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Playing games with algorithms: Algorithmic combinatorial game theory
 IN: PROC. 26TH SYMP. ON MATH FOUND. IN COMP. SCI., LECT. NOTES IN COMP. SCI., SPRINGERVERLAG
, 2001
"... Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, ..."
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Cited by 46 (11 self)
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Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, which analyzes ideal play in perfectinformation games. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomialtime algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.
A polynomialtime theory of blackbox groups I
, 1998
"... We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic o ..."
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Cited by 40 (6 self)
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We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic obstacles such as factoring integers and discrete logarithm. While these and other “abelian obstacles ” persist, we demonstrate that the “nonabelian normal structure ” of matrix groups over finite fields can be mapped out in great detail by polynomialtime randomized (Monte Carlo) algorithms. The methods are based on statistical results on finite simple groups. We indicate the elements of a project under way towards a more complete “recognition” of such groups in polynomial time. In particular, under a now plausible hypothesis, we are able to determine the names of all nonabelian composition factors of a matrix group over a finite field. Our context is actually far more general than matrix groups: most of the algorithms work for “blackbox groups ” under minimal assumptions. In a blackbox group, the group elements are encoded by strings of uniform length, and the group operations are performed by a “black box.”
Fast management of permutation groups I
, 1997
"... We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play ..."
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Cited by 21 (3 self)
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We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.
New Methods for Using Cayley Graphs in Interconnection Networks
 DISCRETE APPLIED MATHEMATICS
, 1992
"... A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more timeefficient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayl ..."
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Cited by 16 (5 self)
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A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more timeefficient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayley graphs in a spaceefficient manner (log 2 (3) bits per node), so that copies could be cheaply stored at each node of an interconnection network. The second algorithm is especially useful for providing a compact encoding of an optimal routing table (for example, a 13 kilobyte optimal table for 64,000 nodes). The algorithm relies on using a compact encoding of group elements known from computational group theory. Generalizations of all of the above are presented for Schreier coset graphs.
Efficient representation of perm groups
 Combinatorica
, 1991
"... Abstract: This note presents an elementary version of Sims’s algorithm for computing strong generators of a given perm group, together with a proof of correctness and some notes about appropriate lowlevel data structures. Upper and lower bounds on the running time are also obtained. (Following a su ..."
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Cited by 12 (1 self)
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Abstract: This note presents an elementary version of Sims’s algorithm for computing strong generators of a given perm group, together with a proof of correctness and some notes about appropriate lowlevel data structures. Upper and lower bounds on the running time are also obtained. (Following a suggestion of Vaughan Pratt, we adopt the convention that perm = permutation, perhaps thereby saving millions of syllables in future research.) 1. A data structure for perm groups. A “perm, ” for the purposes of this paper, is a onetoone mapping of a set onto itself. If α and β are perms such that α takes i ↦ → j and β takes j ↦ → k, the product αβ takes i ↦ → k. We write α − for the inverse of the perm α; hence αβ = γ iff α = γβ −. Let Π(k) be the set of all perms of the positive integers that fix all points> k. Consider the following data structure: For 1 ≤ j ≤ k, either σkj = ∅ or σkj is a perm of Π(k) that takes k ↦ → j. Let Σ(k) be the set of all non ∅ perms σkj. We assume that σkk is the identity perm; hence Σ(k) is always nonempty. We write Γ(k) for the set of all perms that can be written as products of the form σ1... σk where each σi is in Σ(i). There is an easy way to test if a given perm π ∈ Π(k) is a member of Γ(k): Let π take k ↦ → j. Then if σkj = ∅ we have π � ∈ Γ(k); otherwise if k = 1 we have π ∈ Γ(k);
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.
Generalizing boolean satisfiability II: Theory
, 2004
"... This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contai ..."
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Cited by 11 (2 self)
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This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying zap, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the DavisPutnamLogemannLoveland inference procedure to this broader setting, and show that earlier computational improvements are either subsumed or left intact by the new method. The third paper in this series discusses zap’s implementation and presents experimental performance results. 1.
Computational Pólya theory
 In Surveys in combinatorics
, 1995
"... A permutation group G of degree n has a natural induced action on words of length n over a finite alphabet \Sigma , in which the image x of x under permutation g 2 G is obtained by permuting the positions of symbols in x according to g. The key result in "P'olya theory" is that the number of ..."
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Cited by 7 (1 self)
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A permutation group G of degree n has a natural induced action on words of length n over a finite alphabet \Sigma , in which the image x of x under permutation g 2 G is obtained by permuting the positions of symbols in x according to g. The key result in "P'olya theory" is that the number of orbits of this action is given by an evaluation of the cycleindex polynomial PG (z 1 ; : : : ; z n ) of G at the point z 1 = \Delta \Delta \Delta = z n = j\Sigma j.
Towards a practical, theoretically sound algorithm for random generation in finite groups
"... This work presents a new, simple O(log 2 G) algorithm, the Fibonacci cube algorithm, for producing random group elements in black box groups. After the initial O(log 2 G) group operations, εuniform random elements are produced using O((log 1/ε)log G) operations each. This is the first major a ..."
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Cited by 5 (1 self)
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This work presents a new, simple O(log 2 G) algorithm, the Fibonacci cube algorithm, for producing random group elements in black box groups. After the initial O(log 2 G) group operations, εuniform random elements are produced using O((log 1/ε)log G) operations each. This is the first major advance over the ten year old result of Babai [Bab91], which had required O(log 5 G) group operations. Preliminary experimental results show the Fibonacci cube algorithm to be competitive with the product replacement algorithm. The new result leads to an amusing reversal of the state of affairs for permutation group algorithms. In the past, the fastest random generation for permutation groups was achieved as an application of permutation group membership algorithms and used deep knowledge about permutation representations. The new black box random generation algorithm is also valid for permutation groups, while using no knowledge that is specific to permutation representations. As an application, we demonstrate a new algorithm for permutation group membership that is asymptotically faster than all previously known algorithms. 1
Reliability Modeling of Structured Systems: Exploring Symmetry in StateSpace Generation
 Pacific Rim International Symposium on FaultTolerant Systems
"... A large number of systems are implemented using regular interconnected topologies. Markov analysis of such systems results in large state spaces. We explore symmetry, in particular rotational and permutational, of such systems to achieve a significant reduction in the size of the state space require ..."
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Cited by 3 (0 self)
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A large number of systems are implemented using regular interconnected topologies. Markov analysis of such systems results in large state spaces. We explore symmetry, in particular rotational and permutational, of such systems to achieve a significant reduction in the size of the state space required to analyze them. The resulting much smaller state spaces allow analyses of very large systems. We define equivalent classes of states and develop an algorithm to generate small state spaces and the corresponding Markov chain for systems with permutation symmetries. The state space generation process is also simplified. We demonstrate our technique using several examples. Our technique is very useful in the exact analysis of large systems. This research was supported in part by NASA under NAS119480 while the author was on sabbatical at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681. i 1 Introduction A large number of...