The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R to equal the minimum number of line segments or links needed to construct a path in R between the point pair. The minimum rectilinear link path problem considered here is to compute a rectilinear path consisting of the minimum number of links between two points in R, when R is inside an n-sided rectilinear simple polygon. In this paper we present optimal sequential and parallel algorithms to compute a minimum rectilinear link path in a trapezoided region R. Our parallel algorithm requires O(log n) time using a total of O(n) operations. The complexity of our algorithm matches that of the algorithm of McDonald and Peters [19]. By exploiting the dual structure of the trapezoidation of R, we obtain a conceptually simple and easy to implement algorithm. As applications of our techniques we provide an optimal solution to the minimum nested polygon problem and the minimum polygon separation problem. The minimum nested polygon problem asks for finding a rectilinear polygon, with minimum number of sides, that is nested between two given rectilinear polygons one of which is contained in the other. The minimum polygon separation problem asks for computing a minimum number of orthogonal lines and line segments that separate two given non-intersecting simple rectilinear polygons. All parallel algorithms are deterministic, designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM), and are optimal.

Parallel Processing Letters
School of Computer Science

Maheshwari, A, & Sack, J.-R. (1999). Simple optimal algorithms for rectilinear link path and polygon separation problems. Parallel Processing Letters, 9(1), 31–42.