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 $O(\lg ^{2+\epsilon } n)$ amortized time, for any constant $\epsilon \in (0, 1)$. Obvious solutions based on directly dynamizing solutions to the static version of this problem all require $\Omega ((\lg n / \lg \lg n)^2)$ 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.

This work was supported by NSERC and the Canada Research Chairs Program.