Succinct Representations of Permutations and Functions

We investigate the problem of succinctly representing an arbitrary permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for any i and any (positive or negative) integer power k. A representation taking (1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in constant time, for any positive constant \epsilon <= 1. A representation taking the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers in O(lg n / lg lg n) time. We then consider the more general problem of succinctly representing an arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed quickly for any i and any integer power k. We give a representation that takes (1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and computes arbitrary positive powers in constant time. It can also be used to compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time. We place emphasis on the redundancy, or the space beyond the information-theoretic lower bound that the data structure uses in order to support operations efficiently. A number of lower bounds have recently been shown on the redundancy of data structures. These lower bounds confirm the space-time optimality of some of our solutions. Furthermore, the redundancy of one of our structures "surpasses" a recent lower bound by Golynski [Golynski, SODA 2009], thus demonstrating the limitations of this lower bound.Comment: Preliminary versions of these results have appeared in the Proceedings of ICALP 2003 and 2004. However, all results in this version are improved over the earlier conference versio

GraCT: A Grammar based Compressed representation of Trajectories

We present a compressed data structure to store free trajectories of moving objects (ships over the sea, for example) allowing spatio-temporal queries. Our method, GraCT, uses a $k^2$-tree to store the absolute positions of all objects at regular time intervals (snapshots), whereas the positions between snapshots are represented as logs of relative movements compressed with Re-Pair. Our experimental evaluation shows important savings in space and time with respect to a fair baseline.Comment: This research has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie Actions H2020-MSCA-RISE-2015 BIRDS GA No. 69094

Covariant - tensor method for quantum groups and applications I: SU(2)qSU(2)_{q}

A covariant - tensor method for $SU(2)_{q}$ is described. This tensor method is used to calculate q - deformed Clebsch - Gordan coefficients. The connection with covariant oscillators and irreducible tensor operators is established. This approach can be extended to other quantum groups.Comment: 18 page

Automorphism Groups of Geometrically Represented Graphs

We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. Using this, we characterize automorphism groups of interval, permutation and circle graphs. We combine techniques from group theory (products, homomorphisms, actions) with data structures from computer science (PQ-trees, split trees, modular trees) that encode all geometric representations. We prove that interval graphs have the same automorphism groups as trees, and for a given interval graph, we construct a tree with the same automorphism group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982]. For permutation and circle graphs, we give an inductive characterization by semidirect and wreath products. We also prove that every abstract group can be realized by the automorphism group of a comparability graph/poset of the dimension at most four

Python for education: permutations

Python implementation of permutations is presented. Three classes are introduced: Perm for permutations, Group for permutation groups, and PermError to report any errors for both classes. The class Perm is based on Python dictionaries and utilize cycle notation. The methods of calculation for the perm order, parity, ranking and unranking are given. A random permutation generation is also shown. The class Group is very simple and it is also based on dictionaries. It is mainly the presentation of the permutation groups interface with methods for the group order, subgroups (normalizer, centralizer, center, stabilizer), orbits, and several tests. The corresponding Python code is contained in the modules perms and groups.Comment: 26 pages, 1 figure, 2 table

Fast Fourier Transforms for Finite Inverse Semigroups

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest--specifically, for the rook monoid and its wreath products by arbitrary finite groups.Comment: ver 3: Added improved upper and lower bounds for the memory required by the fast zeta transform on the rook monoid. ver 2: Corrected typos and (naive) bounds on memory requirements. 30 pages, 0 figure

SgpDec : Cascade (de)compositions of finite transformation semigroups and permutation groups

We describe how the SgpDec computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products.Final Accepted Versio