Parallel Tempering Algorithms: Optimal Allocation of Replicas to Processors

The optimal allocation of replicas (OAR) algorithm is a subroutine that can be used to minimize CPU idle time in parallel tempering simulations without substantially increasing the wall clock time of the simulation. It is freely distributed under the GNU Public License.

Why is it needed?

The parallel tempering algorithm is a form of Markov chain Monte Carlo simulation. The idea behind the parallel tempering technique is to sample n replica systems, each in the canonical ensemble, and each at a different temperature, Ti. Parallel tempering achieves good sampling by allowing systems to escape from low free energy minima by exchanging configurations with systems at higher temperatures, which are free to sample representative volumes of phase space. Parallel tempering is an exact method in statistical mechanics, in that it satisfies the detailed balance or balance condition, depending on the implementation. The detailed balance version of parallel tempering typically uses a form of the Metropolis Hastings update rule. The balance version is often used in multiprocessor simulations to which OAR applies.

Due to the need to satisfy the balance condition, the n different systems must be synchronized whenever a swap is attempted. This synchronization is in Monte Carlo steps, rather than in real, wall clock time. If we define the simulation events between each attempted swap as a Monte Carlo step, then at the beginning of each Monte Carlo step, each replica must have completed the same number of Monte Carlo steps and be available to swap configurations with other replicas. This constraint introduces a potentially large inefficiency in the simulation, as different replicas are likely to require different amounts of computational processing time in order to complete a Monte Carlo step. This inefficiency is not a problem on a single processor system, as a single processor will simply step through all the replicas to complete the Monte Carlo steps of each. This inefficiency is a significant problem on multi-processor machines, however, where individual CPUs can spend large quantities of time idling as they wait for other CPUs to complete the Monte Carlo steps of other replicas.

Traditionally, each processor on multi-processors machines has been assigned one replica in parallel tempering simulations. It is our experience that this type of assignment is generally highly inefficient, with typical CPU idle times of 40 to 60%. When one takes into account that the lower-temperature systems should have more moves per Monte Carlo step due to the increased correlation times, the idle time rises to intolerable levels that can approach 95%.


This algorithm can be incorporated into any parallel simulation code that is used for parallel tempering simulations. This algorithm provides optimum performance for a cluster of homogeneous processors. See the Appendix in Ref. [1] for the derivation of an algorithm for use with a cluster of heterogeneous processors. The scheduler algorithm requires that the simulator runs a short preliminary simulation for each of the n replica systems to determine the time per Monte Carlo move, α(Ti), where i = 1,..., n. The simulator must also choose a value for the number of Monte Carlo moves, Nmove(Ti), to be attempted per big Monte Carlo step for each of the n systems. (Note: It is often desirable to perform more configurational Monte Carlo moves per Monte Carlo step at lower temperatures because the correlation time is longer than it is at higher temperatures.)

The algorithm can allocate the replicas to the processors in two different ways so that the simulation is conducted either with minimum CPU idle time (theoretically 100% efficiency) or at minimum real wall clock time but with a small loss of efficiency (typically 3 to 8%).

An example of how the algorithm allocates replicas to processors is shown in the figures below. Figure a) shows the usual one-replica-per-processor allocation and Figure b) shows how the algorithm allocates the same jobs to only 12 processors but with only a small increase in real wall clock time (100% efficiency option).

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Download sources

The OAR algorithm is written in ANSI C (download scheduler.tar.gz, last updated 4 December 2003). The function to be called is scheduler. To make use of the subroutine

Also included in the download is an example program called call_test.c that calls scheduler and writes the output from the subroutine to the screen, and a makefile to make the executable for this example program.

We ask that you cite our paper [1] in any publications that result from the use of the scheduler subroutine.

Function call and returned values

When calling scheduler you need to pass as arguments the number of replicas, nsim, the option type, option, which should be set to 0 for maximum efficiency or 1 for minimum wall clock time, and α(Ti) and Nmove(Ti) for each of the replicas. The scheduler algorithm is declared as follows:

void scheduler(int nsim, int option, double *alpha,
               double *Nsteps, int *nproc, int *njobs, 
	       double job[nprocMAX][nsimMAX][2]);
where nprocMAX and nsimMAX are respectively the maximum number of homogeneous processors and replicas allowed in the code (set at 50 for both but can easily be altered in scheduler.h), α[nsimMAX] is a one dimensional array that contains the value of α(Ti) for each of the replicas, Nmove[nsimMAX] is a one dimensional array that contains the value of Nmove(Ti) for each of the replicas, nproc is the number of processors required to perform the simulation, njobs[nprocMAX] is a one dimensional array that contains the number of different replicas that will be simulated on each processor, and job is a three dimensional array that contains the main output from the algorithm.

job[i][j][0] gives the replica number of the jth + 1 job on processor i + 1, job[i][j][1] gives the length of time the processor spends on the replica, and job[i][j][2] gives the number of MC moves that the processor is responsible for on the replica. The maximum value of j to be accessed for each processor, i, is given by njobs[i]-1.


scheduler.c is unsupported software.


1) D. J. Earl and M. W. Deem, ``Optimal Allocation of Replicas to Processors in Parallel Tempering Simulations,'' part of Hans C. Andersen Festschrift, 108 (2004) 6844-6849 (Reprint).