Compact and Fast Sensitivity Oracles for Single-Source Distances

Let s denote a distinguished source vertex of a non-negatively real weighted and undirected graph G with n vertices and m edges. In this paper we present two efficient single-source approximate-distance sensitivity oracles, namely compact data structures which are able to quickly report an approximate (by a multiplicative stretch factor) distance from s to any node of G following the failure of any edge in G. More precisely, we first present a sensitivity oracle of size O(n) which is able to report 2-approximate distances from the source in O(1) time. Then, we further develop our construction by building, for any 0<epsilon<1, another sensitivity oracle having size O(n*1/epsilon*log(1/epsilon)), and is able to report a (1+epsilon)-approximate distance from s to any vertex of G in O(log(n)*1/epsilon*log(1/epsilon)) time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source additively-stretched sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.

[1]  SommerChristian.  Shortest-path queries in static networks , 2014 .

[2]  Gad M. Landau,et al.  On Cartesian Trees and Range Minimum Queries , 2009, ICALP.

[3]  Stefano Leucci,et al.  Fault-Tolerant Approximate Shortest-Path Trees , 2014, ESA.

[4]  Mikkel Thorup,et al.  Oracles for Distances Avoiding a Failed Node or Link , 2008, SIAM J. Comput..

[5]  Fabrizio Grandoni,et al.  Improved Distance Sensitivity Oracles via Fast Single-Source Replacement Paths , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[6]  Gad M. Landau,et al.  On Cartesian Trees and Range Minimum Queries , 2009, Algorithmica.

[7]  Stefano Leucci,et al.  Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees , 2016, STACS.

[8]  Mikkel Thorup,et al.  Approximate distance oracles , 2005, J. ACM.

[9]  David Peleg,et al.  Fault Tolerant Approximate BFS Structures , 2014, SODA.

[10]  Stefano Leucci,et al.  Fault-Tolerant Approximate Shortest-Path Trees , 2017, Algorithmica.

[11]  David R. Karger,et al.  A nearly optimal oracle for avoiding failed vertices and edges , 2009, STOC '09.

[12]  Mattia D'Emidio,et al.  Path-Fault-Tolerant Approximate Shortest-Path Trees , 2015, SIROCCO.

[13]  Surender Baswana,et al.  Approximate Shortest Paths Avoiding a Failed Vertex: Near Optimal Data Structures for Undirected Unweighted Graphs , 2012, Algorithmica.

[14]  Michael Langberg,et al.  f-Sensitivity Distance Oracles and Routing Schemes , 2010, ESA.

[15]  Ran Duan,et al.  Dual-failure distance and connectivity oracles , 2009, SODA.

[16]  Christian Sommer,et al.  Shortest-path queries in static networks , 2014, ACM Comput. Surv..

[17]  Michael Langberg,et al.  f-Sensitivity Distance Oracles and Routing Schemes , 2010, Algorithmica.

[18]  Enrico Nardelli,et al.  Swapping a Failing Edge of a Single Source Shortest Paths Tree Is Good and Fast , 2002, Algorithmica.

[19]  Merav Parter,et al.  Dual Failure Resilient BFS Structure , 2015, PODC.

[20]  Uzi Vishkin,et al.  Finding Level-Ancestors in Trees , 1994, J. Comput. Syst. Sci..

[21]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[22]  Michael A. Bender,et al.  The Level Ancestor Problem Simplified , 2002, LATIN.

[23]  David Peleg,et al.  Sparse Fault-Tolerant BFS Trees , 2013, ESA.