VC-Dimension and Shortest Path Algorithms

We explore the relationship between VC-dimension and graph algorithm design. In particular, we show that set systems induced by sets of vertices on shortest paths have VC-dimension at most two. This allows us to use a result from learning theory to improve time bounds on query algorithms for the point-to-point shortest path problem in networks of low highway dimension, such as road networks. We also refine the definitions of highway dimension and related concepts, making them more general and potentially more relevant to practice. In particular, we define highway dimension in terms of set systems induced by shortest paths, and give cardinality-based and average case definitions.

[1]  Michael T. Goodrich,et al.  Almost optimal set covers in finite VC-dimension , 1995, Discret. Comput. Geom..

[2]  Peter Sanders,et al.  In Transit to Constant Time Shortest-Path Queries in Road Networks , 2007, ALENEX.

[3]  Dorothea Wagner,et al.  Algorithmics of Large and Complex Networks - Design, Analysis, and Simulation [DFG priority program 1126] , 2009, Algorithmics of Large and Complex Networks.

[4]  Haim Kaplan,et al.  Reach for A*: Shortest Path Algorithms with Preprocessing , 2006, The Shortest Path Problem.

[5]  Peter Sanders,et al.  Highway Hierarchies Hasten Exact Shortest Path Queries , 2005, ESA.

[6]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[7]  Ronald J. Gutman,et al.  Reach-Based Routing: A New Approach to Shortest Path Algorithms Optimized for Road Networks , 2004, ALENEX/ANALC.

[8]  Kenneth L. Clarkson,et al.  A Las Vegas algorithm for linear programming when the dimension is small , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[9]  Andrew V. Goldberg,et al.  The shortest path problem : ninth DIMACS implementation challenge , 2009 .

[10]  Edith Cohen,et al.  Reachability and distance queries via 2-hop labels , 2002, SODA '02.

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

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

[13]  Ran Raz,et al.  Distance labeling in graphs , 2001, SODA '01.

[14]  Andrew V. Goldberg,et al.  PHAST: Hardware-Accelerated Shortest Path Trees , 2011, 2011 IEEE International Parallel & Distributed Processing Symposium.

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

[16]  Amos Fiat,et al.  Highway dimension, shortest paths, and provably efficient algorithms , 2010, SODA '10.

[17]  Stefano Leonardi,et al.  Algorithms - ESA 2005, 13th Annual European Symposium, Palma de Mallorca, Spain, October 3-6, 2005, Proceedings , 2005, ESA.

[18]  Kenneth L. Clarkson,et al.  Algorithms for Polytope Covering and Approximation , 1993, WADS.

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

[20]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[21]  Peter Sanders,et al.  Contraction Hierarchies: Faster and Simpler Hierarchical Routing in Road Networks , 2008, WEA.

[22]  Jon M. Kleinberg,et al.  Detecting a Network Failure , 2004, Internet Math..

[23]  Dror Rawitz,et al.  Hitting sets when the VC-dimension is small , 2005, Inf. Process. Lett..