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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| Veröffentlicht in: | Algorithmica Jg. 65; H. 4; S. 772 - 801 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Ozlen, Melih Burton, Benjamin A. |
| Author_xml | – sequence: 1 givenname: Benjamin A. surname: Burton fullname: Burton, Benjamin A. email: bab@maths.uq.edu.au organization: School of Mathematics and Physics, The University of Queensland – sequence: 2 givenname: Melih surname: Ozlen fullname: Ozlen, Melih organization: School of Mathematical and Geospatial Sciences, RMIT University |
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| Keywords | Backtracking 52B55 Normal surfaces Vertex enumeration Linear programming 90C05 57N35 57N10 Tree traversal |
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| References | Hass, Lagarias, Pippenger (CR24) 1999; 46 Megiddo (CR33) 1986; 35 Matveev (CR31) 2003 Tollefson (CR39) 1998; 183 CR16 CR11 Smale (CR36) 1983; 27 Rubinstein (CR35) 1995 Burton (CR13) 2011; 118 Coulson, Goodman, Hodgson, Neumann (CR15) 2000; 9 Avis, Bremner, Seidel (CR2) 1997; 7 Burton (CR8) 2004; 13 de Ghellinck, Vial (CR17) 1986; 1 Kleiner, Lott (CR29) 2008; 12 Bazaraa, Jarvis, Sherali (CR5) 2010 Burton, Rubinstein, Tillmann (CR14) 2012; 364 Karmarkar (CR28) 1984; 4 Jaco, Oertel (CR27) 1984; 23 Bland (CR7) 1977; 2 Haken (CR23) 1962; 80 McMullen (CR32) 1970; 17 Burton (CR10) 2010 Jaco, Letscher, Rubinstein (CR26) 2002 Fukuda, Liebling, François (CR19) 1997; 8 Fukuda, Prodon (CR20) 1996 Avis (CR1) 2000 Kneser (CR30) 1929; 38 Birman (CR6) 1980; 27 Burton (CR12) 2010; 79 Haken (CR22) 1961; 105 CR25 CR21 Terzer, Stelling (CR37) 2010 Tillmann (CR38) 2008; 54 Dyer (CR18) 1983; 8 Balinski (CR4) 1961; 9 Avis, Fukuda (CR3) 1992; 8 Burton (CR9) 2009; 9 Ziegler (CR40) 1995 Motzkin, Raiffa, Thompson, Thrall, Kuhn, Tucker (CR34) 1953 D. Coulson (9645_CR15) 2000; 9 S. Smale (9645_CR36) 1983; 27 B.A. Burton (9645_CR9) 2009; 9 W. Haken (9645_CR23) 1962; 80 H. Kneser (9645_CR30) 1929; 38 M.E. Dyer (9645_CR18) 1983; 8 B.A. Burton (9645_CR10) 2010 J.S. Birman (9645_CR6) 1980; 27 S. Matveev (9645_CR31) 2003 M.S. Bazaraa (9645_CR5) 2010 T.S. Motzkin (9645_CR34) 1953 9645_CR11 W. Haken (9645_CR22) 1961; 105 9645_CR16 D. Avis (9645_CR2) 1997; 7 R.G. Bland (9645_CR7) 1977; 2 9645_CR21 D. Avis (9645_CR3) 1992; 8 B.A. Burton (9645_CR12) 2010; 79 J.H. Rubinstein (9645_CR35) 1995 W. Jaco (9645_CR27) 1984; 23 G. Ghellinck de (9645_CR17) 1986; 1 B. Kleiner (9645_CR29) 2008; 12 S. Tillmann (9645_CR38) 2008; 54 G.M. Ziegler (9645_CR40) 1995 D. Avis (9645_CR1) 2000 N. Karmarkar (9645_CR28) 1984; 4 M.L. Balinski (9645_CR4) 1961; 9 J. Hass (9645_CR24) 1999; 46 W. Jaco (9645_CR26) 2002 K. Fukuda (9645_CR20) 1996 9645_CR25 P. McMullen (9645_CR32) 1970; 17 B.A. Burton (9645_CR13) 2011; 118 K. Fukuda (9645_CR19) 1997; 8 B.A. Burton (9645_CR14) 2012; 364 M. Terzer (9645_CR37) 2010 B.A. Burton (9645_CR8) 2004; 13 N. Megiddo (9645_CR33) 1986; 35 J.L. Tollefson (9645_CR39) 1998; 183 |
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| SubjectTerms | 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 |
| Title | A Tree Traversal Algorithm for Decision Problems in Knot Theory and 3-Manifold Topology |
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