Finding small simple cycle separators for 2-connected planar graphs
Journal of Computer and System Sciences
Fast algorithms for shortest paths in planar graphs, with applications
SIAM Journal on Computing
Applications of generalized matrix searching to geometric algorithms
Discrete Applied Mathematics - Computational combinatiorics
Dynamic algorithms for shortest paths in planar graphs
Theoretical Computer Science
Faster shortest-path algorithms for planar graphs
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Shortest path queries in planar graphs
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Planarization of Graphs Embedded on Surfaces
WG '95 Proceedings of the 21st International Workshop on Graph-Theoretic Concepts in Computer Science
On-Line Algorithms for Shortest Path Problems on Planar Digraphs
WG '96 Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science
Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane
ESA '96 Proceedings of the Fourth Annual European Symposium on Algorithms
Multiple-source shortest paths in planar graphs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Planar graphs, negative weight edges, shortest paths, and near linear time
Journal of Computer and System Sciences - Special issue on FOCS 2001
ACM Transactions on Algorithms (TALG)
Shortest paths in planar graphs with real lengths in O(n log2n/ log log n) time
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
Many Distances in Planar Graphs
Algorithmica
Exact distance oracles for planar graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Submatrix maximum queries in Monge matrices and Monge partial matrices, and their applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Shortest-path queries in static networks
ACM Computing Surveys (CSUR)
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There are several known data structures that answers distance queries between two arbitrary vertices in a planar graph. The tradeoff is among preprocessing time, storage space and query time. In this paper we present three data structures that answer such queries, each with its own advantage over previous data structures. The first one improves the query time of data structures of linear space. The second improves the preprocessing time of data structures with a space bound of O(n4/3) or higher while matching the best known query time. The third data structure improves the query time for a similar range of space bounds, at the expense of a longer preprocessing time. The techniques that we use include modifying the parameters of planar graph decompositions, combining the different advantages of existing data structures, and using the Monge property for finding minimum elements of matrices.