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Approximating Approximate Distance Oracles [article]

Michael Dinitz, Zeyu Zhang
2016 arXiv   pre-print
Given a finite metric space (V,d), an approximate distance oracle is a data structure which, when queried on two points u,v ∈ V, returns an approximation to the the actual distance between u and v which  ...  In particular, we give an O( n)-approximation to the problem of finding the smallest stretch 3 Thorup-Zwick distance oracle, as well as the problem of finding the smallest Pǎtraşcu-Roditty distance oracle  ...  In this paper we ask a natural but very different type of question about approximate distance oracles: can we find (or approximate) the best approximate distance oracle?  ... 
arXiv:1612.05623v1 fatcat:yldtsxpkrjbl7ftgd2orszyf64

Approximate distance oracles

Mikkel Thorup, Uri Zwick
2001 Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01  
The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k −1.  ...  We show that G = (V, E) can be preprocessed in O(kmn 1/k ) expected time, constructing a data structure of size O(kn 1+1/k ), such that any subsequent distance query can be answered, approximately, in  ...  oracles and distance labels.  ... 
doi:10.1145/380752.380798 dblp:conf/stoc/ThorupZ01 fatcat:wvffmpvwnnc5rfsnx2yljzklli

Approximate distance oracles

Mikkel Thorup, Uri Zwick
2005 Journal of the ACM  
The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k −1.  ...  We show that G = (V, E) can be preprocessed in O(kmn 1/k ) expected time, constructing a data structure of size O(kn 1+1/k ), such that any subsequent distance query can be answered, approximately, in  ...  oracles and distance labels.  ... 
doi:10.1145/1044731.1044732 fatcat:fj4vnfij5je3db2wi5aspdbbu4

Approximate Distance Oracles with Improved Query Time [article]

Christian Wulff-Nilsen
2012 arXiv   pre-print
Furthermore, for any $0 < \epsilon \leq 1$, we give an oracle of size $O(kn^{1 + 1/k})$ that answers $((2 + \epsilon)k)$-approximate distance queries in $O(1/\epsilon)$ time.  ...  Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 2$, we show that a $(2k-1)$-approximate distance oracle for $G$ of size $O(kn^{1 +  ...  To our knowledge, the oracle of Mendel and Naor cannot be used to produce approximate shortest paths, only distances. Our second oracle then has the same drawback (due to Lemma 10).  ... 
arXiv:1202.2336v3 fatcat:iwd27w7qsvfkhlay36jcagygb4

Approximate distance oracles for geometric spanners

Joachim Gudmundsson, Christos Levcopoulos, Giri Narasimhan, Michiel Smid
2008 ACM Transactions on Algorithms  
This represents the first data structure that answers (1 + ε)-approximate shortest path queries in constant time, and hence functions as an approximate distance oracle.  ...  Other applications include query versions of closest pair problems, and the efficient computation of the approximate dilations of geometric graphs.  ...  Since the query time is essentially bounded by a constant, Thorup and Zwick refer to their queries as approximate distance oracles.  ... 
doi:10.1145/1328911.1328921 fatcat:icaqazphnndw5ijlrxbno772da

More Compact Oracles for Approximate Distances in Planar Graphs [article]

Christian Sommer
2011 arXiv   pre-print
In FOCS'01, Thorup introduced approximate distance oracles for planar graphs.  ...  Distance oracles are data structures that provide fast (possibly approximate) answers to shortest-path and distance queries in graphs.  ...  Theorem 3 (Additive-Stretch Approximate Distance Oracle).  ... 
arXiv:1109.2641v2 fatcat:ghbbqnpzazb25cpu74pdqdoty4

Approximate Distance Oracles Subject to Multiple Vertex Failures [article]

Ran Duan, Yong Gu, Hanlin Ren
2020 arXiv   pre-print
Previously there are (1+ϵ)-approximate d-edge sensitive distance oracles [Chechik et al. 2017] answering distance queries when d edges fail, which have size O(n^2(log n/ϵ)^d· dlog W) and query time poly  ...  These results are the first approximate distance oracles of poly-logarithmic query time for any constant number of vertex failures in general undirected graphs.  ...  Acknowledgments We are grateful to anonymous reviewers for helpful comments, bringing [59] to our attention, and pointing out the recent work [48] that allows us to derandomize the oracle in Section  ... 
arXiv:2002.06812v2 fatcat:g7cw5tacfjarxb7mep7vkenenm

Approximate Distance Oracle with Constant Query Time [article]

