## 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.

### Citations

193 | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra, Discrete and Computational Geometry 8
- Avis, Fukuda
- 1992
(Show Context)
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... |

152 |
The maximum numbers of faces of a convex polytope
- McMullen
- 1970
(Show Context)
Citation Context ...size of the output), which for bounded polytopes can grow exponentially large in the polytope dimension. Specifically, for a polytope of dimension p with f facets, the upper bound theorem of McMullen =-=[27]-=- shows the worst case to be ( 1 ) ( 1 ) f −⌈ f −⌊ number of vertices ≤ 2 p⌉ ⌊ 1 2 p⌋ + 2 p⌋−1 ⌈ 1 2 p⌉−1 As an exercise, we can estimate this upper bound in the context of normal surface enumeration. ... |

87 | How good are convex hull algorithms
- Avis, Bremner, et al.
- 1997
(Show Context)
Citation Context ...s over subsets S+,S− ⊆ Vi−1, and memory consumption because the entire vertex set Vi must be computed and stored at each stage, ready for use in the subsequent iteration of the main loop. Avis et al. =-=[2]-=- present a series of heuristic options for this ordering and proceed to manufacture cases in which each of them performs poorly; Fukuda and Prodon [12] also experiment with different heuristic orderin... |

69 |
Geschlossene flächen in dreidimensionalen mannigfaltigkeiten. Jahresbericht der Deut
- Kneser
- 1929
(Show Context)
Citation Context ...ion of S 2 × S 1 topic. Thanks must also go to the University of Melbourne for their continued support of the development of Regina. 2. Normal surface theory Normal surfaces were introduced by Kneser =-=[25]-=-, and further developed by Haken [13, 14] for use with the unknot recognition problem and the homeomorphism problem. They are now commonplace in recognition and decomposition algorithms and more recen... |

58 | lrs: A revised implementation of the reverse search vertex enumeration algorithm
- Avis
(Show Context)
Citation Context ...l 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 algorithms typically require only a small number of poly... |

58 | The computational complexity of knot and link problems
- Hass, Lagarias, et al.
- 1999
(Show Context)
Citation Context ...e offer very little topological background, concentrating instead on the linear programming aspects of normal surface theory. For a more extensive review of normal surfaces, the reader is referred to =-=[15]-=- or [16]. 2.1. Triangulations and normal surfaces. The key topological structures that we work with in this paper are triangulations and normal surfaces. We proceed to define each of these in turn. Tr... |

56 | Theorie der Normalflächen - Haken - 1961 |

54 |
Algorithms for the complete decomposition of a closed 3-manifold
- Jaco, Tollefson
- 1995
(Show Context)
Citation Context ...de 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 Oertel [19] and Hemion [16]. Several recurring themes are found in these and many... |

50 |
An algorithm to decide if a 3-manifold is a Haken manifold
- Jaco, Oertel
- 1984
(Show Context)
Citation Context ... 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 Oertel =-=[19]-=- and Hemion [16]. Several recurring themes are found in these and many other topological algorithms: (i) they are extremely slow, (ii) they are extremely difficult to implement, and (iii) they are all... |

46 | 0–efficient triangulations of 3-manifolds
- Jaco, Rubinstein
- 1998
(Show Context)
Citation Context ...e unknot recognition problem and the homeomorphism problem. They are now commonplace in recognition and decomposition algorithms and more recently have found applications in simplification algorithms =-=[20]-=-. From a practical perspective, many of these algorithms are extremely messy and difficult to implement, due to the complex geometric operations involved and the myriad of problematic cases. Some have... |

33 |
The complexity of vertex enumeration methods
- Dyer
- 1983
(Show Context)
Citation Context ...proachable also makes the resulting algorithms impractically slow for all but the smallest 3-manifolds. This is because vertex enumeration can grow exponentially slow in the dimension of the polytope =-=[11]-=-, which equates to exponentially slow in the complexity of the 3-manifold. Any practical implementation therefore requires a highly optimized vertex enumeration algorithm. Vertex enumeration algorithm... |

