Indexed Geometric Jumbled Pattern Matching

Abstract

We consider how to preprocess n colored points in the plane such that later, given a multiset of colors, we can quickly find an axis-aligned rectangle containing a subset of the points with exactly those colors, if one exists. We first give an index that uses o(n ⁴) space and o (n) query time when there are ${\mathcal{O}({1})}$ distinct colors. We then restrict our attention to the case in which there are only two distinct colors. We give an index that uses ${\mathcal{O}({n})}$ bits and ${\mathcal{O}({1})}$ query time to detect whether there exists a matching rectangle. Finally, we give a ${\mathcal{O}({n})}$-space index that returns a matching rectangle, if one exists, in ${\mathcal{O}({\lg ^2 n / \lg \lg n})}$ time.