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All-pairs shortest paths with a sublinear additive error

2011
*
ACM Transactions on Algorithms
*

Our algorithm thus gives

doi:10.1145/2000807.2000813
fatcat:tgauavtctbcenoce7yayap6m3u
*a*smooth qualitative and quantitative transition between the fastest exact*shortest**paths*algorithm, and the fastest approximation algorithm*with**a*linear*additive**error*. ...*pair*of vertices u, v in the graph to within an*additive**error*δ p (u, v), where δ(u, v) is the exact length of the*shortest**path*between u and v. ...*all*we can hope for is to obtain*a*polynomially small*additive**error*. ...##
###
All-Pairs Shortest Paths with a Sublinear Additive Error
[chapter]

2008
*
Lecture Notes in Computer Science
*

Our algorithm thus gives

doi:10.1007/978-3-540-70575-8_51
fatcat:hpohnneaajbfrmhd53yqow2kmi
*a*smooth qualitative and quantitative transition between the fastest exact*shortest**paths*algorithm, and the fastest approximation algorithm*with**a*linear*additive**error*. ...*pair*of vertices u, v in the graph to within an*additive**error*δ p (u, v), where δ(u, v) is the exact length of the*shortest**path*between u and v. ...*all*we can hope for is to obtain*a*polynomially small*additive**error*. ...##
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Breaking the Linear Error Barrier in Differentially Private Graph Distance Release
[article]

2022
*
arXiv
*
pre-print

By adding

arXiv:2204.14247v1
fatcat:7bue6isnf5gqzbvm2kz7qnhu2y
*a*set of shortcut edges to the original graph, we show that any node*pair*has*a**shortest**path**with*link length Õ(n^1/2). ... Computing*all*pairwise distances on the constructed graph only introduces Õ(n^1/2)*error*in answering*all*pairwise*shortest**path*distances for fixed privacy parameter. ... Is this linear dependence on n inherent, or is it possible to release*all*-*pairs*distances*with**error**sublinear*in n ?" ...##
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A Hierarchy of Lower Bounds for Sublinear Additive Spanners
[article]

2017
*
arXiv
*
pre-print

Specifically, for any integer k> 2, any compression scheme

arXiv:1607.07497v3
fatcat:bgi5sjfjy5flrpzgi7tde42nba
*with*size O(n^1+1/2^k-1 - ϵ) has*a**sublinear**additive*stretch function f: f(d) = d + Ω(d^1-1/k). ... This lower bound matches Thorup and Zwick's (2006) construction of*sublinear**additive*emulators. ... We thank Michael Elkin for proposing the question of finding*a*lower bound hierarchy for mixed spanners (as shown here), as well as observing the corresponding upper bounds. ...##
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Sublinear-Space Distance Labeling using Hubs
[article]

2016
*
arXiv
*
pre-print

For

arXiv:1507.06240v3
fatcat:ewkcbc2cdrgixkkgkni5hqz6zm
*a*queried*pair*of nodes (u,v), the length of*a**shortest*u-v-*path*passing through*a*hub node from S(u)∩ S(v) is then used as an upper bound on the distance between u and v. ... To our knowledge, this is the first*additive*scheme*with*constant absolute*error*to use labels of*sublinear*size. The corresponding decoding time is then small (any T=ω(1) is sufficient). ... provide*a**shortest**path*cover*with*sufficiently high probability. ...##
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Sublinear Distance Labeling
[article]

2016
*
arXiv
*
pre-print

For approximate r-

arXiv:1507.02618v2
fatcat:tu57adirl5dddmg4babgsnclse
*additive*labeling schemes, that return distances within an*additive**error*of r we show*a*scheme of size O ( n/r·polylog (r n)/ n ) for r > 2. ...*A*D-preserving distance labeling scheme only returns precise distances between*pairs*of nodes that are at distance at least D from each other. ... Consider*a**pair*of nodes u, v P V*with*dist G pu, vq ě D. Let p be*a**shortest**path*between u and v, then |p| ě D. ...##
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A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

1998
*
Algorithmica
*

Given an

doi:10.1007/pl00009223
fatcat:vrmwlmlchjdx3fugevbo4eshtu
*error*parameter ε such that 0 < ε, our algorithm maintains approximate*all*-*pairs**shortest**paths*in an undirected planar graph G*with*nonnegative edge lengths. ... In this paper we give*a*fully dynamic approximation scheme for maintaining*all*-*pairs**shortest**paths*in planar networks. ... We are grateful to the referees for their comments and for pointing out an*error*in an earlier version of the document. ...##
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Graph Spanners: A Tutorial Review
[article]

