Dynamic Colored Orthogonal Range Searching

Abstract

In the colored orthogonal range reporting problem, we want a data structure for storing n colored points so that given a query axis-aligned rectangle, we can report the distinct colors among the points inside the rectangle. This natural problem has been studied in a series of papers, but most prior work focused on the static case. In this paper, we give a dynamic data structure in the 2D case which can answer queries in O(log1+o(1) n + k log1/2+o(1) n) time, where k denotes the output size (the number of distinct colors in the query range), and which can support insertions and deletions in O(log2+o(1) n) time (amortized) in the standard RAM model. This is the first fully dynamic structure with polylogarithmic update time whose query cost per color reported is sublogarithmic (near √ log n). We also give an alternative data structure with O(log1+o(1) n + k log3/4+o(1) n) query time and O(log3/2+o(1) n) update time (amortized). We also mention extensions to higher constant dimensions. 2012 ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Data structures design and analysis