Permutation Group Algorithms

Seress, Ákos

doi:10.1017/cbo9780511546549

Cited by 254 publications

(281 citation statements)

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“…Sect. 3 shows the optimization rules applied on reversible circuits obtained using Schreier-Sims Algorithm for the strong generating set-finding problem [15] to decrease the quantum cost of the circuits. Sect.…”

Section: Introductionmentioning

confidence: 99%

Integration of Irreversible Gates in Reversible Circuits Using NCT Library

Osman¹,

Younes²

2013

IOSR-JCE

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Section: Introductionmentioning

confidence: 99%

Integration of Irreversible Gates in Reversible Circuits Using NCT Library

Osman¹,

Younes²

2013

IOSR-JCE

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“…many) permutations acting on the color class C i . Applying the standard orbit finding algorithm for permutation groups [17,20] we can compute the hypergraph X i with vertex set V i and multiset E i consisting of all hyperedges E i = {{e π | e ∈ E i and π ∈ Sym(C i ) × Gσ}}. Since Gσ canonizes X i+1 , it follows that |E i | ≤ 2 b · |E i |.…”

Section: Discussionmentioning

confidence: 99%

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

et al. 2013

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We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm. Keywords and phrases IntroductionA hypergraph is an ordered pair X = (V, E) where V is the vertex set and E ⊆ 2 V is the edge set. Two hypergraphs X = (V, E) and X = (V , E ) are said to be isomorphic,Given two hypergraphs X and X , the decision problem Hypergraph Isomorphism (HI) asks whether X ∼ = X . Graph Isomorphism (GI) is obviously polynomial-time reducible to HI. Conversely, HI is also known to be polynomial-time reducible to GI: Given a pair of hypergraphs X = (V, E) and X = (V , E ) as instance for HI, the reduced instance of GI consists of two corresponding bipartite graphs Y and Y defined as follows. The graph Y has vertex set V E and edge set E(Y ) = {{v, e} | v ∈ V, e ∈ E and v ∈ e}, and Y is defined similarly. Here, C D denotes the disjoint union of the sets C and D. It is easy to verify that Y ∼ = Y if and only if X ∼ = X assuming that V can be mapped only to V and E can be mapped only to E . This latter condition is easy to enforce. However, since the above reduction blows up the size of the vertex set in the bipartite encoding, the Zemlyachenko-Luks-Babai graph isomorphism algorithm [3,5,6,25] that runs in time c √ n log n , where n is the size of the vertex set of the graph, does not yield a similar algorithm for hypergraph isomorphism. We note here that the best known hypergraph isomorphism test due to Luks [16] has running time c n . Recently, Babai and Codenotti [4] have shown a 2Õ (k 2 √ n) isomorphism testing algorithm for hypergraphs with hyperedges of size bounded by k.Motivated by this situation, we explore the same algorithmic problem for bounded color class hypergraphs. The input to Colored Hypergraph Isomorphism (CHI) is a pair

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“…For permutation groups G, the Schreier-Sims algorithm and its variants [17], [16] solve the constructive membership problem in G, by computing a strong generating set Y . However, for large-base permutation groups (see Section 3), there are asymptotically more efficient algorithms, which use reduction first.…”

Section: Definitionmentioning

confidence: 99%

A data structure for a uniform approach to computations with finite groups

Neunhöffer

Seress

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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We describe a recursive data structure for the uniform handling of permutation groups and matrix groups. This data structure allows the switching between permutation and matrix representations of segments of the input group, and has wide-ranging applications. It provides a framework to process theoretical algorithms which were considered too complicated for implementation such as the asymptotically fastest algorithms for the basic handling of large-base permutation groups and for Sylow subgroup computations in arbitrary permutation groups. It also facilitates the basic handling of matrix groups. The data structure is general enough for the easy incorporation of any matrix group or permutation group algorithm code; in particular, the library functions of the GAP computer algebra system dealing with permutation groups and matrix groups work with a minimal modification.

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Permutation Group Algorithms

Cited by 254 publications

References 0 publications

Integration of Irreversible Gates in Reversible Circuits Using NCT Library

Integration of Irreversible Gates in Reversible Circuits Using NCT Library

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

A data structure for a uniform approach to computations with finite groups

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