A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons

Seidel, R., A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons, Computational Geometry: Theory and Applications 1 (1991) 51-64. This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log* n). This leads to a simple algorithm of the same complexity for triangulating polygons. More generally, if S is presented as a plane graph with k connected components, then the expected running time of the algorithm is O(n log* n + k log n). As a by-product our algorithm creates a search structure of expected linear size that allows point location queries in the resulting trapezoidation in logarithmic expected time. The analysis of the expected performance is elementary and straightforward. All expectations are with respect to ‘coinflips’ generated by the algorithm and are not based on assumptions about the geometric distribution of the input.