文章基本信息

标题：Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P
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作者：Volker Diekert ; J{\"u}rn Laun ; Alexander Ushakov 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2012
卷号：14
页码：218-229
DOI：10.4230/LIPIcs.STACS.2012.218
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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 in P. 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. (The algorithm has been implemented and is available in the CRAG library.) 3. The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of Power Circuits.
关键词：Algorithmic group theory; Data structures; Compression; Word Problem