Common Intervals of Multiple Permutations

Given k permutations of n elements, a k-tuple of intervals of these permutations consisting of the same set of elements is called a common interval. We present an algorithm that finds in a family of k permutations of n elements all z common intervals in optimal O(kn+z) time and O(n) additional space. Additionally, we show how to adapt this algorithm to multichromosomal and circular permutations.     

Remark on one problem in extremal combinatorics

We reduce the problem of determining the maximum number of permutations of a finite set such that any pair of permutations has at least t common transpositions to the problem of determining the maximum number of permutations of finite set such that any pair has at least t common fixed points. The latter problem was solved by the author in [1].     

A Simplified and Generalized Treatment of DES-Related Ciphers

Abstract

This work is a study of DES-like ciphers where the bitwise exclusive-or (XOR) operation in the underlying Feistel network is replaced by an arbitrary group operation. The authors construct a two-round simplified version of DES that contains all the DES components and show that its set of encryption permutations is not a group under functional composition, it is not a pure cipher, and its set of encryption permutations does not generate the alternating group. They present a non-statistical proof that for n ≤ 4 the set of n-round Feistel permutations over an arbitrary group do not constitute a group under functional composition.     

Exact and approximation algorithms for sorting by reversals,with application to genome rearrangement

Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of genes on corresponding chromosomes, and areversal takes an arbitrary substring of elements, and reverses their order.For this problem, we develop two algorithms: a greedy approximation algorithm, that finds a solution provably close to optimal inO(n ²) time and0(n) space forn-element permutations, and a branch- and-bound exact algorithm, that finds an optimal solution in0(mL(n, n)) time and0(n ²) space, wherem is the size of the branch- and-bound search tree, andL(n, n) is the time to solve a linear program ofn variables andn constraints. The greedy algorithm is the first to come within a constant factor of the optimum; it guarantees a solution that uses no more than twice the minimum number of reversals. The lower and upper bounds of the branch- and-bound algorithm are a novel application of maximum-weight matchings, shortest paths, and linear programming.In a series of experiments, we study the performance of an implementation on random permutations, and permutations generated by random reversals. For permutations differing byk random reversals, we find that the average upper bound on reversal distance estimatesk to within one reversal fork<1/2n andn<100. For the difficult case of random permutations, we find that the average difference between the upper and lower bounds is less than three reversals forn<50. Due to the tightness of these bounds, we can solve, to optimality, problems on 30 elements in a few minutes of computer time. This approaches the scale of mitochondrial genomes.This research was supported by a postdoctoral fellowship from the Program in Mathematics and Molecular Biology of the University of California at Berkeley under National Science Foundation Grant DMS-8720208, and by a fellowship from the Centre de recherches mathématiques of the Université de Montréal.This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada, and the Fonds pour la formation de chercheurs et l'aide à la recherche (Québec). The author is a fellow of the Canadian Institute for Advanced Research.     

A Cost-Optimal Pipeline Algorithm for Permutation Generation in Lexicographic Order

In this paper we solve the open problem of designing a cost-optimal parallel algorithm for generating permutations ofMelements out of the set {0, 1, …,N− 1}, in lexicographic order. Our algorithm runs on the simplest model of parallel computation, i.e., a linear array of sizeM, where each processor PE(i) needs only a constant number of words of length logN, and is responsible for producing with constant delay, theith components in the successive permutations. This algorithm runs in timeO(N!/(N−M)!) on an array of sizeMand is therefore cost-optimal when the time needed to output the permutations is taken into account. Moreover, by doubling the number of cells, it is possible to implement this algorithm on a unidirectional linear array.     

The design of complete encryption networks using cryptographically equivalent permutations

A design procedure for complete substitution-permutation encryption networks is presented. The cryptographically important property of completeness is achieved after three iterations, the minimum possible number for all networks of size N = i · n² (i = 1,2,…,n), where n is the size of the substitution function used. The permutation stage is constructed by choosing a single member of a class of cryptographically equivalent permutations for all the network rounds, hence having the advantage of simplifying the network implementation. An algorithm for generating members of the class of cryptographically equivalent permutations is given.     

