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

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 ${\mathcal{A}}_{HC}$ 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.

17 pages; 4 figures