31 |
The double description method,” in “Contribution to the Theory of
- Motzkin, Raiffa, et al.
- 1953
(Show Context)
Citation Context ...tation therefore requires a highly optimized vertex enumeration algorithm. Vertex enumeration algorithms fall into two broad categories: those based on the double description method of Motzkin et al. =-=[28]-=-, and those based on pivoting, as described for example by Dyer [11]. Both classes of algorithms have been analyzed and optimized in the literature; see for instance the Received by the editor August ... |

30 | Symmetries, isometries and length spectra of closed hyperbolic three-manifolds
- Hodgson, Weeks
- 1994
(Show Context)
Citation Context ...ayered loops are used for extreme “stress testing”. In detail: • The closed hyperbolic triangulations are drawn arbitrarily from the Hodgson-Weeks census of small-volume closed hyperbolic 3-manifolds =-=[17]-=-. These include six smaller cases (9 ≤ n ≤ 13) for use with standard coordinates and five larger cases (16 ≤ n ≤ 20) for use with quadrilateral coordinates. • An n-tetrahedron twisted layered loop is ... |

28 | What Every Programmer Should Know About Memory
- Drepper
(Show Context)
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... |

27 |
The classification of knots and 3-dimensional spaces
- Hemion
- 1992
(Show Context)
Citation Context ...nd 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 Oertel [19] and Hemion =-=[16]-=-. Several recurring themes are found in these and many other topological algorithms: (i) they are extremely slow, (ii) they are extremely difficult to implement, and (iii) they are all based on normal... |

27 | A census of cusped hyperbolic 3-manifolds
- Callahan, Hildebrand, et al.
- 1999
(Show Context)
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. Proof The following argument assumes the double description method is used with vertex filtering. For the non-filtered case, simply remove all references to fil... |

24 |
An algorithm to recognize the 3-sphere
- Rubinstein
- 1994
(Show Context)
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... |

20 |
Thin position and the recognition problem for S 3
- Thompson
- 1994
(Show Context)
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... |

17 |
Über das Homöomorphieproblem der 3-Mannigfaltigkeiten
- Haken
- 1962
(Show Context)
Citation Context ...instein 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 Oertel [19] and Hemion [16]. Several recurring themes are found in these and many other topological algorithms: (i) they are extremely slow, (ii) they are extremely difficul... |

16 |
Regina (normal surface and 3-manifold topology software
- Burton
- 1999
(Show Context)
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 ... |

11 |
Double Description Method Revisited”, in Combinatorics and Computer
- Fukuda, Prodon
- 1996
(Show Context)
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... |

11 | Normal Surface QTheory
- Tollefson
- 1998
(Show Context)
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... |

11 |
Algorithms for essential surfaces
- Jaco, Letscher, et al.
- 2002
(Show Context)
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 ... |

10 |
Rubinstein: Ideal triangulations of 3–manifolds I; spun normal surface theory
- Kang, Hyam
- 2004
(Show Context)
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... |

9 |
A census of cusped hyperbolic
- Callahan, Hildebrand, et al.
- 1999
(Show Context)
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... |

9 | Normal surfaces in topologically finite 3-manifolds. L’Ens
- Tillmann
(Show Context)
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... |

8 |
the 3-manifold topology software, Experiment
- Regina
(Show Context)
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 ... |

7 | Theorie der Normalflächen, Acta Mathematica 105 - Haken - 1961 |

6 | Algorithms for Essential Surfaces in 3Manifolds
- Jaco, Letscher, et al.
- 2002
(Show Context)
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 ... |

6 |
Tables of 3-Manifolds up to Complexity 6.” Max-Planck-Institut für
- Matveev
- 1998
(Show Context)
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 ≤... |

4 |
minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds, Geometric Topology
- Polyhedral
- 1993
(Show Context)
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 |
FXrays: Extremal ray enumeration software, http://www
- Culler, Dunfield
(Show Context)
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 |
Khaled Elbassioni, and Vladimir Gurvich, Generating all vertices of a polyhedron is hard, Discrete Comput
- Khachiyan, Boros, et al.
(Show Context)
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... |

1 | E-mail address: bab@debian.org - Warren, delight, et al. - 2002 |