A technique for adding range restrictions to generalized searching problems

Abstract

In a generalized searching problem, a set $S$ of $n$ colored geometric objects has to be stored in a data structure, such that for any given query object $q$, the distinct colors of the objects of $S$ intersected by $q$ can be reported efficiently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (resp.\ fat triangles) with a fat triangle (resp.\ point). For both problems, a data structure is obtained having size $O(n^{1+\epsilon})$ and query time $O((\log n)^2 + C)$. Here, $C$ denotes the number of colors reported by the query, and $\epsilon$ is an arbitrarily small positive constant