Shiri Chechik
2013 arXiv   pre-print
An approximate distance oracle is a succinct data structure that provides fast answers to distance queries between any two nodes.  ...  In this paper we consider approximate distance oracles for general undirected graphs with non-negative edge weights with constant query time.  ...  To overcome these drawbacks, much of the work on distance oracles considers approximated distances.  ... 
arXiv:1305.3314v1 fatcat:dsud4ukekrffbhwdkzlwug5fei

Approximate Distance Oracles with Improved Query Time [chapter]

Christian Wulff-Nilsen
2014 Encyclopedia of Algorithms  
Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 2, we show that a (2k − 1)-approximate distance oracle for G of size O(kn 1+1/k ) and with  ...  It is therefore natural to consider approximate distance oracles where some error in the reported distances is allowed.  ...  Since its query time is independent of the size of the graph (when k is), we refer to it as an approximate distance oracle.  ... 
doi:10.1007/978-3-642-27848-8_568-1 fatcat:p5xmycz6dbemjhgp5du2altgeu

Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs [article]

Itay Laish, Shay Mozes
2017 arXiv   pre-print
A Vertex-Labeled Approximate Distance Oracle is a data structure that, given a vertex v and a label λ, returns a (1+ε)-approximation of the distance from v to the closest vertex with label λ in G.  ...  In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements.  ...  We focus on approximate vertex-labeled distance oracles for fixed parameter ε ≥ 0. When queried, such oracle returns at least the true distance, but not more than (1 + ε) times the true distance.  ... 
arXiv:1707.02414v2 fatcat:zw3jpwi6nvdkbi5ehqrecs2hxy

Space-Efficient Path-Reporting Approximate Distance Oracles [article]

Michael Elkin, Ofer Neiman, Christian Wulff-Nilsen
2014 arXiv   pre-print
For distance oracles, we show how to break the n n space bound of Thorup and Zwick if approximate paths rather than distances need to be reported.  ...  We consider approximate path-reporting distance oracles, distance labeling and labeled routing with extremely low space requirement, for general undirected graphs.  ...  Conclusions We gave space-efficient approximate distance oracles, distance labeling, and labeled routing for undirected graphs.  ... 
arXiv:1410.0768v1 fatcat:betu3eo5pray7nrcdrt2454wei

Approximate Distance Oracles with Improved Query Time [chapter]

Christian Wulff-Nilsen
2016 Encyclopedia of Algorithms  
Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 2, we show that a (2k − 1)-approximate distance oracle for G of size O(kn 1+1/k ) and with  ...  It is therefore natural to consider approximate distance oracles where some error in the reported distances is allowed.  ...  Since its query time is independent of the size of the graph (when k is), we refer to it as an approximate distance oracle.  ... 
doi:10.1007/978-1-4939-2864-4_568 fatcat:adaclkt7trefbokrjcxaccz3g4

Approximate Distance Oracles with Improved Query Time [chapter]

Christian Wulff-Nilsen
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 2, we show that a (2k − 1)-approximate distance oracle for G of size O(kn 1+1/k ) and with  ...  It is therefore natural to consider approximate distance oracles where some error in the reported distances is allowed.  ...  Since its query time is independent of the size of the graph (when k is), we refer to it as an approximate distance oracle.  ... 
doi:10.1137/1.9781611973105.39 dblp:conf/soda/Wulff-Nilsen13 fatcat:cmzpmxyeivcjrnsg3cel4wpak4

Approximate Distance Oracles with Improved Preprocessing Time [article]

Christian Wulff-Nilsen
2011 arXiv   pre-print
Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 1$, we show that for some universal constant $c$, a $(2k-1)$-approximate distance  ...  We also give an oracle which is faster for smaller $k$.  ...  A (2k − 1)-Approximate Distance Oracle In this section, we present a (2k − 1)-approximate distance oracle with subquadratic preprocessing time for k ≥ 6.  ... 
arXiv:1109.4156v1 fatcat:idmtsb5yurd5bnjb3jszeh4f3y

Approximate Distance Oracles for Planar Graphs with Subpolynomial Error Dependency [article]

Hung Le
2022 arXiv   pre-print
Thorup [FOCS'01, JACM'04] and Klein [SODA'01] independently showed that there exists a (1+ϵ)-approximate distance oracle for planar graphs with O(n (log n)ϵ^-1) space and O(ϵ^-1) query time.  ...  While the dependency on n is nearly linear, the space-query product of their oracles depend quadratically on 1/ϵ.  ...  All approximate distance oracles, including ours, use balanced shortest path separators.  ... 
arXiv:2207.05659v1 fatcat:2vie7v36arfexdjmzm5aeqvn2y
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