4-(<i>N<SUP>2</SUP></i>-1) puzzle: parallelization and performance on clusters

Abstract

In this paper, an analysis of the 4-(<i>N<SUP>2</SUP></i>-1) Puzzle, which is a generalization of the (<i>N<SUP>2</SUP></i>-1) Puzzle, is presented. This problem is of interest due to its algorithmic and computational complexity and its applications to robot movements with several objectives. Taking the formal definition as a starting point, 4 heuristics that can be used to predict the best achievable objective and to estimate the number of steps required to reach a solution state from a given configuration are analyzed. By selecting the objective, a sequential and parallel solution over a cluster is presented for the (<i>N<SUP>2</SUP></i>-1) Puzzle, based on the heuristic search algorithm A*. Also, variations of the classic heuristic are analyzed. The experimental work focuses on analyzing the possible superlinearity and the scalability of the parallel solution on clusters, by varying the physical configuration and the dimension of the problem. Finally, the suitability of the heuristic used to assess the best achievable objective in the 4-(<i>N<SUP>2</SUP></i>-1) Puzzle is analyzed.