On the hardness of approximating label-cover
Information Processing Letters
Studying (non-planar) road networks through an algorithmic lens
Proceedings of the 16th ACM SIGSPATIAL international conference on Advances in geographic information systems
Highway dimension, shortest paths, and provably efficient algorithms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
VC-dimension and shortest path algorithms
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Preprocessing speed-up techniques is hard
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
Exact Routing in Large Road Networks Using Contraction Hierarchies
Transportation Science
Search-Space size in contraction hierarchies
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Shortest-path queries in static networks
ACM Computing Surveys (CSUR)
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For some graph classes, most notably real-world road networks, shortest path queries can be answered very efficiently if the graph is preprocessed into a contraction hierarchy. The preprocessing algorithm contracts nodes in some order, adding new edges (shortcuts) in the process. While preprocessing and query algorithm work for any contraction ordering, it is desirable to use one that produces as few shortcuts as possible. It is known that the problem of minimizing the size (number of edges) of a given graph's contraction hierarchy is APX-hard. Also, any graph can be processed into a contraction hierarchy with at most O(nh log D) edges, where n, D, and h are the number of nodes, the diameter, and the highway dimension of the original graph, respectively. In this paper we show that the O(nh log D) bound is tight for a wide range of parameters n, D, and h. We also show that planar graphs, despite having highway dimension Ω(√n), can be preprocessed into a graph of size O(n log n). Finally, we present a simpler proof of APX-hardness.