A Distributed Algorithm for Finding Hamiltonian Cycles in Random Graphs in O(log n) Time

TL;DR

This paper introduces a distributed algorithm that efficiently finds Hamiltonian cycles in random graphs with certain edge probabilities, operating in logarithmic rounds in a distributed setting.

Contribution

It presents the first distributed algorithm capable of finding Hamiltonian cycles in random graphs within O(log n) rounds for p in ilde{ ilde{ ext{Omega}}}(1/\sqrt{n}), a significant improvement over prior methods.

Findings

Algorithm finds Hamiltonian cycles w.h.p. in O(log n) rounds.

Works in the synchronous model with message size O(log n).

Uses O(log n) memory per node.

Abstract

It is known for some time that a random graph $G(n,p)$ contains w.h.p. a Hamiltonian cycle if $p$ is larger than the critical value $p_{crit}= (\log n + \log \log n + \omega_n)/n$. The determination of a concrete Hamiltonian cycle is even for values much larger than $p_{crit}$ a nontrivial task. In this paper we consider random graphs $G(n,p)$ with $p$ in $\tilde{\Omega}(1/\sqrt{n})$, where $\tilde{\Omega}$ hides poly-logarithmic factors in $n$. For this range of $p$ we present a distributed algorithm ${\cal A}_{HC}$ that finds w.h.p. a Hamiltonian cycle in $O(\log n)$ rounds. The algorithm works in the synchronous model and uses messages of size $O(\log n)$ and $O(\log n)$ memory per node.