Fast Algorithms to Enumerate All Common Intervals of Two Permutations

Given two permutations of n elements, a pair of intervals of these permutations consisting of the same set of elements is called a common interval . Some genetic algorithms based on such common intervals have been proposed for sequencing problems and have exhibited good prospects. In this paper we propose three types of fast algorithms to enumerate all common intervals: (i) a simple O(n ² ) time algorithm (LHP), whose expected running time becomes O(n) for two randomly generated permutations, (ii) a practically fast O(n ² ) time algorithm (MNG) using the reverse Monge property, and (iii) an O(n+K) time algorithm (RC), where K is the number of common intervals. It will also be shown that the expected number of common intervals for two random permutations is O(1) . This result gives a reason for the phenomenon that the expected time complexity O(n) of the algorithm LHP is independent of K . Among the proposed algorithms, RC is most desirable from the theoretical point of view; however, it is quite complicated compared with LHP and MNG. Therefore, it is possible that RC is slower than the other two algorithms in some cases. For this reason, computational experiments for various types of problems with up to n=10 ⁶ are conducted. The results indicate that (i) LHP and MNG are much faster than RC for two randomly generated permutations, and (ii) MNG is rather slower than LHP for random inputs; however, there are cases in which LHP requires Ω(n ² ) time, but MNG runs in o(n ² ) time and is faster than both LHP and RC. Received December 21, 1996; revised June 2, 1998.     

Exact Largest and Smallest Size of Components

Golomb and Gaal [15] study the number of permutations on n objects with largest cycle length equal to k . They give explicit expressions on ranges n/(i+1) < k ≤ n/i for i=1,2, \ldots, derive a general recurrence for the number of permutations of size n with largest cycle length equal to k , and provide the contribution of the ranges (n/(i+1),n/i] for i=1,2,\ldots, to the expected length of the largest cycle. We view a cycle of a permutation as a component. We provide exact counts for the number of decomposable combinatorial structures with largest and smallest components of a given size. These structures include permutations, polynomials over finite fields, and graphs among many others (in both the labelled and unlabelled cases). The contribution of the ranges (n/(i+1),n/i] for i=1,2,\ldots, to the expected length of the smallest and largest component is also studied. Received June 27, 2000; revised October 8, 2000.     

Parallel permutation generation on linear array

Given n items, a parallel algorithm for generating all the n! permutations is presented. The computational model used is a linear array which consists of n identical processing elements with a simple structure. One permutation is produced at each other time step. The elapsed time to produce a permutation is independent of the integer n. The basic idea used is the iterative method and the exchange of two consecutive components in an existing permutation. The design procedures of this algorithm are considered in detail. The ranking and unranking functions of the required permutations are also discussed.     

Permutation equivalence of cubic rotation symmetric Boolean functions

Rotation symmetric Boolean functions have been extensively studied for about 15 years because of their applications in cryptography and coding theory. Until recently little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. For the much more complicated case of cubic rotation symmetric functions generated by a single monomial, the affine equivalence classes under permutations which preserve rotation symmetry were determined in 2011. It was conjectured then that the cubic equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. This conjecture is probably difficult, but here we take a step towards it by proving that the cubic affine equivalence classes found in 2011 are the same if all permutations, not just those preserving rotation symmetry, are allowed. The needed new idea uses the theory of circulant matrices.     

Sorting Unsigned Permutations by Weighted Reversals,Transpositions, and Transreversals

Reversals, transpositions and transreversals are common events in genome rearrangement. The genome rearrangement sorting problem is to transform one genome into another using the minimum number of given rearrangement operations. An integer permutation is used to represent a genome in many cases. It can be divided into disjoint strips with each strip denoting a block of consecutive integers. A singleton is a strip of one integer. And the genome rearrangement problem turns into the problem of sorting a permutation into the identity permutation equivalently. Hannenhalli and Pevzner designed a polynomial time algorithm for the unsigned reversal sorting problem on those permutations with O(log n) singletons. In this paper, first we describe one case in which Hannenhalli and Pevzner’s algorithm may fail and propose a corrected approach. In addition, we propose a (1+ε)-approximation algorithm for sorting unsigned permutations with O(log n) singletons by reversals of weight 1 and transpositions/transreversals of weight 2.     

On a gauntlet thrown by David Gries

Summary It is requested to design a program that will generate the N! permutations of the values from 0 through N — 1 in such an order that the transition from one permutation to the next is always performed by exactly one swap of two neighbours.     

