Optimistic Parallel Discrete Relaxation* Kinson Ho and Paul N. Hilfinger Computer Science Division University of California at Berkeley, U. S. A. Hans W. Guesgen Computer Science Department University of Auckland, New Zealand {ho,hilfingr}@cs.berkeley.edu, hans@cs.auckland.ac.nz Abstract Discrete relaxation is frequently used to com- pute the fixed point of a discrete system where / is monotonic with respect to some partial order Given an appropriate initial value for X, discrete relaxation repeats the assignment until a fixed point for / is found. Monotonicity of / with respect to is a sufficient (but in general not neces- sary) condition for iterative, hill-climbing tech- niques such as discrete relaxation to find the fixed point of /. In this paper we introduce monotonic asyn- chronous iteration as a novel way of imple- menting parallel discrete relaxation in prob- lem domains for which monotonicity is a neces- sary condition. This is an optimistic technique that maintains monotonicity without limiting concurrency, resulting in good parallel perfor- mance. We illustrate this technique with the parallel implementation of a constraint satisfac- tion system that computes globally consistent solutions, and present performance numbers for experiments on a shared-memory implementa- tion. The performance numbers show that it is indeed possible to obtain a reasonable speedup when parallelizing global constraint satisfac- tion. We believe that monotonic asynchronous iteration is applicable to parallel discrete relax- ation in general. 1 Introduction Discrete relaxation is frequently used to compute the fixed point of a discrete system where / is monotonic with respect to some partial order <. Given *Ho and Hilfinger are supported by NSF Grant CCR-84- 51213. Guesgen performed part of this work while at the Ger- man National Research Center for Computer Science (GMD). in St. Augustin, Germany, and the International Computer Science Institute in Berkekey, California. At the GMD he was supported by the German Federal Ministry for Research and Technology (BMFT) within the joint projects TEX-B (grant ITW8506D) and TASSO (grant ITW8900A7). an appropriate initial value for X, discrete relaxation re- peats the assignment until a fixed point for / is found. Monotonicity of / with respect to < is a suf- ficient (but in general not necessary) condition for itera- tive, hill-climbing techniques such as discrete relaxation to find the fixed point of / [Parker, 1987]. Discrete relaxation is widely used in the solution of constraint satisfaction problems (CSPs), and many par- allel implementations of discrete relaxation for CSPs have been reported [Kasif and Rosenfeld, 1983; Rosen- feld et a/., 1976]. These attempts have all focused on CSP solvers that compute locally consistent (arc con- sistent) solutions, which are relatively straightforward to parallelize as the computations are inherently mono- tonic. On the other hand, discrete relaxation algorithms used in CSP solvers that compute globally consistent so- lutions are very difficult to parallelize because for this class of problems, monotonicity is a necessary correct- ness condition that is not automatically satisfied. The need to maintain monotonicity (for correctness) often limits the amount of concurrency available in a paral- lel implementation, degrading the performance signifi- cantly. In this paper we introduce monotonic asynchronous iteration as a novel way of implementing parallel dis- crete relaxation in problem domains for which mono- tonicity is a necessary condition. This is an optimistic technique that maintains monotonicity without limiting concurrency, resulting in good parallel performance. We illustrate this technique with the parallel implementa- tion of CONSAT [Guesgen, 1989], a constraint satisfac- tion system that computes globally consistent solutions, and describes an experiment on a shared-memory im- plementation. The performance numbers show that it is indeed possible to obtain a reasonable speedup when par- allelizing global constraint satisfaction, and thus proving that Kasif's conjecture is correct [1990]. We believe that monotonic asynchronous iteration is applicable to paral- lel discrete relaxation in general. 2 Discrete Relaxation Consider the problem of finding the fixed point of a dis- crete system . For our purposes, the system has the following properties: 268 Constraint Satisfaction Problems