Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Computing Envelopes in Four Dimensions with Applications
SIAM Journal on Computing
Handbook of discrete and computational geometry
Two-point Euclidean shortest path queries in the plane
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
SIAM Journal on Computing
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
The geodesic diameter of polygonal domains
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Link distance and shortest path problems in the plane
Computational Geometry: Theory and Applications
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We consider a variant of two-point Euclidean shortest path query problem: given a polygonal domain, build a data structure for two-point shortest path query, provided that query points always lie on the boundary of the domain. As a main result, we show that a logarithmic-time query for shortest paths between boundary points can be performed using $\tilde{O}(n^5)$ preprocessing time and $\tilde{O}(n^5)$ space where n is the number of corners of the polygonal domain and the $\tilde{O}$-notation suppresses the polylogarithmic factor. This is realized by observing a connection between Davenport-Schinzel sequences and our problem in the parameterized space. We also provide a tradeoff between space and query time; a sublinear time query is possible using O(n 3 + 驴 ) space. Our approach also extends to the case where query points should lie on a given set of line segments.