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Voronoi diagrams were introduced by R. Klein (1988) as an axiomatic basis of Voronoi diagrams.We show how to construct abstract Voronoi diagrams in time O(n log n) by a randomized algorithm, which is based on Clarkson and Shor's randomized incremental construction technique (1989). The new algorithm has the following advantages over previous algorithms:l It can handle a much wider class of abstract Voronoi diagrams than the algorithms presented in by Klein (1989)

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We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed off-line. Our analysis is doubly founded on a novel geometric structure called the angle hull. Let D be a connected region inside a simple polygon, P. We define the angle hull of D, AH(D), to be the set of all points in P that can see two points of D at a right angle. We show that the perimeter of AH(D) cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight.