Solving visibility and separability problems on a mesh-of-processors

Abstract

In this paper we study parallel algorithms for the Mesh-of-Processors architecture to solve visibility and related separability problems for sets of simple polygons in the plane. In particular, we present the following algorithms: -An O(]//N time algorithm for computing on a Mesh-of-Processors of size N the visibility polygon from a point located in an N-vertex polygon, possibly with holes. -O(~/N) time algorithms for computing on a Mesh-of-Processors of size N the set of all points on the boundary of an N-vertex polygon P which are visible in a given direction d as well as the visibility hull of IP for a given direction d. An O (VN) time algorithm for detecting on a Mesh-of-Processors of size 2N whether two N-vertex polygons are separable in a given direction and an O(~) time algorithm for detecting on a Mesh-of-Processors of size MN whether M N-vertex polygons are sequentially separable in a given direction.All proposed algorithms are asymptotically optimal (for the Mesh-of-Processors) with respect to time and number of processors.

Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments and Motion Planning in the Plane

“…The new set of edges found in steps (5) and (6), along with the edges of E 1 (E 2 ) that were retained in step (3) (step (4)) give us our new edge set E. There will be a constant number of edges of E per PE. Since the merge step takes O(n) time on a p n p n mesh, the total run-time of this step will be O(n).…”

“…Parallel algorithms, however, o er the possibility of signi cant speed-ups. Our research aims at obtaining e cient parallel algorithms for such motion planning problems and will be aided by the considerable progress that has been made in the area of parallel algorithms for computational geometry in recent years ( 1,4,6,9,13,19,23], for example). In this paper we develop e cient parallel mesh algorithms for two di erent techniques of planning motion for an object with two degrees of freedom moving in the plane among polygonal obstacles.…”

“…Note that it may not be possible to de ne a total order on the elements of the set L E. If this were possible, then a simple sort step, followed by a selected broadcasting step would enable us to nd e a and e b 6. If jLj is less than or equal to p n, then all we have to do is make copies of the rst row in every row of the mesh, and then pass each li through its entire row so that each e 2 E can determine e a and e b .…”