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Distance Oracles for Sparse Graphs

2009
*
2009 50th Annual IEEE Symposium on Foundations of Computer Science
*

Naturally, the following question arises: what happens

doi:10.1109/focs.2009.27
dblp:conf/focs/SommerVY09
fatcat:deu5xejmezb7xcnd7qp7j7lshy
*for**sparse**graphs*? In this paper we give a new lower bound*for*approximate*distance**oracles*in the cell-probe model. ... Our results in fact show that*for*any high-girth regular*graph*, an approximate*distance**oracle*that supports efficient queries*for*all subgraphs of G must obey Eq. (1). ... The first author thanks the Institute*for*Theoretical Computer Science at Tsinghua University*for*their invitation and their very kind hospitality. ...##
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On the hardness of distance oracle for sparse graph
[article]

2010
*
arXiv
*
pre-print

In this paper we show that set-intersection is harder than

arXiv:1006.1117v1
fatcat:bli3723ajrdfjhmhkyphsv623u
*distance**oracle*on*sparse**graphs*. ... We show that if one can build*distance**oracle**for**sparse**graph*G=(V,E), which requires s(|V|,|E|) space and answers a (2-\epsilon,c)-approximate*distance*query in time t(|V|,|E|) where (2-\epsilon) is ... In this paper we show a reduction from the set intersection problem to*distance**oracle*on*sparse**graphs*. ...##
###
A New Infinity of Distance Oracles for Sparse Graphs

2012
*
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
*

*For*space in terms of m, this is the first hardness matching the space of a non-trivial/sub-quadratic

*distance*

*oracle*. ... Given a weighted undirected

*graph*, our basic goal is to represent all pairwise

*distances*using much less than quadratic space, such that we can estimate the

*distance*between query vertices in constant ... We are interested in the inherent trade-offs between stretch and the space needed to store the

*distance*

*oracles*

*for*

*sparse*

*graphs*with m = O(n) edges. ...

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

2014
*
arXiv
*
pre-print

We consider approximate path-reporting

arXiv:1410.0768v1
fatcat:betu3eo5pray7nrcdrt2454wei
*distance**oracles*,*distance*labeling and labeled routing with extremely low space requirement,*for*general undirected*graphs*. ...*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. ... It might be possible to improve preprocessing of our*distance**oracles*, e.g., by using techniques from [WN12]*for**graphs*that are not too*sparse*. ...##
###
Faster Approximate Distance Queries and Compact Routing in Sparse Graphs
[article]

2012
*
arXiv
*
pre-print

We present

arXiv:1201.2703v1
fatcat:chzbi4rpevghfovztoqilx4b5i
*distance**oracles*that,*for**sparse**graphs*, substantially break the lower bound barrier at the expense of higher query time. ... Any*distance**oracle*that returns paths of worst-case stretch (2k-1) must require space Ω(n^1 + 1/k)*for**graphs*of n nodes. ... evaluate three schemes: the stretch-3 scheme of Thorup and Zwick (TZ) [14, 15] ; Reduced approximation ratio (REAR): the stretch-2 scheme from §6 with α = n; and Reduced space (RES): the stretch-3 scheme (*for*...##
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More Compact Oracles for Approximate Distances in Undirected Planar Graphs
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

ACM 2004), Thorup introduced approximate

doi:10.1137/1.9781611973105.40
dblp:conf/soda/KawarabayashiST13
fatcat:bw7b44ybkna23nk7sjyx3hxyw4
*distance**oracles**for*planar*graphs*(concurrent with Klein, SODA 2002). ... Thorup proved that,*for*any > 0 and*for*any undirected planar*graph*G = (V, E) on n = |V | nodes, there exists a (1 + )-approximate*distance**oracle*using space O(n −1 log n) such that approximate*distance*... Award*for*Young Scientists. ...##
###
A Simple Algorithm for the Graph Minor Decomposition − Logic meets Structural Graph Theory–
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

ACM 2004), Thorup introduced approximate

doi:10.1137/1.9781611973105.30
dblp:conf/soda/GroheKR13
fatcat:hzy72uuhk5c3vjliv2izydtbqe
*distance**oracles**for*planar*graphs*(concurrent with Klein, SODA 2002). ... Thorup proved that,*for*any > 0 and*for*any undirected planar*graph*G = (V, E) on n = |V | nodes, there exists a (1 + )-approximate*distance**oracle*using space O(n −1 log n) such that approximate*distance*... Award*for*Young Scientists. ...##
###
Distance Oracles for Stretch Less Than 2
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

*For*the realistic case of

*sparse*

*graphs*, our

*distance*

*oracles*exhibit a smooth three-way trade-off between space, stretch and query time -a phenomenon that does not occur in dense

*graphs*. ... We present

*distance*

*oracles*

*for*weighted undirected

*graphs*that return

*distances*of stretch less than 2. ... The authors would like to thank the anonymous reviewers

*for*their suggestions. We gratefully acknowledge the support of NSF grant CNS 10-17069. ...

