## OPTIMIZING THE DOUBLE DESCRIPTION METHOD FOR NORMAL SURFACE ENUMERATION

Citations: | 1 - 1 self |

### BibTeX

@MISC{Burton_optimizingthe,

author = {Benjamin A. Burton},

title = {OPTIMIZING THE DOUBLE DESCRIPTION METHOD FOR NORMAL SURFACE ENUMERATION},

year = {}

}

### OpenURL

### Abstract

Abstract. Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina. 1.

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Citation Context ...rders of magnitude as a result. We focus only on the double description method in this paper. Pivoting algorithms are certainly appealing, particularly because of their extremely low memory footprint =-=[1, 3]-=-. However, it is difficult to exploit the embedded surface constraints with these algorithms. We discuss this in more detail in Section 3. On a practical note, there are two well-known implementations... |

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Citation Context ... Section 6 concludes with a summary of our findings. The author is indebted to Bernard Blackham for his helpful suggestions regarding micro-optimization, and for highlighting the excellent references =-=[10, 34]-=- on thisOPTIMIZING THE DOUBLE DESCRIPTION METHOD 455 Figure 1. A two-tetrahedron triangulation of S 2 × S 1 topic. Thanks must also go to the University of Melbourne for their continued support of th... |

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Citation Context ...cally equivalent. Much progress has been made on these problems; notable examples include the unknot recognition algorithm of Haken [13], the 3-sphere recognition algorithm of Rubinstein and Thompson =-=[29, 30, 31]-=-, the connected sum and JSJ decomposition algorithms of Jaco and Tollefson [22], and the solution to the homeomorphism problem for Haken manifolds, developed by Haken [14] and completed by Jaco and Oe... |

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Citation Context ...thms. We discuss this in more detail in Section 3. On a practical note, there are two well-known implementations for the enumeration of normal surfaces: FXrays [9], by Culler and Dunfield, and Regina =-=[4, 5]-=-, by the author. Both are freely available under the GNU General Public License. David Letscher wrote a proof-of-concept program Normal in 1999 that preceded both implementations, but his software is ... |

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Citation Context ...gorithm, have never been implemented at all. Recent techniques have been developed to reduce both the difficulty and inefficiency of these algorithms; examples include Tollefson’s quadrilateral space =-=[33]-=-, the crushing method of Jaco and Rubinstein [20], and the “guts” analysis of Jaco et al. [18]. Since the focus of this paper is on the double description method, we offer very little topological back... |

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Citation Context ...oth the difficulty and inefficiency of these algorithms; examples include Tollefson’s quadrilateral space [33], the crushing method of Jaco and Rubinstein [20], and the “guts” analysis of Jaco et al. =-=[18]-=-. Since the focus of this paper is on the double description method, we offer very little topological background, concentrating instead on the linear programming aspects of normal surface theory. For ... |

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Citation Context ...n estimate this upper bound in the context of normal surface enumeration. Consider a one-vertex triangulation of a closed 3-manifold containing n tetrahedra. Extending a result of Kang and Rubinstein =-=[23]-=-, Tillmann [32] shows that the matching equations have a solution space of dimension 2n +1. Takingthe intersection with the projective hyperplane and the non-negative orthant in Rd , it follows that t... |

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Citation Context ... pseudo-separating hyperplanes as a result (and thereby strengthen our dimensional filtering). Indeed, this behaviour is observed for many ideal triangulations in the cusped census of Callahan et al. =-=[8]-=-. We proceed with a proof of Lemma 4.3.470 BENJAMIN A. BURTON Proof. The following argument assumes the double description method is used with vertex filtering. For the non-filtered case, simply remo... |

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Citation Context ...double description method. 453 c○2009 American Mathematical Society Reverts to public domain 28 years from publication454 BENJAMIN A. BURTON optimized double description methods of Fukuda and Prodon =-=[12]-=-, or the recent lexicographic pivoting method of Avis [1]. If we restrict our focus to topological problems, there are further gains to be had. Essentially, we can exploit the fact that normal surface... |

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Citation Context ... upper bound in the context of normal surface enumeration. Consider a one-vertex triangulation of a closed 3-manifold containing n tetrahedra. Extending a result of Kang and Rubinstein [23], Tillmann =-=[32]-=- shows that the matching equations have a solution space of dimension 2n +1. Takingthe intersection with the projective hyperplane and the non-negative orthant in Rd , it follows that the projective s... |

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Citation Context ...thms. We discuss this in more detail in Section 3. On a practical note, there are two well-known implementations for the enumeration of normal surfaces: FXrays [9], by Culler and Dunfield, and Regina =-=[4, 5]-=-, by the author. Both are freely available under the GNU General Public License. David Letscher wrote a proof-of-concept program Normal in 1999 that preceded both implementations, but his software is ... |

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Citation Context ...oth the difficulty and inefficiency of these algorithms; examples include Tollefson’s quadrilateral space [33], the crushing method of Jaco and Rubinstein [20], and the “guts” analysis of Jaco et al. =-=[18]-=-. Since the focus of this paper is on the double description method, we offer very little topological background, concentrating instead on the linear programming aspects of normal surface theory. For ... |

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Citation Context ...nates. • An n-tetrahedron twisted layered loop is an extremely well-structured triangulation of the quotient space S3 /Q4n. Twisted layered loops are conjectured by Matveev to have minimal complexity =-=[26]-=-, and a proof of this claim has recently been announced by Jaco, Rubinstein and Tillmann [21]. Here we include four smaller cases (9 ≤ n ≤ 18) for standard coordinates, and four larger cases (30 ≤ n ≤... |

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Citation Context ...cally equivalent. Much progress has been made on these problems; notable examples include the unknot recognition algorithm of Haken [13], the 3-sphere recognition algorithm of Rubinstein and Thompson =-=[29, 30, 31]-=-, the connected sum and JSJ decomposition algorithms of Jaco and Tollefson [22], and the solution to the homeomorphism problem for Haken manifolds, developed by Haken [14] and completed by Jaco and Oe... |

3 |
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Citation Context ...ed surface constraints with these algorithms. We discuss this in more detail in Section 3. On a practical note, there are two well-known implementations for the enumeration of normal surfaces: FXrays =-=[9]-=-, by Culler and Dunfield, and Regina [4, 5], by the author. Both are freely available under the GNU General Public License. David Letscher wrote a proof-of-concept program Normal in 1999 that preceded... |

2 |
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Citation Context ... rooted in the problem it aims to solve. Dyer shows that counting the vertices of an arbitrary polytope is NP-hard [11], and Khachiyan et al. show that vertex enumeration over a polyhedron is NP-hard =-=[24]-=-; these results do not bode well. At the very least, the running time is bounded below by the number of vertices (i.e., the size of the output), which for bounded polytopes can grow exponentially larg... |

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