Parallel Symmetry-Breaking in Sparse Graphs

This paper describes efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. The primary technique allows us to 3-color a rooted tree in $O( \lg^* n )$ time on an EREW PRAM using a linear number of processors. These techniques are used to construct fast linear processor algorithms for several problems, including the problem of $( \Delta + 1)$-coloring constant-degree graphs and 5-coloring planar graphs. Lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs are also proved.

[1]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[2]  Maurice Queyranne,et al.  A network flow solution to some nonlinear 0-1 programming problems, with applications to graph theory , 1982, Networks.

[3]  Robert E. Tarjan,et al.  Two linear-time algorithms for five-coloring a planar graph , 1980 .

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

[5]  Richard Cole,et al.  Parallel merge sort , 1988, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[6]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[7]  Pierre A. Humblet,et al.  A Distributed Algorithm for Minimum-Weight Spanning Trees , 1983, TOPL.

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

[9]  Johan Håstad,et al.  Optimal bounds for decision problems on the CRCW PRAM , 1987, STOC.

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

[11]  Amos Israeli,et al.  An Improved Parallel Algorithm for Maximal Matching , 1986, Inf. Process. Lett..

[12]  S. Cook,et al.  Lower bounds in parallel machine computation , 1987 .

[13]  Steven Fortune,et al.  Parallelism in random access machines , 1978, STOC.

[14]  Frank Harary,et al.  Graph Theory , 2016 .

[15]  Mark K. Goldberg,et al.  A new parallel algorithm for the maximal independent set problem , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[16]  John H. Reif,et al.  An optimal parallel algorithm for integer sorting , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[17]  Gary L. Miller,et al.  Parallel tree contraction and its application , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[18]  Norishige Chiba,et al.  A Linear 5-Coloring Algorithm of Planar Graphs , 1981, J. Algorithms.

[19]  Gregory E. Shannon.  Parallel Independent Set Algorithms for Sparse Graphs , 1986 .

[20]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1984, STOC '84.

[21]  Bruce M. Maggs,et al.  Communication-Efficient Parallel Graph Algorithms , 1986, ICPP.

[22]  Allan Borodin,et al.  Routing, merging and sorting on parallel models of computation , 1982, STOC '82.

[23]  Philip N. Klein,et al.  An efficient parallel algorithm for planarity , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).