N ov 2 00 5 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 variables-nodes can decide to freeze in a self-organized way, each variable taking its decision on the basis of purely local information. The second solver, called the “SP reinforcement algorithm”, makes use of time-dependent external forcings on each variable, which are adapted in time in such a way that the algorithm approaches its estimated closest solution. Both methods allow 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.  Construction and VHDL Implementation of a Fully Local Network with Good Reconstruction Properties of the Inputs , 2005, IWINAC.

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

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

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

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

[6]  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.

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

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

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

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

[11]  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).

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

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