Connected component and simple polygon intersection searching

Abstract

Efficient data structures are given for the following two query problems: (i) preprocess a setP of simple polygons with a total ofn edges, so that all polygons ofP intersected by a query segment can be reported efficiently, and (ii) preprocess a setS ofn segments, so that the connected components of the arrangement ofS intersected by a query segment can be reported quickly. In these problems we do not want to return the polygons or connected components explicitly (i.e., we do not wish to report the segments defining the polygon, or the segments lying in the connected components). Instead, we assume that the polygons (or connected components) are labeled and we just want to report their labels. We present data structures of sizeO(n 1+e) that can answer a query in timeO(n 1+e+k), wherek is the output size. If the edges ofP (or the segments inS) are orthogonal, the query time can be improved toO(logn+k) usingO(n logn) space. We also present data structures that can maintain the connected components as we insert new segments. For arbitrary segments the amortized update and query time areO(n 1/2+e) andO(n 1/2+e+k), respectively, and the space used by the data structure isO(n 1+e. If we allowO(n 4/3+e space, the amortized update and query time can be improved toO(n 1/3+e andO(n 1/3+e+k, respectively. For orthogonal segments the amortized update and query time areO(log2 n) andO(log2 n+klogn), and the space used by the data structure isO (n logn). Some other related results are also mentioned.