Survey-propagation decimation through distributed local computations

We discuss the implementation of two distributed solvers of the random K-SAT problem, based on some development of the recently introduced survey propagation (SP) algorithm. The first solver, called the ‘SP diffusion algorithm’, diffuses as dynamical information the maximum bias over the system, so that variable nodes can decide to freeze in a self-organized way, each variable making its decision on the basis of purely local information. The second solver, called the ‘SP reinforcement algorithm’, makes use of time-dependent external forcing messages on each variable, which are adapted in time in such a way that the algorithm approaches its estimated closest solution. Both methods allow us to find a solution of the random 3-SAT problem in a range of parameters comparable with the best previously described serialized solvers. The simulated time of convergence towards a solution (if these solvers were implemented on a fully parallel device) grows as log(N).

[1]  Riccardo Zecchina,et al.  Lossy data compression with random gates. , 2005, Physical review letters.

[2]  Riccardo Zecchina,et al.  Construction and VHDL Implementation of a Fully Local Network with Good Reconstruction Properties of the Inputs , 2005, IWINAC.

[3]  Thierry Mora,et al.  Clustering of solutions in the random satisfiability problem , 2005, Physical review letters.

[4]  R. Zecchina,et al.  Source coding by efficient selection of ground-state clusters. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Riccardo Zecchina,et al.  Survey propagation: An algorithm for satisfiability , 2002, Random Struct. Algorithms.

[6]  Riccardo Zecchina,et al.  Exact Probing of Glassy States by Survey Propagation , 2005 .

[7]  G. Parisi.  A backtracking survey propagation algorithm for K-satisfiability , 2003, cond-mat/0308510.

[8]  M. Mézard,et al.  Two Solutions to Diluted p-Spin Models and XORSAT Problems , 2003 .

[9]  Giorgio Parisi.  Some remarks on the survey decimation algorithm for K-satisfiability , 2003, ArXiv.

[10]  R. Monasson,et al.  Rigorous decimation-based construction of ground pure states for spin-glass models on random lattices. , 2002, Physical review letters.

[11]  M. Mézard,et al.  Random K-satisfiability problem: from an analytic solution to an efficient algorithm. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[13]  X. Jin.  Factor graphs and the Sum-Product Algorithm , 2002 .

[14]  R. Zecchina,et al.  Phase transitions in combinatorial problems , 2001 .

[15]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

[16]  Daniel A. Spielman,et al.  Improved low-density parity-check codes using irregular graphs and belief propagation , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[17]  Brendan J. Frey,et al.  Factor graphs and the sum-product algorithm , 2001, IEEE Trans. Inf. Theory.

[18]  Herman Schmit,et al.  Implementation of near Shannon limit error-correcting codes using reconfigurable hardware , 2000, Proceedings 2000 IEEE Symposium on Field-Programmable Custom Computing Machines (Cat. No.PR00871).

[19]  W. Freeman,et al.  Generalized Belief Propagation , 2000, NIPS.

[20]  D.J.C. MacKay,et al.  Good error-correcting codes based on very sparse matrices , 1997, Proceedings of IEEE International Symposium on Information Theory.

[21]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.