On the Complexity of Distributed Network Decomposition

In this paper, we improve the bounds for computing a network decomposition distributively and deterministically. Our algorithm computes an (n?(n),n?(n))-decomposition innO(?(n))time, whereformula. As a corollary we obtain improved deterministic bounds for distributively computing several graph structures such as maximal independent sets and ?-vertex colorings. We also show that the class of graphs G whose maximum degree isnO(?(n))where ?(n)=1/log lognis complete for the task of computing a near-optimal decomposition, i.e., a (logn, logn)-decomposition, in polylog(n) time. This is a corollary of a more general characterization, which pinpoints the weak points of existing network decomposition algorithms. Completeness is to be intended in the following sense: if we have an algorithmAthat computes a near-optimal decomposition in polylog(n) time for graphs inG, then we can compute a near-optimal decomposition in polylog(n) time for all graphs.

[1]  Michael Luby.  Removing Randomness in Parallel Computation without a Processor Penalty , 1993, J. Comput. Syst. Sci..

[2]  Alessandro Panconesi.  Locality in Distributed Computing , 1993 .

[3]  Manhoi Choy,et al.  Efficient fault tolerant algorithms for resource allocation in distributed systems , 1992, STOC '92.

[4]  Richard Cole,et al.  Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking , 2018, Inf. Control..

[5]  Joan Boyar,et al.  Coloring Planar Graphs in Parallel , 1987, J. Algorithms.

[6]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1985, JACM.

[7]  Andrew V. Goldberg,et al.  Network decomposition and locality in distributed computation , 1989, 30th Annual Symposium on Foundations of Computer Science.

[8]  Howard J. Karloff.  A las vegas rnc algorithm for maximum matching , 1986, Comb..

[9]  Andrew V. Goldberg,et al.  Parallel Symmetry-Breaking in Sparse Graphs , 1988, SIAM J. Discret. Math..

[10]  David B. Shmoys,et al.  Efficient Parallel Algorithms for Edge Coloring Problems , 1987, J. Algorithms.

[11]  Michael E. Saks,et al.  Decomposing graphs into regions of small diameter , 1991, SODA '91.

[12]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[13]  Michael Luby.  A Simple Parallel Algorithm for the Maximal Independent Set Problem , 1986, SIAM J. Comput..

[14]  Lenore Cowen,et al.  Fast network decomposition , 1992, PODC '92.

[15]  Nancy A. Lynch,et al.  Upper Bounds for Static Resource Allocation in a Distributed System , 1981, J. Comput. Syst. Sci..

[16]  Eugene Styer,et al.  Improved algorithms for distributed resource allocation , 1988, PODC '88.

[17]  David K. Garnick,et al.  Locality in distributed computations , 1988, CSC '88.

[18]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[19]  Aravind Srinivasan,et al.  The local nature of Δ-coloring and its algorithmic applications , 1995, Comb..

[20]  Baruch Awerbuch,et al.  Complexity of network synchronization , 1985, JACM.

[21]  Joseph Naor,et al.  A Fast Parallel Algorithm to Color a Graph with Delta Colors , 1988, J. Algorithms.

[22]  Howard J. Karloff.  An NC Algorithm for Brooks' Theorem , 1989, Theor. Comput. Sci..

[23]  Baruch Awerbuch,et al.  Routing with Polynomial Communication-Space Trade-Off , 1992, SIAM J. Discret. Math..