New methods for computing visibility graphs

Authors:
M. H. Overmars;E. Welzl
Affiliations:
tDept, of Computer Science, University of Utrecht, P.O.Box 80.012, 3508 TA Utecht, the Netherlands;Dept. of Mathematics, Free University Berlin, Arnimallee 2-6, 1000 Berlin 33, West Germany
Venue:
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Year:
1988

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Abstract

Let S be a set of n non-intersecting line segments in the plane. The visibility graph GS of S is the graph that has the endpoints of the segments in S as nodes and in which two nodes are adjacent whenever they can “see” each other (i.e., the open line segment joining them is disjoint from all segments or is contained in a segment). Two new methods are presented to construct GS. Both methods are very simple to implement. The first method is based on a new solution to the following problem: given a set of points, for each point sort the other points around it by angle. It runs in time &Ogr;(n2). The second method uses the fact that visibility graphs often are sparse and runs in time &Ogr;(m log n) where m is the number of edges in GS. Both methods use only Ogr;(n) storage.