Untangled Monotonic Chains and Adaptive Range Search

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 in the plane, it can answer range queries in O(k logn+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( √ n) in the worst case. If k = o( √ n/ logn), our result surpasses the performance of optimal-time linear-space data structures. Our data structure is also implicit, requiring no extra space beyond that of the data points themselves. We also 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+n logn) time.