On k-Path Covers and their Applications

For a directed graph G with vertex set V we call a subset C ⊆ V a k-(All-)Path Cover if C contains a node from any path consisting of k nodes. This paper considers the problem of constructing small k-Path Covers in the context of road networks with millions of nodes and edges. In many application scenarios the set C and its induced overlay graph constitute a very compact synopsis of G which is the basis for the currently fastest data structure for personalized shortest path queries, visually pleasing overlays of subsampled paths, and efficient reporting, retrieval and aggregation of associated data in spatial network databases. Apart from a theoretical investigation of the problem, we provide efficient algorithms that produce very small k-Path Covers for large real-world road networks (with a posteriori guarantees via instance-based lower bounds).

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

[2]  Amos Fiat,et al.  VC-Dimension and Shortest Path Algorithms , 2011, ICALP.

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

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

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

[6]  Dorothea Wagner,et al.  User-Constrained Multi-Modal Route Planning , 2012, ALENEX.

[7]  Daniel Delling,et al.  Customizing Driving Directions with GPUs , 2014, Euro-Par.

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

[9]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[10]  Ben Strasser,et al.  Customizable Contraction Hierarchies , 2014, SEA.

[11]  Jian Pei,et al.  On k-skip shortest paths , 2011, SIGMOD '11.

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

[13]  Stefan Funke,et al.  Polynomial-Time Construction of Contraction Hierarchies for Multi-Criteria Objectives , 2013, SOCS.

[14]  Bostjan Bresar,et al.  Minimum k-path vertex cover , 2010, Discret. Appl. Math..

[15]  Yan Huang,et al.  Distance Preserving Graph Simplification , 2011, 2011 IEEE 11th International Conference on Data Mining.

[16]  David Haussler,et al.  Epsilon-nets and simplex range queries , 1986, SCG '86.

[17]  Subhash Khot.  On the power of unique 2-prover 1-round games , 2002, STOC '02.

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

[19]  Ben Strasser,et al.  Erratum: Customizable Contraction Hierarchies , 2014, SEA.