The Space-Stretch-Time Tradeoff in Distance Oracles

We present new distance oracles for computing distances of stretch less than 2 on general weighted undirected graphs. For the realistic case of sparse graphs and for any integer k, the new oracles return paths of stretch 1 + 1/k and exhibit a smooth three-way tradeoff of S ×T 1/k = O(n 2) between space S, stretch and query time T. This significantly improves the state-of-the-art for each point in the space-stretch-time tradeoff space, and matches the known space-time curve for stretch 2 and larger. We also present new oracles for stretch 1 + 1/(k + 0.5). A particularly interesting case is of stretch 5/3, where improving the query time of our oracles from T to T 1 − e for any e > 0 would lead to the first purely o(mn)-time combinatorial algorithm for Boolean Matrix Multiplication, a longstanding open problem.

[1]  Mihaela Enachescu,et al.  Reducing Maximum Stretch in Compact Routing , 2008, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications.

[2]  Aristides Gionis,et al.  Fast shortest path distance estimation in large networks , 2009, CIKM.

[3]  Mikkel Thorup,et al.  Compact routing schemes , 2001, SPAA '01.

[4]  Mikkel Thorup,et al.  A New Infinity of Distance Oracles for Sparse Graphs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[5]  Mihai Patrascu,et al.  Distance Oracles beyond the Thorup-Zwick Bound , 2014, SIAM J. Comput..

[6]  Gerhard Weikum,et al.  Fast and accurate estimation of shortest paths in large graphs , 2010, CIKM.

[7]  Uri Zwick,et al.  Replacement paths and k simple shortest paths in unweighted directed graphs , 2005, TALG.

[8]  Ken-ichi Kawarabayashi,et al.  Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus and Minor-Free Graphs , 2011, ICALP.

[9]  Giri Narasimhan,et al.  Fast algorithms for constructing t-spanners and paths with stretch t , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[10]  Ben Y. Zhao,et al.  Shortest paths in less than a millisecond , 2012, WOSN '12.

[11]  Sandeep Sen,et al.  Approximate distance oracles for unweighted graphs in expected O(n2) time , 2006, TALG.

[12]  Mikkel Thorup.  Compact oracles for reachability and approximate distances in planar digraphs , 2004, JACM.

[13]  Lenore Cowen,et al.  Near-Linear Time Construction of Sparse Neighborhood Covers , 1999, SIAM J. Comput..

[14]  Brighten Godfrey,et al.  Approximate distance queries and compact routing in sparse graphs , 2011, 2011 Proceedings IEEE INFOCOM.

[15]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[16]  Mikkel Thorup,et al.  Approximate distance oracles , 2001, JACM.

[17]  Telikepalli Kavitha,et al.  Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[18]  Wei Chen,et al.  A compact routing scheme and approximate distance oracle for power-law graphs , 2012, TALG.

[19]  Ken-ichi Kawarabayashi,et al.  More Compact Oracles for Approximate Distances in Undirected Planar Graphs , 2013, SODA.

[20]  Ely Porat,et al.  Preprocess, Set, Query! , 2013, Algorithmica.

[21]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[22]  Wei Yu,et al.  Distance Oracles for Sparse Graphs , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[23]  Brighten Godfrey,et al.  Distance Oracles for Stretch Less Than 2 , 2013, SODA.

[24]  Assaf Naor,et al.  Ramsey partitions and proximity data structures , 2006, FOCS.

[25]  Brighten Godfrey,et al.  Brief announcement: a simple stretch 2 distance oracle , 2013, PODC '13.

[26]  Shiri Chechik,et al.  Approximate Distance Oracle with Constant Query Time , 2013, ArXiv.

[27]  Camil Demetrescu,et al.  Algorithms – ESA 2011 , 2011, Lecture Notes in Computer Science.

[28]  J. Matousek,et al.  On the distortion required for embedding finite metric spaces into normed spaces , 1996 .

[29]  Guy E. Blelloch,et al.  A New Combinatorial Approach for Sparse Graph Problems , 2008, ICALP.

[30]  Christian Wulff-Nilsen,et al.  Approximate Distance Oracles with Improved Query Time , 2012, Encyclopedia of Algorithms.

[31]  Uri Zwick,et al.  All-Pairs Almost Shortest Paths , 1997, SIAM J. Comput..

[32]  Christian Wulff-Nilsen,et al.  Approximate distance oracles with improved preprocessing time , 2011, SODA.

[33]  Nikhil Bansal,et al.  Regularity Lemmas and Combinatorial Algorithms , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[34]  Ittai Abraham,et al.  On Approximate Distance Labels and Routing Schemes with Affine Stretch , 2011, DISC.