Deterministic Constructions of Approximate Distance Oracles and Spanners

Roditty, Liam; Thorup, Mikkel; Zwick, Uri

doi:10.1007/11523468_22

Cited by 129 publications

(139 citation statements)

References 14 publications

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“…However, most previous work on these problems either focused on efficient performance (stretch, memory) and ignored the preprocessing stage (cf. [6,20,41,46] and references), or provided time-efficient sequential (centralized) preprocessing algorithms [7,8,30,49,49,53]. Relatively little attention was given to distributed preprocessing algorithms, and previous work on such algorithms either ignored time-efficiency (cf.…”

Section: Distributed Construction Of Compact Routing Tablesmentioning

confidence: 99%

Distributed distance computation and routing with small messages

2018

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We consider shortest paths computation and related tasks from the viewpoint of network algorithms, where the n-node input graph is also the computational system: nodes represent processors and edges represent communication links, which can in each time step carry an O(log n)-bit message. We identify several basic distributed distance computation tasks that are highly useful in the design of more sophisticated algorithms and provide efficient solutions. We showcase the utility of these tools by means of several applications.

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Section: Distributed Construction Of Compact Routing Tablesmentioning

confidence: 99%

Distributed distance computation and routing with small messages

2018

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“…It was quickly observed [ADD + 93] by Althöfer et al that one can obtain (2k − 2, 0) spanners on O(n 1+1/k ) edges for any integer k, and that this tradeoff is optimal assuming the popular Girth Conjecture posed by Erdös [Erd64]. The construction time was improved in various ways in subsequent work [RZ04,RTZ05,BS07]. A later direction of research studied mixed spanners, which contain a tradeoff between their α and β term; see [EP04,TZ06,Pet07] and the references therein.…”

Section: Spannersmentioning

confidence: 99%

Better Distance Preservers and Additive Spanners

Bodwin¹,

Williams²

2015

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

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We make improvements to the upper bounds on several popular types of distance preserving graph sketches. These sketches are all various restrictions of the additive pairwise spanner problem, in which one is given an undirected unweighted graph G, a set of node pairs P , and an error allowance +β, and one must construct a sparse subgraphThe first part of our paper concerns pairwise distance preservers, which make the restriction β = 0 (i.e. distances must be preserved exactly). Our main result here is an upper bound of |H| = O(n 2/3 |P | 2/3 + n|P | 1/3 ) when G is undirected and unweighted. This improves on existing bounds whenever |P | = ω(n 3/4 ), and it is the first such improvement in the last ten years.We then devise a new application of distance preservers to graph clustering algorithms, and we apply this algorithm to subset spanners, which require P = S × S for some node subset S, and (standard) spanners, which require P = V × V . For both of these objects, our construction generalizes the best known bounds when the error allowance is constant, and we obtain the strongest polynomial error/sparsity tradeoff that has yet been reported (in fact, for subset spanners, ours is the first nontrivial construction that enjoys improved sparsity from a polynomial error allowance).We leave open a conjecture that O(n 2/3 |P | 2/3 + n) pairwise distance preservers are possible for undirected unweighted graphs. Resolving this conjecture in the affirmative would improve and simplify our upper bounds for all the graph sketches mentioned above. * gbodwin@stanford.edu †

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“…This result was improved by Baswana and Sen [2] and Roditty et al [16], who showed that for every integer t ≥ 3, any connected weighted graph with n vertices contains a tspanner with O(tn 1+2/(t+1) ) edges. The following lower bound was proved by Althöfer et al [1]: For every real number t > 1, there exists a connected weighted graph G with n vertices, such that every t-spanner of G contains Ω(n 1+4/(3(t+2)) ) edges.…”

Section: Introductionmentioning

confidence: 98%

On Spanners of Geometric Graphs

Gudmundsson

Smid

2006

Algorithm Theory – SWAT 2006

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Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every t with 1 < t < 1 4 log n, there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n 1+1/t ) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(tn 1+2/(t+1) ) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

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Deterministic Constructions of Approximate Distance Oracles and Spanners

Cited by 129 publications

References 14 publications

Distributed distance computation and routing with small messages

Distributed distance computation and routing with small messages

Better Distance Preservers and Additive Spanners

On Spanners of Geometric Graphs

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