(1 + ε)-Distance Oracles for Vertex-Labeled Planar Graphs

We consider vertex-labeled graphs, where each vertex v is attached with a label from a set of labels. The vertex-to-label distance query desires the length of the shortest path from the given vertex to the set of vertices with the given label. We show how to construct an oracle for a vertex-labeled planar graph, such that \(O(\frac{1}{\epsilon}n\log n)\) storing space is needed, and any vertex-to-label query can be answered in \(O(\frac{1}{\epsilon}\log n\log \Delta)\) time with stretch 1 + e. Here, Δ is the hop-diameter of the given graph. For the case that Δ = O(logn), we construct a distance oracle that achieves \(O(\frac{1}{\epsilon}\log n)\) query time, without changing space usage.

[1]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[2]  Mikkel Thorup,et al.  Undirected single source shortest paths in linear time , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

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

[4]  Christian Sommer,et al.  Exact distance oracles for planar graphs , 2010, SODA.

[5]  Volker Heun,et al.  A New Succinct Representation of RMQ-Information and Improvements in the Enhanced Suffix Array , 2007, ESCAPE.

[6]  Christian Sommer,et al.  More Compact Oracles for Approximate Distances in Planar Graphs , 2011, ArXiv.

[7]  Mike Paterson,et al.  Combinatorics, Algorithms, Probabilistic and Experimental Methodologies, First International Symposium, ESCAPE 2007, Hangzhou, China, April 7-9, 2007, Revised Selected Papers , 2007, ESCAPE.

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

[9]  Richard Cole,et al.  Searching dynamic point sets in spaces with bounded doubling dimension , 2006, STOC '06.

[10]  Philip N. Klein,et al.  Preprocessing an undirected planar network to enable fast approximate distance queries , 2002, SODA '02.

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

[12]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

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

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

[15]  Raphael Yuster,et al.  Distance Oracles for Vertex-Labeled Graphs , 2011, ICALP.

[16]  Moshe Lewenstein,et al.  Fast, precise and dynamic distance queries , 2011, SODA '11.

[17]  Stavros Papadopoulos,et al.  Nearest keyword search in XML documents , 2011, SIGMOD '11.

[18]  Shiri Chechik.  Improved Distance Oracles for Vertex-Labeled Graphs , 2011, ArXiv.