Kruskal-EDS: Edge Dynamic Stratification

TL;DR

Kruskal-EDS is an adaptive MST algorithm that replaces global sorting with a stratification approach based on distribution estimation, achieving significant speedups especially on sparse or heavy-tailed graphs.

Contribution

It introduces a distribution-adaptive stratification method for Kruskal's MST algorithm, reducing sorting complexity and improving performance on specific graph types.

Findings

Achieves up to 10x speedup over standard Kruskal

Effective on sparse graphs and heavy-tailed distributions

Demonstrates correctness and efficiency across multiple graph families

Abstract

We introduce \textbf{Kruskal-EDS} (\emph{Edge Dynamic Stratification}), a distribution-adaptive variant of Kruskal's minimum spanning tree (MST) algorithm that replaces the mandatory $\Theta(m\log m)$ global sort with a three-phase procedure inspired by Birkhoff's ergodic theorem. In Phase 1, a sample of $\sqrt{m}$ edges estimates the weight distribution in $\Theta(\sqrt{m}\log m)$ time. In Phase 2, all $m$ edges are assigned to $k$ strata in $\Theta(m\log k)$ time via binary search on quantile boundaries -- no global sort. In Phase 3, strata are sorted and processed in order; execution halts as soon as $n{-}1$ MST edges are accepted. We prove an effective complexity of $\Theta(m + p\cdot(m/k)\log(m/k))$, where $p \leq k$ is the number of strata actually processed. On sparse graphs or heavy-tailed weight distributions, $p \ll k$ and the algorithm achieves near-$\Theta(m)$ behaviour. We further derive the optimal strata count $k^* = \lceil\sqrt{m/\ln(m+1)}\,\rceil$, balancing partition overhead against intra-stratum sort cost. An extensive benchmark on 14 graph families demonstrates correctness on 12 test cases and practical speedups reaching $\mathbf{10\times}$ in wall-clock time and $\mathbf{33\times}$ in sort operations over standard Kruskal. A 3-dimensional TikZ visualisation of the complexity landscape illustrates the algorithm's adaptive behaviour as a function of graph density and weight distribution skewness.