A compact data structure and parallel algorithms for permutation graphs (original) (raw)

An Algorithamic Approach to Permutation Graphs and Its Complexities LathaMohanan

Asian Resonance, 2014

Permutation graph was first introduced by Chartrand and Harary in 1967 and their purpose was to study the cycle permutation graph. This paper definespermutation graph and study its properties and characterisationthrough theorems .It alsodiscuss characterisation of permutation labelling by a theorem .It gives real life application of permutation graph as a class of intersection graphs. It gives a sorting permutation using queues in parallel. It also gives canonical colouring of a permutation graph which gives a minimal colouring.This has given through an algorithm and discuss its complexity.

Parallel algorithms for solvable permutation groups

Journal of Computer and System Sciences, 1988

A number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and tinding a composition series. Additionally, for nilpotent groups, one can, in NC, find a central composition series, and find pointwise stabilizers of sets. The latter is applied to an instance of graph isomorphism. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to NC.

Coloring permutation graphs in parallel

Discrete Applied Mathematics, 2002

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having speciÿc key properties. We propose an e cient parallel algorithm which colors an n-node permutation graph in O(log 2 n) time using O(n 2 =log n) processors on the CREW PRAM model. Speciÿcally, given a permutation we construct a tree T * [ ], which we call coloring-permutation tree, using certain combinatorial properties of. We show that the problem of coloring a permutation graph is equivalent to ÿnding vertex levels in the coloring-permutation tree.

A Vector Space Framework for Parallel Stable Permutations

Formal Methods, 1995

We establish a formal foundation for stable permutations in the domain of a parallel model of computation applicable to a customized set of complexity metrics. By means of vector spaces, we develop an algebrao{geometric representation that is expressive, exible and simple to use, and present a taxonomy categorizing stable permutations into classes of index{digit, linear, translation, a ne and polynomial permutations. For each class, we demonstrate its general behavioral properties and then analyze particular examples in each class, where we derive results about its inverse, xed instances, number of instances local and nonlocal to a processor, as well as its compositional relationships to other permutations. Such examples are bit{reversal, radix{ Q exchange, radix{Q shu e and unshu e within the index{digit class, radix{Q butter y and 1's complement within the translation class, binary{to{Gray and Gray{to{binary within the linear class, and arithmetic add 1, arithmetic subtract 1 and 2's complement in the polynomial class. These were primarily chosen due to their importance in implementing orthogonal transforms, such as Cooley{Tukey Fast Fourier Transforms (FFT), real{to{complex (complex{to{real) FFT, and sine and cosine transforms on distributed memory parallel processing systems.

A Parallel Algorithm for Transitive Closure

Parallel and Distributed Computing Systems, 2002

We present a parallel algorithm for the problem of computing the transitive closure for an acyclic digraph with Ò vertices and Ñ edges. We use the BSP/CGM model of parallel computing. Our algorithm uses Ç´ÐÓ Ôµ rounds of communications with Ô processors, where Ô Ò, and each processor has Ç´Ñ Ò Ô µ local memory. The local computation of each processor is equal to the product of the number of edges and vertices of that are stored in Ô.

On the shuffle-exchange permutation network

The shuffle-exchange permutation network (SEP n) is a fixed degree Cayley graph which has been proposed as a basis for massively parallel systems. We propose a routing algorithm with an upper bound of (5/8)n 2 + O(n), where n is the length of the permutation. (This improves on a (9/8)n 2 routing algorithm described earlier [5].) Thus, the diameter of SEP n is at most (5/8) n 2 + O(n). We also show that the diameter is at least n 2 / 2-O(n). We demonstrate that SEP n has a Hamilton cycle, for n 3, left open in [5], and describe embeddings of variable-degree Cayley networks, such as bubble-sort networks [1], star networks [2] and pancake networks [4] into SEP n. Our embeddings for these networks are substantial improvements of earlier results stated in [5].

A reduction algorithm for large-base primitive permutation groups

LMS J. Comput. Math, 2006

We present a nearly linear-time Las Vegas algorithm that, given a large-base primitive permutation group, constructs its natural imprimitive representation. A large-base primitive permutation group is a subgroup of a wreath product of symmetric groups Sn and Sr in product action on r-tuples of k-element subsets of {1,..., n}, containing Ar n. The algorithm is a randomised speed-up of a deterministic algorithm of Babai, Luks, and Seress.

Parallel permutation and sorting algorithms and a new generalized connection network

Journal of the ACM, 1982

O(k log N) algonthms are obtained to permute and sort N data items on cube and perfect shuffle computers with N 1+1/k processing elements, 1 __<_ k -< log N These algorithms lead directly to a generahzedconnecuon-network construction having O(klog N) delay and O(kNX+l/klog N) contact pairs. This network has the advantage that the switches can be set m O(klog N) t~me by either a cube or perfect shuffle computer with N ~+~/h processing elements. Categories and SubJect Descriptors. B 4 3 [Input~Output and Data Communications] Interconnecuons (Subsystems)--topology, C. 1.2 [Processor Architectures]. Multiple Data Stream Architectures (MulUprocessors)--mterconnect:on architectures, parallel processors, single-instruct:on-stream multiple-data-stream architectures (SIMD); F 2 2 [Analysis of Algorithms and Problem Complexity] Nonnumencal Algorithms and Problems--routing and layout, sorting and searching

On the Treewidth and Pathwidth of Permutation Graphs

Unknown Publisher eBooks, 1992

In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[7r] is a permutation graph with treewidth k, then we show that the pathwidth of G[7r] is at most 2k, and we give an algorithm which constructs a path-decomposition with width at most 2k in time O( nk). We assume that the permutation 7r is given. For permutation graphs of which the treewidth is bounded by some constant, this result, together with previous results ([9], ), shows that the treewidth and pathwidth can be computed in linear time.