On Dynamic Approximate Shortest Paths for Planar Graphs with Worst-Case Costs

Given a base weighted planar graph Ginput on n nodes and parameters M, e we present a dynamic distance oracle with 1 + e stretch and worst case update and query costs of e--3M4 · poly-log(n). We allow arbitrary edge weight updates as long as the shortest path metric induced by the updated graph has stretch of at most M relative to the shortest path metric of the base graph Ginput. For example, on a planar road network, we can support fast queries and dynamic traffic updates as long as the shortest path from any source to any target (including using arbitrary detours) is between, say, 80 and 3 miles-per-hour. As a warm-up we also prove that graphs of bounded treewidth have exact distance oracles in the dynamic edge model. To the best of our knowledge, this is the first dynamic distance oracle for a non-trivial family of dynamic changes to planar graphs with worst case costs of o(n1/2) both for query and for update operations.

[1]  Philip N. Klein,et al.  Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time , 2011, FOCS.

[2]  Peter Sanders,et al.  Dynamic Highway-Node Routing , 2007, WEA.

[3]  Dorothea Wagner,et al.  Engineering multilevel overlay graphs for shortest-path queries , 2009, JEAL.

[4]  Philip N. Klein,et al.  Structured recursive separator decompositions for planar graphs in linear time , 2012, STOC '13.

[5]  Philip N. Klein,et al.  An O (n log n) algorithm for maximum st-flow in a directed planar graph , 2006, SODA '06.

[6]  Christos D. Zaroliagis,et al.  Geometric containers for efficient shortest-path computation , 2005, JEAL.

[7]  Peter Sanders,et al.  Engineering Route Planning Algorithms , 2009, Algorithmics of Large and Complex Networks.

[8]  Stefan Funke,et al.  Ultrafast Shortest-Path Queries via Transit Nodes , 2006, The Shortest Path Problem.

[9]  Andrew V. Goldberg,et al.  A Hub-Based Labeling Algorithm for Shortest Paths in Road Networks , 2011, SEA.

[10]  Mikkel Thorup,et al.  Worst-case update times for fully-dynamic all-pairs shortest paths , 2005, STOC '05.

[11]  David Eisenstat,et al.  Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs , 2013, STOC '13.

[12]  Robert E. Tarjan,et al.  A data structure for dynamic trees , 1981, STOC '81.

[13]  Bruno Courcelle,et al.  Compact Forbidden-Set Routing , 2007, STACS.

[14]  Dorothea Wagner,et al.  Landmark-Based Routing in Dynamic Graphs , 2007, WEA.

[15]  Andrew V. Goldberg,et al.  Hierarchical Hub Labelings for Shortest Paths , 2012, ESA.

[16]  Hans L. Bodlaender,et al.  NC-Algorithms for Graphs with Small Treewidth , 1988, WG.

[17]  Bruno Courcelle,et al.  Constrained-Path Labellings on Graphs of Bounded Clique-Width , 2010, Theory of Computing Systems.

[18]  Karsten Weihe,et al.  Dijkstra's algorithm on-line: an empirical case study from public railroad transport , 2000, JEAL.

[19]  A. Goldberg,et al.  TRANSIT: Ultrafast Shortest-Path Queries with Linear-Time Preprocessing , 2006 .

[20]  Andrew V. Goldberg,et al.  Customizable Route Planning , 2011, SEA.

[21]  Aaron Bernstein,et al.  Fully Dynamic (2 + epsilon) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[22]  Peter Sanders,et al.  Exact Routing in Large Road Networks Using Contraction Hierarchies , 2012, Transp. Sci..

[23]  Ittai Abraham,et al.  Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels , 2012, STOC '12.

[24]  Mikkel Thorup,et al.  Fully-Dynamic All-Pairs Shortest Paths: Faster and Allowing Negative Cycles , 2004, SWAT.

[25]  Sakti Pramanik,et al.  An Efficient Path Computation Model for Hierarchically Structured Topographical Road Maps , 2002, IEEE Trans. Knowl. Data Eng..

[26]  Christos D. Zaroliagis,et al.  Using Multi-level Graphs for Timetable Information in Railway Systems , 2002, ALENEX.

[27]  Peter Sanders,et al.  Fast Routing in Road Networks with Transit Nodes , 2007, Science.

[28]  Ramesh Hariharan,et al.  Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths , 2007, J. Algorithms.

[29]  Philip N. Klein,et al.  A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs , 1998, Algorithmica.

[30]  Giuseppe F. Italiano,et al.  A new approach to dynamic all pairs shortest paths , 2004, JACM.

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

[32]  Frank Schulz,et al.  Using Multi-Level Graphs for Timetable Information , 2001 .

[33]  Uri Zwick,et al.  Dynamic Approximate All-Pairs Shortest Paths in Undirected Graphs , 2004, FOCS.