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Solving Difficult Instances of Boolean Satisfiability in the Presence of Symmetry
, 2002
"... Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large siz ..."
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Cited by 44 (17 self)
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Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large size, but are still solved in minutes. Yet, small and difficult SAT instances must exist if P##NP. To this end, our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from Very Large Scale Integration (VLSI) routing problems. With an efficient implementation to solve the graph automorphism problem [39, 50, 51], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.
Counting cases in marching cubes: Toward a generic algorithm for producing substitopes
 Proceedings of the 14th IEEE Visualization 2003 (VIS'03) 8
, 2003
"... Distinct cases of colorings for a square, assigning one color to each vertex. Top row: seven cases result from using four colors (fluid, bone, tissue, lesion) when applying Separating Surfaces to a square. Bottom row: thirteen cases result from using three colors (+ – =) when applying Marching Cubes ..."
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Cited by 14 (1 self)
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Distinct cases of colorings for a square, assigning one color to each vertex. Top row: seven cases result from using four colors (fluid, bone, tissue, lesion) when applying Separating Surfaces to a square. Bottom row: thirteen cases result from using three colors (+ – =) when applying Marching Cubes to a square. This paper describes a technique for counting the cases that arise in a family of visualization techniques. This family includes Marching Cubes, Sweeping Simplices, Contour Meshing, Interval Volumes, and Separating Surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes (geometric substitutions of polytopes). To count the cases, we observe that discrete “color ” values are assigned to the vertices of a polytope. A group of symmetries acts on the colorings to produce equivalence classes called orbits, each of which corresponds to a single case. Using a software system (“GAP”) for computational group theory, we calculate the cases that arise in the family of visualization techniques. These casecounts are organized into a table that provides a taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculation confirms previously reported casecounts for large dimensions that are too large to check by hand, and predicts the number of cases that will arise in algorithms that have not yet been invented.
Simple Groups in Computational Group Theory
 INTERNATIONAL CONGRESS OF MATHEMATICANS
, 1998
"... This note describes recent research using structural properties of finite groups to devise efficient algorithms for group computation. ..."
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Cited by 12 (2 self)
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This note describes recent research using structural properties of finite groups to devise efficient algorithms for group computation.
Polynomialtime computation in matrix groups
, 1999
"... This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods that e ..."
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Cited by 4 (3 self)
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This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods that enable such efficiency.
Construction of Co3. An example of the use of an integrated system for computational group theory
 Groups St Andrews 1997 in
, 1999
"... This paper aims to demonstrate, by example, a small sample of the capabilities of the GAP system [S+ 97] for computational algebra. We specifically focus on the advantages arising from the use of an integrated system such as GAP, which allows the easy combination of techniques ..."
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Cited by 2 (0 self)
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This paper aims to demonstrate, by example, a small sample of the capabilities of the GAP system [S+ 97] for computational algebra. We specifically focus on the advantages arising from the use of an integrated system such as GAP, which allows the easy combination of techniques
Quadrilaterals Subdivided by Triangles in the Hyperbolic Plane
 Plane, RoseHulman Math. Sci. Tech Rep
, 1998
"... In this paper, we consider trianglequadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. These tilings are called divisible tilings or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condit ..."
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Cited by 1 (1 self)
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In this paper, we consider trianglequadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. These tilings are called divisible tilings or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condition, in which a single triangle occurs at each vertex of the quadrilateral. All possible such divisible tilings are catalogued as well as determining the minimal genus surface on which the divisible tiling exists. The tiling groups of these surfaces are also determined. 1. Introduction Let S be a surface which is a twodimensional object such as the covering of a sphere or torus. Every surface may be represented as a sphere with handles where is the genus of the surface. For example, a sphere (Figure 1) has genus 0 while a torus (Figure 2) has genus 1. A genus 2 surface is pictured in Figure 3. A surface can be tiled if it can be completely covered by a series of nonoverlapping polygons,...
Divisible Tilings in the Hyperbolic Plane
"... Abstract. We consider trianglequadrilateral pairs in the hyperbolic plane which “kaleidoscopically ” tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of1, 2, and 3 par ..."
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Cited by 1 (0 self)
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Abstract. We consider trianglequadrilateral pairs in the hyperbolic plane which “kaleidoscopically ” tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of1, 2, and 3 parameter families as well as a finite number ofexceptional cases.
New YorkJourn of Mathematics
, 2000
"... . We consider trianglequadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisibletiling All possible such divisible tilings are classified. There are a finite number of 1, 2, and 3 parameter f ..."
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. We consider trianglequadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisibletiling All possible such divisible tilings are classified. There are a finite number of 1, 2, and 3 parameter families as well as a finite number of exceptional cases. Co0,0, 1. IntroducA:: 238 2. Tilings and Tiling Groups 240 2.1. Tiling Groups 241 2.2. Divisible Tilings 243 3. Overview of QuadrilateralSearc h 246 3.1. Free Verticx 246 3.2. Constrained VerticW 248 3.3. The TwoSearc h Methods 250 4. Direc Construcx9R Method (K # 12) 251 4.1. PolygonConstrucHxS and EliminationNo Interior Hubs 253 4.2. Computer Algorithm and Extension to Interior Hubs. 256 5. BoundaryConstrucLLA Method (K>12) 257 5.1. Geometric Quadrilateral Test 259 5.2. Boundary Word Test 259 5.3. Example: Failure of (2, 3, 7) to Tile (7, 7, 7, 7) 262 5.4. Example:Sucle:Sxx Tiling of (5, 5, 5, 5) by (2, 4, 5) 262 6. Catalogue o...
The SchreierSims algorithm for matrix groups
, 2004
"... Email address: henrik.baarnhielm@imperial.ac.ukAbstract. This is the report of a project with the aim to make a new implementation of the SchreierSims algorithm in GAP, specialized for matrix groups. The standard SchreierSims algorithm is described in some detail, followed by descriptions of the ..."
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Email address: henrik.baarnhielm@imperial.ac.ukAbstract. This is the report of a project with the aim to make a new implementation of the SchreierSims algorithm in GAP, specialized for matrix groups. The standard SchreierSims algorithm is described in some detail, followed by descriptions of the probabilistic SchreierSims algorithm and the SchreierToddCoxeterSims algorithm. Then we discuss our implementation and some optimisations, and finally we report on the performance of our implementation, as compared to the existing implementation in GAP, and we give benchmark results. The conclusion is that our implementation in some cases