Untangled Monotonic Chains and Adaptive Range Search

Abstract

We present the first adaptive data structure for two-dimensional orthogonal range search. Our data structure is adaptive in the sense that it gives improved search performance for data with more inherent sortedness. Given n points on the plane, the linear-space data structure can answer range queries in O(logn + k + m) time, where m is the number of points in the output and k is the minimum number of monotonic chains into which the point set can be decomposed, which is $O(\sqrt{n})$ in the worst case. Our result matches the worst-case performance of other optimal-time linear-space data structures, or surpasses them when $k=o(\sqrt{n})$. Our data structure can also be made implicit, requiring no extra space beyond that of the data points themselves, in which case the query time becomes O(k logn + m). We present a novel algorithm of independent interest to decompose a point set into a minimum number of untangled, same-direction monotonic chains in O(kn + nlogn) time.

Funding for this research was made possible by NSERC Discovery Grants, the Canada Research Chairs Program, and the NSERC Strategic Grant on Optimal Data Structures for Organization and Retrieval of Spatial Data.