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Distance Oracles for Sparse Graphs

Christian Sommer, Elad Verbin, Wei Yu
2009 2009 50th Annual IEEE Symposium on Foundations of Computer Science  
Naturally, the following question arises: what happens 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.  ... 
doi:10.1109/focs.2009.27 dblp:conf/focs/SommerVY09 fatcat:deu5xejmezb7xcnd7qp7j7lshy

On the hardness of distance oracle for sparse graph [article]

Hagai Cohen, Ely Porat
2010 arXiv   pre-print
In this paper we show that set-intersection is harder than 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.  ... 
arXiv:1006.1117v1 fatcat:bli3723ajrdfjhmhkyphsv623u

A New Infinity of Distance Oracles for Sparse Graphs

Mihai Patrascu, Liam Roditty, Mikkel Thorup
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.  ... 
doi:10.1109/focs.2012.44 dblp:conf/focs/PatrascuRT12 fatcat:yoqvg43jmvbhfi5gkajgn6cwpm

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

Michael Elkin, Ofer Neiman, Christian Wulff-Nilsen
2014 arXiv   pre-print
We consider approximate path-reporting 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.  ... 
arXiv:1410.0768v1 fatcat:betu3eo5pray7nrcdrt2454wei

Faster Approximate Distance Queries and Compact Routing in Sparse Graphs [article]

Rachit Agarwal, P. Brighten Godfrey, Sariel Har-Peled
2012 arXiv   pre-print
We present 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  ... 
arXiv:1201.2703v1 fatcat:chzbi4rpevghfovztoqilx4b5i

More Compact Oracles for Approximate Distances in Undirected Planar Graphs [chapter]

Ken-ichi Kawarabayashi, Christian Sommer, Mikkel Thorup
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
ACM 2004), Thorup introduced approximate 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.  ... 
doi:10.1137/1.9781611973105.40 dblp:conf/soda/KawarabayashiST13 fatcat:bw7b44ybkna23nk7sjyx3hxyw4

A Simple Algorithm for the Graph Minor Decomposition − Logic meets Structural Graph Theory– [chapter]

Martin Grohe, Ken-ichi Kawarabayashi, Bruce Reed
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
ACM 2004), Thorup introduced approximate 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.  ... 
doi:10.1137/1.9781611973105.30 dblp:conf/soda/GroheKR13 fatcat:hzy72uuhk5c3vjliv2izydtbqe

Distance Oracles for Stretch Less Than 2 [chapter]

Rachit Agarwal, P. Brighten Godfrey
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.  ... 
doi:10.1137/1.9781611973105.38 dblp:conf/soda/AgarwalG13 fatcat:rtskv7wtfraann2cwararo5g7e

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

Surender Baswana, Akshay Gaur, Sandeep Sen, Jayant Upadhyay
2008 Lecture Notes in Computer Science  
The data structure is termed as approximate 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.  ... 
doi:10.1007/978-3-540-70575-8_50 fatcat:erqtijtkmzfxzj5othkv73zuz4

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.  ...  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.  ... 
arXiv:1109.2641v2 fatcat:ghbbqnpzazb25cpu74pdqdoty4

Distance Oracles beyond the Thorup-Zwick Bound

Mihai Patrascu, Liam Roditty
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.  ... 
doi:10.1109/focs.2010.83 dblp:conf/focs/PatrascuR10 fatcat:krrgiaeaq5dvdbreosiw7oi6p4

Sublinear-Space Distance Labeling using Hubs [article]

Paweł Gawrychowski, Adrian Kosowski, Przemysław Uznański
2016 arXiv   pre-print
Our simple approach is thus the first sublinear-space 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.  ... 
arXiv:1507.06240v3 fatcat:ewkcbc2cdrgixkkgkni5hqz6zm

Spanners with Slack [chapter]

T. -H. Hubert Chan, Michael Dinitz, Anupam Gupta
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.  ... 
doi:10.1007/11841036_20 fatcat:e773zu7tibb2za2cru6kn5lfvq

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

Timothy M. Chan, Dimitrios Skrepetos, Marc Herbstritt
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.  ... 
doi:10.4230/lipics.socg.2018.24 dblp:conf/compgeom/ChanS18 fatcat:acijbbi4zve4tmyk25mdoqbzzq

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

Hung Le
2019 arXiv   pre-print
This construction is based on a necessary and sufficient condition given by 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.  ... 
arXiv:1804.01588v3 fatcat:tdrgbwz54rev5gl55t7lxrgxqm
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