Computing minimal triangulations in time O(nα log n) = o(n2.376)

  • Authors:

  • Pinar Heggernes;Jan Arne Telle;Yngve Villanger

  • Affiliations:

  • University of Bergen, Bergen, Norway;University of Bergen, Bergen, Norway;University of Bergen, Bergen, Norway

  • Venue:
  • SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2005

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Abstract

The problem of computing minimal triangulations of graphs, also called minimal fill, was introduced and solved in 1976 by Rose, Tarjan, and Lueker [17] in time O(nm), thus O(n3) for dense graphs. Although the topic has received increasing attention since then, and several new results on characterizing and computing minimal triangulations have been presented, this first time bound has remained the best. In this paper we introduce an O(n α log n) time algorithm for computing minimal triangulations, where O(nα) is the time required to multiply two n × n matrices. The current best known α is less than 2.376, and thus our result breaks the long standing asymptotic time complexity bound for this problem. To achieve this result, we introduce and combine several techniques that are new to minimal triangulation algorithms, like working on the complement of the input graph, graph search for a vertex set A that bounds the size of the connected components when A is removed, and matrix multiplication.