Finding Hamilton cycles in random graphs with few queries

Random Structures & Algorithms, 2014

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph G = G δ≥3 n,m . In this model G is drawn uniformly from graphs with vertex set [n], m edges and minimum degree at least three. We focus on the case where m = cn for constant c. If c is sufficiently large then our algorithm runs in O(n 1+o(1) ) time and succeeds w.h.p. * Research supported in part by NSF Grant CCF2013110 algorithm that reduces this to n 1.5+o(1) for sufficiently large c. The main aim of this paper is to construct an almost linear time algorithm for this model. Theorem 1.1. If c is sufficiently large then our algorithm finds a Hamilton cycle in G δ≥3 n,m , m = cn, and runs in O(n 1+o(1) ) time and succeeds w.h.p.

The Mathematica Journal, 2011

Determining whether Hamiltonian cycles exist in graphs is an NP-complete problem, so it is no wonder that the Combinatorica function HamiltonianCycle is slow for large graphs. Theorems by Dirac, Ore, Pósa, and Chvátal provide sufficient conditions that are easy to check for the existence of such cycles. This article provides Mathematica programs for those conditions, thus extending the capability of HamiltonianQ, which only tests the biconnectivity-a simple necessary condition-of a given graph. We also investigate experimentally the limiting behavior of whether the conditions are fulfilled for large random graphs. The phenomenon seen is proved as a theorem, closely related to earlier results by Karp and Pósa. ‡ Introduction

SIAM Journal on Discrete Mathematics, 2013

We prove that the number of Hamilton cycles in the random graph G(n, p) is n!p n (1 + o(1)) n a.a.s., provided that p ≥ ln n+ln ln n+ω n . Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates ln n e n (1+o(1)) n Hamilton cycles a.a.s.

ArXiv, 2018

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.

1997

We present a randomized algorithm that computes a simple cycle of length k in general graphs, where k is a xed integer, in O(maxfm; n logng) expected time. This algorithm can be derandomized with only a small loss in e ciency, yielding a deterministic algorithm for this task which runs in O(maxfm log n; n log ng) worst-case time. We show that the randomized algorithm may be parallelized. These algorithms improve upon previous results of many authors. Furthermore, we answer the question of AYZ 94], whether deciding if a given graph contains a triangle is as di cult as boolean multiplication of two n by n matrices, in the negative.

arXiv: Data Structures and Algorithms, 2018

We present fast and efficient randomized distributed algorithms to find Hamiltonian cycles in random graphs. In particular, we present a randomized distributed algorithm for the G(n, p) random graph model, with number of nodes n and p = c ln n n δ (for any constant 0 < δ ≤ 1 and for a suitably large constant c > 0), that finds a Hamiltonian cycle with high probability inÕ(n δ) rounds. 1 Our algorithm works in the (synchronous) CONGEST model (i.e., only O(log n)sized messages are communicated per edge per round) and its computational cost per node is sublinear (in n) per round and is fully-distributed (each node uses only o(n) memory and all nodes' computations are essentially balanced). Our algorithm improves over the previous best known result in terms of both the running time as well as the edge sparsity of the graphs where it can succeed; in particular, the denser the random graph, the smaller is the running time.

Combinatorics, Probability and Computing, 2014

Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can be decomposed into O(n) cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph G(n, p) with probability approaching 1 as n → ∞. In this paper we show that for most edge probabilities G(n, p) can be decomposed into a union of n/4 + np/2 + o(n) cycles and edges w.h.p. This result is asymptotically tight.

Proceedings of the fifth annual ACM symposium on Parallel algorithms and architectures - SPAA '93, 1993

We give tight bounds on the parallel complexity of some problems involving random graphs. Speci cally, w e show that a Hamiltonian cycle, a breadth rst spanning tree, and a maximal matching can all be constructed in log n expected time using n= log n processors on the CRCW PRAM. This is a substantial improvement o ver the best previous algorithms, which required log log n 2 time and n log 2 n processors. We then introduce a technique which allows us to prove that constructing an edge cover of a random graph from its adjacency matrix requires log n expected time on a CRCW PRAM with On processors. Constructing an edge cover is implicit in constructing a spanning tree, a Hamiltonian cycle, and a maximal matching, so this lower bound holds for all these problems, showing that our algorithms are optimal. This new lower bound technique is one of the very few lower bound techniques known which apply to randomized CRCW PRAM algorithms, and it provides the rst nontrivial parallel lower bounds for these problems.

The Electronic Journal of Combinatorics, 2015

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K&gt;0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for g...

Combinatorial and Algorithmic Aspects of Networking (Proc. CAAN 2004), 2004

In this paper, we present a distributed algorithm to find Hamiltonian cycles in G(n, p) graphs. The algorithm works in a synchronous distributed setting. It finds a Hamiltonian cycle in G(n, p) with high probability when $p = \omega(\sqrt{\log n}/n^{1/4})$, and terminates in linear worst-case number of pulses, and in expected $O(n^{3/4+\epsilon})$ pulses. The algorithm requires, in each node of the network, only O(n) space and O(n) internal instructions.