🤖 AI Summary
This work addresses the inefficiency of the classical Kruskal’s algorithm, whose Θ(m log m) time complexity—stemming from global edge sorting—becomes prohibitive on sparse graphs or those with heavy-tailed weight distributions. The authors propose a dynamic layered variant of Kruskal’s algorithm that circumvents full sorting through a three-stage pipeline: first, sampling to estimate the weight distribution; second, adaptively constructing a layered structure inspired by the Birkhoff ergodic theorem to dynamically determine the optimal number of layers k*; and third, processing edges layer-by-layer with early termination. By integrating statistical sampling, binary search over quantile-based boundaries, and per-layer sorting, the method achieves near-linear complexity on sparse or skewed-weight graphs. Experiments across 14 graph types confirm correctness in 12 cases, demonstrating up to a 10× speedup over standard Kruskal and a 33× reduction in sorting operations.
📝 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 $Θ(m\log m)$ 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 $Θ(\sqrt{m}\log m)$ time. In Phase 2, all $m$ edges are assigned to $k$ strata in $Θ(m\log k)$ 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 $Θ(m + p\cdot(m/k)\log(m/k))$, 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-$Θ(m)$ 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.