##
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Distance Oracles for Unweighted Graphs: Breaking the Quadratic Barrier with Constant Additive Error
[chapter]

2008
*
Lecture Notes in Computer Science
*

The data structure is termed as approximate

doi:10.1007/978-3-540-70575-8_50
fatcat:erqtijtkmzfxzj5othkv73zuz4
*distance**oracle*. ...*For*a class of*graphs*, these algorithms indeed run in Θ(n 2 ) time. ... Our preprocessing algorithm*for*(3, c)-approximate*distance**oracle*, where c = 8, will employ the*sparse*subgraph (V, E S ) and an emulator of the given*graph*G. ...##
###
More Compact Oracles for Approximate Distances in Planar Graphs
[article]

2011
*
arXiv
*
pre-print

In FOCS'01, Thorup introduced approximate

arXiv:1109.2641v2
fatcat:ghbbqnpzazb25cpu74pdqdoty4
*distance**oracles**for*planar*graphs*. ... He proved that,*for*any eps>0 and*for*any planar*graph*on n nodes, there exists a (1+eps)-approximate*distance**oracle*using space O(n eps^-1 log n) such that approximate*distance*queries can be answered ... Acknowledgments Many thanks to Mikkel Thorup*for*encouraging our research on these tradeoffs and*for*the valuable comments on an earlier version of this work. ...##
###
Distance Oracles beyond the Thorup-Zwick Bound

2010
*
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
*

*For*weighted

*graphs*with m = n 2 /α edges, our

*distance*

*oracle*has size O(n 2 / 3 √ α) and returns a factor 2 approximation. ...

*For*unweighted

*graphs*, our

*distance*

*oracle*has size O(n 5/3 ) = O(n 1.66··· ) and, when queried about vertices at

*distance*d, returns a path of length 2d + 1. ...

*For*a

*distance*

*oracle*,

*sparse*

*graphs*can be hard. ...

##
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Sublinear-Space Distance Labeling using Hubs
[article]

2016
*
arXiv
*
pre-print

Our simple approach is thus the first sublinear-space

arXiv:1507.06240v3
fatcat:ewkcbc2cdrgixkkgkni5hqz6zm
*distance*labeling*for**sparse**graphs*that simultaneously admits small decoding time (*for*constant Δ, we can achieve any T=ω(1) while maintaining L=o ... We present a hub labeling which allows us to decode exact*distances*in*sparse**graphs*using labels of size sublinear in the number of nodes. ... The size of the*distance**oracle*from [24] is optimal assuming girth-conjecture.*For**sparse**graphs*, [21] design*distance**oracle*of size O(n 4/3 m 1/3 ) and stretch 2 in constant time. ...##
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Spanners with Slack
[chapter]

2006
*
Lecture Notes in Computer Science
*

*oracles*and

*distance*labelings with slack. ... Given a metric (V, d), a spanner is a

*sparse*

*graph*whose shortest-path metric approximates the

*distance*d to within a small multiplicative distortion. ... spanners, where the class C is the class of metrics generated by

*sparse*

*graphs*. ...

##
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Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs

2018
*
International Symposium on Computational Geometry
*

*For*example, Thorup [25] gave (1 + ε)approximate

*distance*

*oracles*

*for*weighted, undirected planar

*graphs*with O (n polylog n) preprocessing time and space, and O (1) query time,

*for*any constant ε > 0, ... Indeed, by grid rounding, we can construct a

*sparse*weighted

*graph*G, such that it (i) approximately preserves

*distances*in the original unit-disk

*graph*G (e.g., see the proof of Lemma 2), and (ii) is ...

*For*the sake of simplicity, we did not optimize the poly(1/ε, log(nW )) factors. ...

##
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A PTAS for subset TSP in minor-free graphs
[article]

2019
*
arXiv
*
pre-print

This construction is based on a necessary and sufficient condition given by

arXiv:1804.01588v3
fatcat:tdrgbwz54rev5gl55t7lxrgxqm
*sparse*spanner*oracles*: light subset spanners exist if and only if*sparse*spanner*oracles*exist. ... We give the first PTAS*for*the subset Traveling Salesperson Problem (TSP) in H-minor-free*graphs*. ... Acknowledgement: We thank Cora Borradaile*for*constructive comments. ...
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