2020
*
arXiv
*
pre-print

As an

arXiv:1909.03152v2
fatcat:vqme4ewuvbbd5ac5erinyykwsy
*additional*contribution, we offer*a*list of open problems on graph spanners. ... This tutorial review provides*a*guiding reference to researchers who want to have an overview of the large body of literature about graph spanners. ... In fact, the latter construction has*a*stronger property: for each node*pair*u, v, the*additive**error*scales*with*W = W (u, v) the maximum edge weight along the true u v*shortest**path*in the input graph ...##
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Analysis and Experimental Evaluation of Time-Dependent Distance Oracles
[chapter]

2014
*
2015 Proceedings of the Seventeenth Workshop on Algorithm Engineering and Experiments (ALENEX)
*

The core of this preprocessing phase is based on

doi:10.1137/1.9781611973754.13
dblp:conf/alenex/KontogiannisMPPWZ15
fatcat:fw63juzbqvgpdohr6zu3x5onji
*a*novel, quite efficient and simple oneto-*all*approximation method for creating approximations of*shortest*travel-time functions. ... We conduct an extensive, comparative experimental study*with**all*query algorithms and six landmark sets. ... The relative*error*for*a*given od−*path*p is the percentage of surplus from the exact*shortest*travel-time (as computed by TDD), i.e.:*With*respect to the observed query times, in*all*cases FCA is the fastest ...##
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Spanners and emulators with sublinear distance errors

2006
*
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
*

These are the first such results

doi:10.1145/1109557.1109645
fatcat:nsfq2xb5mncxjj344vjuhwokem
*with**additive**error*terms of the form o(d), i.e.,*additive**error*terms that are*sublinear*in the distance being approximated. ... We show that any undirected and unweighted graph G = (V, E) on n vertices has*a*subgraph G = (V, E )*with*O(kn 1+1/k ) edges such that for any two vertices . ... distances*with**a**sublinear**error*term in the distance. ...##
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Distance Oracles for Time-Dependent Networks
[article]

2015
*
arXiv
*
pre-print

Our oracle uses subquadratic space and time preprocessing, and provides two

arXiv:1309.4973v3
fatcat:heo247oqxnbirhjil64ir6j7zq
*sublinear*-time query algorithms that deliver constant and (1+σ)-approximate*shortest*-travel-times, respectively, for arbitrary ... origin-destination*pairs*in the network, for any constant σ > ϵ. ... Introduction Distance oracles are succinct data structures encoding*shortest**path*information among*a*carefully selected subset of*pairs*of vertices in*a*graph. ...##
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Approximate Online Pattern Matching in Sublinear Time

2019
*
Foundations of Software Technology and Theoretical Computer Science
*

We design an algorithm that upon arrival of the t-th symbol of T computes kt approximately within O(1)multiplicative factor and m 8/9 -

doi:10.4230/lipics.fsttcs.2019.10
dblp:conf/fsttcs/Chakraborty0K19
fatcat:zsaqdbpi45ey7dq5gt3l7i2s5i
*additive**error*. ... The goal is to find*all*the positions j in T such that there is*a*substring of T ending at j which has edit distance at most k from the pattern P . ... Acknowledgements Authors would like to thank anonymous reviewers for many helpful suggestions and comments on an earlier version of this paper. 10:14 Approximate Online Pattern Matching in*Sublinear*...##
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The 4/3 additive spanner exponent is tight

2016
*
Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016
*

Intuitively, it takes two graphs G, H and produces

doi:10.1145/2897518.2897555
dblp:conf/stoc/AbboudB16
fatcat:laikermn4ndobcf7kxyolytv5q
*a*new graph G ⊗ H whose*shortest**paths*structure looks locally like H but globally like G. ... of*additive**error*is allowed. ... G is the union of many*shortest**paths*between*a*set of node*pairs*P ,*with*the following properties: • Each*pair*p ∈ P has distance k • Each*pair*p ∈ P has*a*unique*shortest**path*between its endpoints ...##
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The 4/3 Additive Spanner Exponent is Tight
[article]

2020
*
arXiv
*
pre-print

of

arXiv:1511.00700v2
fatcat:neuh6znrezgodahkwbgf3r4sqm
*additive**error*is allowed. ...*A*central open question in the field is to prove or disprove whether such*a*tradeoff exists also in the regime of*additive**error*. ... G is the union of many*shortest**paths*between*a*set of node*pairs*P ,*with*the following properties: • Each*pair*p ∈ P has distance k • Each*pair*p ∈ P has*a*unique*shortest**path*between its endpoints ...##
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Preserving Distances in Very Faulty Graphs
[article]

2017
*
arXiv
*
pre-print

Preservers and

arXiv:1703.10293v1
fatcat:wtfba5d25ncanpljtadwfdzcmq
*additive*spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given*pairs*of nodes exactly or*with*some small*additive**error*, respectively. ... We show that this is necessary even in undirected unweighted graphs, and even if you allow for*a*small*additive**error*: If you aim at size O(n^2-ϵ) for ϵ>0, then the*additive**error*has to be Ω( f). ...*additive**error*n δ . ...
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