Dynamic stabbing queries with sub-logarithmic local updates for overlapping intervals: Proc. 12th International Computer Science Symposium in Russia

Abstract

We present a data structure to maintain a set of intervals on the real line subject to fast insertions and deletions of the intervals, stabbing queries, and local updates. Intuitively, a local update replaces an interval by another one of roughly the same size and location. We investigate whether local updates can be implemented faster than a deletion followed by an insertion. We present the first results for this problem for sets of possibly overlapping intervals. If the maximum depth of the overlap (a.k.a. ply) is bounded by a constant, our data structure performs insertions, deletions and stabbing queries in time O(logn)O(log⁡n) , and local updates in time O(logn/loglogn)O(log⁡n/log⁡log⁡n) , where n is the number of intervals. We also analyze the dependence on the ply when it is not constant. Our results are adaptive: the times depend on the current ply at the time of each operation.