Kruskal-EDS: Edge Dynamic Stratification: A Distribution-Adaptive Minimum Spanning Tree Algorithm Inspired by Ergodic Theory

Abstract

Kruskal-EDS: Edge Dynamic StratificationInternational audienceWe introduce Kruskal-EDS (Edge Dynamic Stratification), a distribution-adaptive variant ofKruskal’s minimum spanning tree (MST) algorithm that replaces the mandatory Θ(m log m)global sort with a three-phase√procedure inspired by Birkhoff’s ergodic theorem. In Phase 1, a sample of m edges estimates the weight distribution in O (m log m) time. In Phase 2, all m edges are assigned to k strata in O(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 O(m + p · (m/k) log(m/k)), where p ≤ k is the number of strata actually processed. On sparse graphs or heavy-tailed weight distributions, p ≪ k and the algorithm achieves near-O(m) behaviour. We further derive the optimalp strata count k∗ = ⌈ m/ ln(m + 1) ⌉, balancing partition overhead against intra-stratum sortcost. An extensive benchmark on 14 graph families demonstrates correctness on 12 test casesand practical speedups reaching 10× in wall-clock time and 33× in sort operations overstandard Kruskal. A 3-dimensional TikZ visualisation of the complexity landscape illus-trates the algorithm’s adaptive behaviour as a function of graph density and weight distri-bution skewness