Maintenance of a piercing set for intervals with applications

Abstract

We show how to efficiently maintain a minimum piercing set for a set S of intervals on the line, under insertions and deletions to/from S. A linear-size dynamic data structure is presented, which enables us to compute a new minimum piercing set following an insertion or deletion in time O(c(S) log |S|), where c(S) is the size of the new minimum piercing set. We also show how to maintain a piercing set for S of size at most (1+ε)c(S), for 0 < ε ≤ 1, in (formula presented) amortized time per update. We then apply these results to obtain efficient (sometimes improved) solutions to the following three problems: (i) the shooter location problem, (ii) computing a minimum piercing set for arcs on a circle, and (iii) dynamically maintaining a box cover for a d-dimensional point set.