Simpler Analyses of Union-Find

TL;DR

This paper presents simplified analyses of the union-find data structure using potential functions inspired by continuous algorithms, providing alternate proofs for various near-logarithmic amortized bounds.

Contribution

It introduces a novel potential function approach to analyze union-find, simplifying proofs of its amortized bounds and offering new insights into its efficiency.

Findings

Provided alternate proofs for $O(\log\log n)$, $O(\log^* n)$, $O(\log^{**} n)$, and $O(\alpha(n))$ bounds.

Used potential functions based on subtree sizes to analyze path compression.

Showed that potential increases are bounded by $O(n)$, leading to simplified amortized analyses.

Abstract

We analyze union-find using potential functions motivated by continuous algorithms, and give alternate proofs of the $O(\log\log{n})$, $O(\log^{*}n)$, $O(\log^{**}n)$, and $O(\alpha(n))$ amortized cost upper bounds. The proof of the $O(\log\log{n})$ amortized bound goes as follows. Let each node's potential be the square root of its size, i.e., the size of the subtree rooted from it. The overall potential increase is $O(n)$ because the node sizes increase geometrically along any tree path. When compressing a path, each node on the path satisfies that either its potential decreases by $\Omega(1)$, or its child's size along the path is less than the square root of its size: this can happen at most $O(\log\log{n})$ times along any tree path.