Sensitivity Analysis of Minimum Spanning Trees in Sub-Inverse-Ackermann Time

TL;DR

This paper introduces a deterministic algorithm for MST sensitivity analysis with improved time complexity using a novel split-findmin data structure, advancing the efficiency of such computations.

Contribution

It presents the first superlinear yet sub-inverse-Ackermann split-findmin algorithm and reduces MST sensitivity to the MST problem, enabling faster analysis.

Findings

Achieved $O(m\log\alpha(m,n))$ time for MST sensitivity.

Developed a novel split-findmin data structure with superlinear but sub-inverse-Ackermann complexity.

Enabled a randomized linear time MST sensitivity algorithm by reduction to MST.

Abstract

We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in $O(m\log\alpha(m,n))$ time, where $\alpha$ is the inverse-Ackermann function. This improves upon a long standing bound of $O(m\alpha(m,n))$ established by Tarjan. Our algorithms are based on an efficient split-findmin data structure, which maintains a collection of sequences of weighted elements that may be split into smaller subsequences. As far as we are aware, our split-findmin algorithm is the first with superlinear but sub-inverse-Ackermann complexity. We also give a reduction from MST sensitivity to the MST problem itself. Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensitivity algoritm.