Pii: S0925-7721(00)00011-0

Computing simple paths among obstacles

Given a set X of points in the plane, two distinguished points s; t 2 X, and a set of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in. We present two results: (1) We show that nding such simple paths among arbitrary obstacles is NP-complete. (2) We give a polynomial-time algorithm t...

Translating a Convex Polygon to Contain a Maximum Number of Points

Pii: S0925-7721(00)00010-9

We improve previous lower bounds on the number of simple polygonizations, and other kinds of crossing-free subgraphs, of a set of N points in the plane by analyzing a suitable configuration. We also prove that the number of crossing-free perfect matchings and spanning trees is minimum when the points are in convex position.  2000 Elsevier Science B.V. All rights reserved.

Pii: S0925-7721(99)00011-5

An efficient technique to solve precision problems consists in using exact computations. For geometric predicates, using systematically expensive exact computations can be avoided by the use of filters. The predicate is first evaluated using rounding computations, and an error estimation gives a certificate of the validity of the result. In this note, we study the statistical efficiency of filt...

Pii: S0925-7721(00)00023-7

It is shown that in any polygonal art gallery of n sides it is possible to place bn/3c point guards whose range of vision is 180◦ so that every interior point of the gallery can be seen by at least one of them. The guards can be stationed at any point of the art gallery. This settles an open problem posed by J. Urrutia.  2000 Elsevier Science B.V. All rights reserved.