Fast and Efficient Distributed Computation of Hamiltonian Cycles in Random Graphs

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 in Õ(n^δ) rounds. 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.

[1]  Béla Bollobás,et al.  The Diameter of Random Graphs , 1981 .

[2]  Béla Bollobás,et al.  Random Graphs , 1985 .

[3]  Colin McDiarmid,et al.  Algorithmic theory of random graphs , 1997 .

[4]  Hartmut Klauck,et al.  Can quantum communication speed up distributed computation? , 2012, PODC.

[5]  Linyuan Lu,et al.  Complex Graphs and Networks (CBMS Regional Conference Series in Mathematics) , 2006 .

[6]  Victor Klee,et al.  Diameters of Random Graphs , 1981, Canadian Journal of Mathematics.

[7]  Guy Louchard,et al.  A distributed algorithm to find Hamiltonian cycles in G(n, p) random graphs , 2005 .

[8]  B. Bollobás.  The evolution of random graphs , 1984 .

[9]  Alan M. Frieze,et al.  Finding hamilton cycles in sparse random graphs , 1987, J. Comb. Theory, Ser. B.

[10]  B. Bollobás,et al.  An algorithm for finding hamilton paths and cycles in random graphs , 1987 .

[11]  Quentin F. Stout,et al.  Optimal parallel construction of Hamiltonian cycles and spanning trees in random graphs , 1993, SPAA '93.

[12]  Leslie G. Valiant,et al.  Fast probabilistic algorithms for hamiltonian circuits and matchings , 1977, STOC '77.

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[14]  Guy Louchard,et al.  A Distributed Algorithm to Find Hamiltonian Cycles in Random Graphs , 2004, CAAN.

[15]  Alan M. Frieze,et al.  On the number of hamilton cycles in a random graph , 1989, J. Graph Theory.

[16]  Mikhail J. Atallah,et al.  Algorithms and Theory of Computation Handbook , 2009, Chapman & Hall/CRC Applied Algorithms and Data Structures series.

[17]  Vijaya Ramachandran,et al.  The diameter of sparse random graphs , 2007, Random Struct. Algorithms.

[18]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[19]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[20]  Hartmut Klauck,et al.  Distributed Computation of Large-scale Graph Problems , 2015, SODA.

[21]  Sergei Vassilvitskii,et al.  A model of computation for MapReduce , 2010, SODA '10.

[22]  E. Palmer.  Graphical evolution: an introduction to the theory of random graphs , 1985 .

[23]  Alan M. Frieze.  Parallel Algorithms for Finding Hamilton Cycles in Random Graphs , 1987, Inf. Process. Lett..

[24]  Michael Krivelevich,et al.  On the Number of Hamilton Cycles in Sparse Random Graphs , 2013, SIAM J. Discret. Math..