A Tree Traversal Algorithm for Decision Problems in Knot Theory and 3-Manifold Topology

In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms...

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Bibliographic Details
Published inAlgorithmica Vol. 65; no. 4; pp. 772 - 801
Main Authors Burton, Benjamin A., Ozlen, Melih
Format Journal Article
LanguageEnglish
Published New York Springer-Verlag 01.04.2013
Subjects
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Mathematics of Computing
Theory of Computation
Backtracking
52B55
Normal surfaces
Vertex enumeration
Linear programming
90C05
57N35
57N10
Tree traversal
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ISSN0178-4617
1432-0541
DOI10.1007/s00453-012-9645-3

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Summary:In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been variants of the classical double description method. In this paper we present the first practical normal surface enumeration algorithm that breaks out of the double description paradigm. This new algorithm is based on a tree traversal with feasibility and domination tests, and it enjoys a number of advantages over the double description method: incremental output, significantly lower time and space complexity, and a natural suitability for parallelisation. Experimental comparisons of running times are included.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-012-9645-3