Adaptive Flip Graph Algorithm for Matrix Multiplication

Yamato Arai, Yuma Ichikawa, Koji Hukushima

Abstract: This study proposes the “adaptive flip graph algorithm”, which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication. The adaptive flip graph algorithm addresses the inherent limitations of exploration and inefficient search encountered in the original flip graph algorithm, particularly when dealing with large matrix multiplication. For the limitation of exploration, the proposed algorithm adaptively transitions over the flip graph, introducing a flexibility that does not strictly reduce the number of multiplications. Concerning the issue of inefficient search in large instances, the proposed algorithm adaptively constraints the search range instead of relying on a completely random search, facilitating more effective exploration. In particular, a formal proof is provided that the introduction of plus transitions in the proposed algorithm ensures the connectivity of any node in the flip graph, which represents a method of matrix multiplication. Numerical experimental results demonstrate the effectiveness of the adaptive flip graph algorithm, which involves applying matrices calculated in characteristic 2. This algorithm reduces the number of multiplications for a 4 × 5 matrix multiplied by a 5 × 5 matrix from 76 to 73 and that for a 5 × 5 matrix multiplied by another 5 × 5 matrix from 95 to 94.