International Journal of Computational Geometry & Applications

The problem of finding the shortest watchman route in a simple polygon P through a point s on its boundary is considered. A route is a watchman route if every point inside P can be seen from at least one point along the route. We present an incremental algorithm that constructs the shortest watchman route in O(n³) time for a simple polygon with n edges. This improves the previous O(n⁴) bound.

Two parallel geometric algorithms based on the idea of point domination are presented. The first algorithm solves the d-dimensional isothetic rectangles intersection counting problem of input size N/2^d, where d>1 and N is a multiple of 2^d, in O(log^d−1 N) time and O(N log N) space. The second algorithm solves the direct dominance reporting problem for a set of N points in the plane in O(log N+J) time and O(N log N) space, where J denotes the maximum of the number of direct dominances reported by any single point in the set. Both algorithms make use of the EREW PRAM (Exclusive Read Exclusive Write Parallel Random Access Machine) consisting of O(N) processors as the computational model.

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with. no self-intersections are NP-hard.

Let P be a set of n points located inside a d-box (rectangle if d=2) R. We study the problem of partitioning R into d-boxes by introducing a set of hyperplane (line if d=2) segments of least total (d-1)–volume (length if d=2). Each of the resulting d-boxes in a valid partition cannot contain points from P as interior points. Since this problem is computationally intractable (NP-hard) even when d=2, we present an efficient approximation algorithm for its solution. The partition generated by our approximation algorithm is guaranteed to be within 2d times the optimal solution value. We also present a problem instance for each d≥2 for which the approximation bound is tight for the algorithm. The time complexity for our algorithm is O(dn log n).

The main concern of this paper is the detection of double contact configurations for some polygons moving in translation in a polygonal environment. We first establish some general properties about such configurations and give conditions of existence of double contacts for two or three objects.

For three convex polygons moving in a polygonal environment or three simple polygons moving in a rectangle there always exists a double contact. Two examples without possibility of double contacts are given, one with three polygons (not convex) moving in a polygonal environment, and one with four convex polygons moving in a rectangle. We deduce an algorithm detecting a double contact position in time O(n²) (resp. O(n³)) for two (resp. three) convex polygons of constant sizes moving in a non-convex polygon of size n.

In this paper we show that it is impossible to solve a number of “natural” two-dimensional geometric problems in polylog time with a polynomial number of processors (unless P=NC). Thus, we disprove a popular belief that there are no natural P-complete geometric problems in the plane. The problems we address include instances of polygon triangulation, planar partitioning, and geometric layering. Our results are based on non-trivial reductions from the monotone circuit value and planar circuit value problems.

Open problems from the Fifth Canadian Conference on Computational Geometry (August 1993) are listed tersely.