On the limits of cache-oblivious rational permutations

Permuting a vector is a fundamental primitive which arises in many applications. In particular, rational permutations, which are defined by permutations of the bits of the binary representations of the vector indices, are widely used. Matrix transposition and bit-reversal are notable examples of rational permutations. In this paper we contribute a number of results regarding the execution of these permutations in cache hierarchies, with particular emphasis on the cache-oblivious setting. We first bound from below the work needed to execute a rational permutation with an optimal cache complexity. Then, we develop a cache-oblivious algorithm to perform any rational permutation, which exhibits optimal work and cache complexities under the tall cache assumption. We finally show that for certain families of rational permutations (including matrix transposition and bit reversal) no cache-oblivious algorithm can exhibit optimal cache complexity for all values of the cache parameters. This latter result specializes the one proved by Brodal and Fagerberg for general permutations to the case of rational permutations, and provides further evidence that the tall cache assumption is often necessary to attain cache optimality in the context of cache-oblivious algorithms.     

Star graph automorphisms and disjoint Hamilton cycles

The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies. We define automorphisms for star graphs St _n of degree n?1, for every positive odd integer n, which yield permutations of labels for the edges of St _n taken from the set of integers between 1 and ? n/2 ?. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known Hamilton cycle in St _n . Our method produces a better than two-fold improvement from ? ? (n)/10 ? to ? 2? (n)/9 ?, where ? is the Euler function, for the known number of edge-disjoint Hamilton cycles in St _n for all odd integers n. For prime n, the improvement is from ? n/8 ? to ? n/5 ?, and we can extend this result to the case when n is the power of a prime greater than 7.     

Minimizing total weighted completion time with uncertain data: A stability approach

A single-machine scheduling problem is investigated provided that the input data are uncertain: The processing time of a job can take any real value from the given segment. The criterion is to minimize the total weighted completion time for the n jobs. As a solution concept to such a scheduling problem with an uncertain input data, it is reasonable to consider a minimal dominant set of job permutations containing an optimal permutation for each possible realization of the job processing times. To find an optimal or approximate permutation to be realized, we look for a permutation with the largest stability box being a subset of the stability region. We develop a branch-and-bound algorithm to construct a permutation with the largest volume of a stability box. If several permutations have the same volume of a stability box, we select one of them due to one of two simple heuristics. The efficiency of the constructed permutations (how close they are to a factually optimal permutation) and the efficiency of the developed software (average CPU-time used for an instance) are demonstrated on a wide set of randomly generated instances with 5 ≤ n ≤ 100.     

The p–p Rearrangement and Failure-Tolerance of Double p-ary Multirings and Generalized Hypercubes

It is shown that an arbitrary grouped p-element permutation can be implemented in a conflict-free way through the commutation of channels on the double p-ary multiring or the double p-ary hypercube. It is revealed that in arbitrary single-element permutations, these commutators display the property of the (p - 1)-nodal failure-tolerance and the generalized hypercube displays in addition the property of the (p - 1)-channel failure-tolerance.     

Fast Generation of Random Permutations Via Networks Simulation

We consider the problem of generating random permutations with uniform distribution. That is, we require that for an arbitrary permutation π of n elements, with probability 1/n! the machine halts with the i th output cell containing π(i) , for 1 ≤ i ≤ n . We study this problem on two models of parallel computations: the CREW PRAM and the EREW PRAM. The main result of the paper is an algorithm for generating random permutations that runs in O(log log n) time and uses O(n ^1+o(1) ) processors on the CREW PRAM. This is the first o(log n) -time CREW PRAM algorithm for this problem. On the EREW PRAM we present a simple algorithm that generates a random permutation in time O(log n) using n processors and O(n) space. This algorithm outperforms each of the previously known algorithms for the exclusive write PRAMs. The common and novel feature of both our algorithms is first to design a suitable random switching network generating a permutation and then to simulate this network on the PRAM model in a fast way. Received November 1996; revised March 1997.     

A q -Analogue of the Path Length of Binary Search Trees

A reformulation of the path length of binary search trees is given in terms of permutations, allowing us to extend the definition to the instance of words, where the letters are obtained by independent geometric random variables (with parameter q ). In this way, expressions for expectation and variance are obtained which in the limit for q→1 are the classical expressions. Received June 6, 2000; revised October 14, 2000.     

Whole mirror duplication-random loss model and pattern avoiding permutations

In this paper we study the problem of the whole mirror duplication-random loss model in terms of pattern avoiding permutations. We prove that the class of permutations obtained with this model after a given number p of duplications of the identity is the class of permutations avoiding the alternating permutations of length p²+1. We also compute the number of duplications necessary and sufficient to obtain any permutation of length n. We provide two efficient algorithms to reconstitute a possible scenario of whole mirror duplications from identity to any permutation of length n. One of them uses the well-known binary reflected Gray code (Gray, 1953) [10]. Other relative models are also considered.