A Dynamic Programming Based Method for Optimum Linear Decomposition of Index Generation Functions

Abstract

The problem addressed in this paper is minimization of the number of linear functions in a linear decomposition. This paper proposes an exact minimization method based on dynamic programming for index generation functions. The proposed method searches for an optimum solution while recursively dividing an index set of a given index generation function. To use partial solutions efficiently in solution search, the proposed method represents partitions of an index set compactly and uniquely by zero-suppressed binary decision diagrams (ZDDs). Existing methods based on a branch-and-bound approach search for a solution sequentially in a depth-first manner. On the other hand, the proposed method searches for multiple partial solutions in parallel in a breadth-first manner. Thus, once a solution is found, we can terminate the search process. This is because the depth of searches corresponds to the number of linear functions. Experimental results using benchmark index generation functions show the effectiveness of the proposed method.