices A 1 , ..., A d . We show that the task of maintaining f (A 1 , ..., A d ) under updates to A 1 , ..., A d can be reduced to the much simpler problem of maintaining some matrix inverse M −1 under updates to M . The latter is a well studied problem called dynamic matrix inverse. By applying our reduction and using known algorithms for dynamic matrix inverse we can obtain fast data structures and iterative algorithms for much more general problems.

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“…Decremental APSP problem. For the decremental APSP problem, a plethora of algorithms is known [Kin99, BHS02, DI04, RZ04, Tho05, BR11, RZ12, AC13, HKN14a, Ber16, HKN16, ACK17, Che18, BN19,PW20b]. We want to point out in particular the exact deterministic decremental APSP algorithm for weighted digraphs by Demetrescu and Italiano [DI04] with running time Õ(n 3 ) and the deterministic (1 + )-approximate decremental APSP algorithm by Henzinger, Forster and Nanongkai [HKN16] with total update time Õ(mn) on undirected, unweighted graphs.…”

Section: Introductionmentioning

confidence: 99%

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

Gutenberg¹,

Wulff-Nilsen²

2020

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

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In the decremental (1 + )-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph G = (V, E) with n = |V |, m = |E|, undergoing edge deletions, and a distinguished source s ∈ V , and we are asked to process edge deletions efficiently and answer queries for distance estimates dist G (s, v) for each v ∈ V , at any stage, such that s, v). In the decremental (1 + )-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates dist G (u, v) for every u, v ∈ V . In this article, we consider the problems for undirected, unweighted graphs.We present a new deterministic algorithm for the decremental (1 + )-approximate SSSP problem that takes total update time O(mn 0.5+o (1) ). Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update timeÕ(n 2 ) and the best existing algorithm for sparse graphs with running timeÕ(n 1.25 √ m) [SODA 2017] whenever m = O(n 1.5−o(1) ). In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic sparse emulator in the deterministic setting.As a by-product, we also obtain a new simple deterministic algorithm for the decremental (1+ )-approximate APSP problem with near-optimal total running timeÕ(mn/ ) matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].

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Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

Cited by 31 publications

References 60 publications

On the Complexity of Weight-Dynamic Network Algorithms

On the Complexity of Weight-Dynamic Network Algorithms

Unifying Matrix Data Structures: Simplifying and Speeding up Iterative Algorithms

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

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