Steiner Minimal Trees in Simple Polygons

An O(n log n) time and O(n) space algorithm for the Euclidean Steiner tree problem with four terminals in a simple polygon with n vertices is given. Its applicability to the problem of determining good quality solutions for any number of terminals is discussed. 1 Introduction We consider the following variant of the Euclidean Steiner tree problem (ESTP): ffl Given: A simple polygon P and a set T = ft i ; t j ; t k ; t l g of four terminals in P . ffl Find: Euclidean Steiner minimal tree (ESMT) spanning T in P . We present an O(n log n) time and O(n) space algorithm for this problem. Our interest in this special case is due to the fact that it is one of the steps toward a heuristic for the ESTP inside a polygon for any number of terminals. More specifically, concatenation of small ESMTs (with up to 4 terminals) proved to yield good quality solutions for the obstacle-free case [9, 1]. Similar approach seems to be applicable when the terminals are inside a simple polygon or amidst po..

Parallel algorithms for polygonal and rectilinear geometry.

Parallel algorithms are presented for geometric problems of two types: problems about simple polygons and problems dealing with the rectilinear metric. Of the first type, an important result presented is an efficient parallel algorithm to compute shortest paths and shortest path trees in simple polygons. Applications of this shown include an efficient parallel algorithm to compute farthest neighbors of vertices of a simple polygon (both with distances measured by shortest paths internal and external to the polygon), an efficient parallel algorithm to determine if two simple polygons may be separated by a single translation, and an NC algorithm to compute the Voronoi diagram of a set of points inside a simple polygon. Another result presented is an optimal parallel algorithm to decide if a simple polygon is monotone. Of the second type, the important result obtained is an optimal parallel algorithm to compute the Voronoi diagram of a set of points on the plane under the rectilinear metric. Another result is an NC algorithm to find the shortest rectilinear path between two points on the plane avoiding rectangular barriers. The third result presented is an optimal parallel algorithm to bipartition a planar point set into two subsets of prescribed rectilinear diameter.Ph.D.Computer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105696/1/9208547.pdfDescription of 9208547.pdf : Restricted to UM users only