Data Structures for Mergeable Trees

TL;DR

This paper introduces efficient data structures for mergeable trees, enabling fast merging and updating operations, with applications in computational topology and theoretical bounds analysis.

Contribution

It presents novel data structures supporting merge operations in O(log^2 n) time, and optimized variants for special cases with improved bounds, advancing dynamic tree management techniques.

Findings

Supports merge operations in O(log^2 n) amortized time

Provides specialized data structures with O(log n) bounds for restricted cases

Includes lower bounds analysis for the problem

Abstract

Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant requires merging two paths in a single operation. In contrast to the standard problem, in which only one tree arc changes at a time, a single merge operation can change many arcs. In spite of this, we develop a data structure that supports merges on an n-node forest in O(log^2 n) amortized time and all other standard tree operations in O(log n) time (amortized, worst-case, or randomized depending on the underlying data structure). For the special case that occurs in the motivating application, in which arbitrary arc deletions (cuts) are not allowed, we give a data structure with an O(log n) time bound per operation. This is asymptotically optimal under certain assumptions. For the even-more special case in which both cuts and parent queries are disallowed, we give an alternative O(log n)-time solution that uses standard dynamic trees as a black box. This solution also applies to the motivating application. Our methods use previous work on dynamic trees in various ways, but the analysis of each algorithm requires novel ideas. We also investigate lower bounds for the problem under various assumptions.