Querying two boundary points for shortest paths in a polygonal domain

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 O˜(n5) preprocessing time and O˜(n5) space where n   is the number of corners of the polygonal domain and the 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(n3+ϵ)O(n3+ϵ) space. Our approach also extends to the case where query points should lie on a given set of line segments.