Dynamic path counting and reporting in linear space

Abstract

In the path reporting problem, we preprocess a tree on n nodes each of which is assigned a weight, such that given an arbitrary path and a weight range, we can report the nodes whose weights are within the range. We consider this problem in dynamic settings, and propose the first non-trivial linear-space solution that supports path reporting in O((lg n/ lg lg n)2 + occ lg n/ lg lg n) time, where occ is the output size, and the insertion and deletion of a node of an arbitrary degree in O(lg2+ϵ n) amortized time, for any constant ϵ ∈ (0, 1). Obvious solutions based on directly dynamizing solutions to the static version of this problem all require Ω((lg n/ lg lg n)2) time for each node reported, and thus our query time is much faster. For the counting version of this problem, we design a structure that supports path counting in O((lg n/ lg lg n)2) time, and insertion and deletion in O((lg n/ lg lg n)2) amortized time. This matches the current best result for 2D dynamic range counting, which can be viewed as a special case of path counting.