Superlinear Speedup in Parallel State-Space Search

When N processors perform depth-first search on-disjoint parts of a state space tree to find a solution, the speedup can be superlinear (i.e., > N) or sublinear (i.e., < N) depending upon when a solution is first encountered in the space by one of the processors. It may appear that on the average, the speedup would be either linear or sublinear. Using an analytical model, we show that if the search space has more than one solution and if these solutions are randomly distributed in a relatively small region of the search space, then the average speedup in parallel depth-first search can be superlinear. If all the solutions (one or more) are uniformly distributed over the whole search space, then the average speedup is linear. This model is validated by our experiments on synthetic state-space trees and the 15-puzzle problem. The same model predicts average superlinear speedup in parallel best-first branchand-bound algorithms on suitable problems. 1 I n t r o d u c t i o n Consider the problem of finding a solution in a state-space tree containing one or more solutions [3,22,21]. Depth-first search (DFS) is a widely used technique for solving such problems[7,22]. A number of parallel formulations of depth-first search have been developed by various researchers [18,10,6,1,17,11]. In one such formulation[10], N processors concurrently perform depth-first search in disjoint parts of a state-space tree to find a solution in the search space. The parts of the state-space searched by different processors are determined dynamically, and are roughly of equal sizes. Since rally one solution is needed, the search terminates whenever any of the processors encounters a solution. Depending upon when a solution is first encountered in the space by the processors, the speedup can be superlinear (i.e., > N) or sublinear (i.e., < N) 1. This phenomenon of speedup being greater than N on N processors in isolated executions of parallel depth-first search has been reported by many researchers [6,18,17,1,25]. The speedup can differ greatly from one execution to another, as the actual parts of the search space searched by different processors are determined dynamically, and can be different for different executions. Hence *This work was supported by Army Research Office grant # DAAG29-84-K-0060 to the Artificial Intelligence Laboratory, and Office of Naval Research Grant N00014-86-K-0763 to the computer science department at the University of Texas at Aust'm. 1This phenomenon is also referred to as 'speedup anomalies'.