Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.30625

Optimal Mixing for Local Edge-Coloring Dynamics on Trees

Haoran wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the mixing time of two local Markov chains for uniformly sampling proper edge colorings of trees. For a tree with $n$ vertices and maximum degree $Δ$, we establish optimal mixing-time bounds for the single-edge Glauber dynamics when $q\geΔ+2$ and for the neighboring-edge heat-bath dynamics when $q\geΔ+1$. Both chains mix in $O_Δ(n(\log n+\log(1/\varepsilon)))$ steps, uniformly over the number of colors $q$. These bounds are tight in their dependence on $n$ for fixed $Δ$. Our proof establishes log-Sobolev inequalities through local color release, reusable recoloring buffers, and component entropy mixing. The two dynamics require different local constructions: a two-spare-color buffer for single-edge updates and a second-layer controller for neighboring-edge updates. The resulting entropy analysis applies to general bounded-degree trees, including regimes where worst-case spatial mixing fails.

Explore related subjects

cs.DSmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Haoran wang. 2026-10-22. Optimal Mixing for Local Edge-Coloring Dynamics on Trees. https://arxiv.org/abs/2610.30625

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Compressed Set Representations based on Set Difference

We introduce a compressed representation of sets of sets that exploits how similar they are to each other. Our representation supports access, membership, predecessor and successor queries on the sets within logarithmic time. It also computes the support of any element (i.e., the sets containing it) in optimal time. In addition, we give a new MST-based construction algorithm for the representation that outperforms standard ones.

cs.DS↗
arXiv2026

Fast Approximation Algorithms for Balanced Edge and Vertex Separators

We present a family of fast pseudo-approximation algorithms for the minimum balanced element cut problem in a graph, which includes vertex and edge separators. Given a graph $G=(V,E)$ with polynomially-bounded finite integer element weights with $n$ vertices and $m$ edges, and a (constant) balance parameter $c\in(0,1/2)$, where $G$ has some (unknown) $c$-balanced element cut of size $OPT_c$, we give a (Monte-Carlo randomized) algorithm running in $O(n^{\varepsilon}m^{1+o(1)})$ time that produces a $Θ(1)$-balanced element cut of size $O(OPT_c\cdot\sqrt{(\log n)/\varepsilon})$ for any value $\varepsilon\in[Θ(1/\log(n)),Θ(1)]$. Our algorithms are based on a semidefinite programming (SDP) relaxation of the problem. A standard approach to solving an SDP relaxation for this kind of problem would use the primal-dual Matrix Multiplicative Weights Update framework of Arora and Kale, but this framework does not seem to apply, since we cannot express all of our SDPs primal variables explicitly. Instead, we present a new MMWU framework for our problem that allows us to represent some of our variables implicitly. We also extend our framework to a general "Partial-variable MMWU framework", which we expect will be useful for future work. Our oracle for the balanced element cut problem within our framework uses $O(n^{\varepsilon}\text{polylog}(n))$ almost-linear time maximum-flow computations, and would be sped up if the time complexity of maximum-flow improves. For graphs with padding bound $β$, another oracle gives an $O(\sqrtβ/\varepsilon)$ approximation in $m^{1+o(1)}n^{\varepsilon}\operatorname{polylog}(n)$ time. It is known that $β\le O(\log h)$ for $K_h$-minor-free graphs (and in particular $β\le O(\log n)$).

cs.DS↗
arXiv2026

Optimal-Time Move Structure Construction

The move structure represents a permutation $π$ of $[0,n)$ by partitioning the domain into $O(r)$ disjoint, contiguously permuted intervals, with $r$ being the minimum number of such intervals. This data structure occupies $O(r)$ words of space and enables $O(1)$-time computation of $π(i)$ given the interval that contains $i$. For permutations where $r \ll n$, this provides an efficient, compressed representation for navigation. While the existing best $O(r)$-space construction approaches require $O(r\log r)$-time, we present an optimal $O(r)$-time and space construction algorithm. This is achieved by replacing balanced search trees with pointer-based lists through introducing a bidirectional strategy that synchronizes construction of the structures for $π$ and its inverse $π^{-1}$ in a single, unified pass. By applying this algorithm to an existing result, we achieve the first optimal $O(n)$-time construction of the longest common prefix (LCP) array from a run-length-encoded Burrows-Wheeler transform (RLBWT) of $r$ runs in $O(r)$ working space. Empirical evaluation on pangenome-scale data confirms that our move structure construction algorithm is consistently faster than the previous best, achieving speedups of up to $\sim 2\times$ while using less memory.

cs.DS↗