Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P

Abstract

Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is is decidable in polynomial time. Before that the best known upper bound was non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. 3. The Word Problem of Higman's group is decidable in polynomial time. It is due to the last result that we were forced to advance the theory of power circuits. Thanks: Part of this work was done when the first two authors visited the Stevens Institute of Technology in September 2010 and March 2011. The support and the hospitality of the Stevens Institute is greatly acknowledged. The work of the third author was partially supported by NSF grant DMS-0914773. © V. Diekert, J. Laun, and A. Ushakov.