Querying two boundary points for shortest paths in a polygonal domain

Abstract

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 usingÕ(n 5 ) preprocessing time andÕ(n 5 ) space where n is the number of corners of the polygonal domain and theÕ-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+ǫ ) space. Our approach also extends to the case where query points should lie on a given set of line segments.

Two-Point L1 Shortest Path Queries in the Plane

“…For two-point queries in the Euclidean metric, Chiang and Mitchell [14] constructed a data structure of size O(n 11 ) that answers each query in O(log n) time, and alternatively, a data structure of size O(n 10 log n) with an O(log 2 n) query time; other data structures with trade-off between preprocessing and query time were also given in [14]. If the query points s and t are both restricted to the boundaries of the obstacles of P, Bae and Okamato [1] built a data structure of size O(n 5 poly(log n)) that answers each query in O(log n) time, where poly(log n) is a polylogarithmic factor. Efficient algorithms were also given for the case when the obstacles have curved boundaries [5,10,13,22,25].…”

Sharp Bounds on Davenport-Schinzel Sequences of Every Order

“…Blocked sequences, on the other hand, have no sparsity criterion. The extremal functions for (standard) Davenport-Schinzel sequences are defined to be λ s (n, m) = Ex( length s + 2 ababa · · · , n, m) and λ s (n) = Ex( length s + 2 ababa · · · , n) Bounds on generalized Davenport-Schinzel sequences are expressed as a function of the inverse-Ackermann function α, yet there is no universally agreed-upon definition of Ackermann's function 2 To cite a fraction of the literature, DS sequences/lower envelopes are routinely applied to problems related to geometric arrangements [61,13,12,31,69,29,30,19,42,44,54,40,32,55,80, 3,46], in kinetic data structures and dynamic geometric algorithms [6, 39, 47, 1, 5, 14, 43, 85], in visibility [25, 76, 62], motion planning [76,58], and geometric containment problems [10,76,77,15], as well as variations on classical problems such as computing shortest paths [17,11,18] and convex hulls [33,16]. They have also been used in some industrial applications [45,20].…”

The Geodesic Diameter of Polygonal Domains

“…Alternatively, one can exploit a two-point query structure only for boundary points on ∂P for Case (B-B). The two-point query structure by Bae and Okamato [6] builds an explicit representation of the graph of the lower envelope of the path-length functions len u,v restricted on ∂P × ∂P in O(n 5 log n log * n) time. 6 Since |V(s * , t * )| ≥ 3 in Case (B-B), such a pair appears as a vertex on the lower envelope.…”