Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations
We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least n/2, where n denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in polynomial time in the classical centralized model.
This paper presents a randomized distributed CONGEST algorithm that finds w.h.p. a Hamiltonian cycle (as well as maximum matching) within O(log n) rounds under the promise that the input graph is a Dirac graph. This upper bound is in contrast to general graphs in which both the decision and search variants of Hamiltonicity require Ω̃(n²) rounds, as shown by Bachrach et al. [PODC'19].
In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL'05]. In Ore graphs, the sum of the degrees of every pair of non-adjacent vertices is at least n, and in Rahman-Kaykobad graphs, the sum of the degrees of every pair of non-adjacent vertices plus their distance is at least n+1. We show how our algorithm for Dirac graphs can be adapted to work for these more general families of graphs.
the CONGEST model
Hamiltonian Path
Hamiltonian Cycle
Dirac graphs
Ore graphs
graph-algorithms
Theory of computation~Distributed algorithms
Theory of computation~Graph algorithms analysis
19:1-19:14
Regular Paper
https://arxiv.org/abs/2302.00742
Noy
Biton
Noy Biton
Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
The author was supported by the Israel Science Foundation under Grant 1867/20.
Reut
Levi
Reut Levi
Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
https://orcid.org/0000-0003-3167-1766
The author was supported by the Israel Science Foundation under Grant 1867/20.
Moti
Medina
Moti Medina
Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel
https://orcid.org/0000-0002-5572-3754
The author was supported by the Israel Science Foundation under Grant 867/19.
10.4230/LIPIcs.MFCS.2023.19
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Noy Biton, Reut Levi, and Moti Medina
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