A New Construction of Permutation Arrays

Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, 1990

We address the fundamental problem of permuting the elements of an array according to some given permutation. Our goal is to perform the permutation quickly using only a polylogarithmic number of bits of extra storage. The main result is an o (n log n ) time, o 008 n ) space worst case method. A simpler method is presented for the case in which both the permutation and its inverse can be computed at (amortised) unit cost. This algorithm requires 0 (n logn) time and 0 (log n ) bits in the worst case. These results are extended to the situation in which we are to apply a power of the permutation. A linear time, 0 (log n ) bit method is presented for the special case in which the data values are all distinct and are either initially in sorted order or will be when permuted.

International Journal of Information and Coding Theory, 2010

We look at some techniques for constructing permutation arrays using projections in finite projective spaces and the geometry of arcs in the finite projective plane. We say a permutation array P A(n, d) has length n and minimum distance d when it consists of a collection of permutations on n symbols that pairwise agree in at most n − d coordinate positions. Such arrays can also be viewed as non-linear codes and are used in powerline communication. While our techniques likely do not produce optimal arrays, we are able to construct examples of codes for certain parameter sets where no known constructions were previously known.

arXiv (Cornell University), 2021

Lecture Notes in Computer Science, 1995

Starting from a permutation of f0; : : : ; n ? 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimension 2 eciently. The parallel algorithms obtained have a workload of O(m + n log n) where m is the number of edges of the permutation graph. They run in time O(log 2 n) on a CREW PRAM.

IEEE Transactions on Information Theory, 2000

A frequency permutation array (FPA) of length = and distance is a set of permutations on a multiset over symbols, where each symbol appears exactly times and the distance between any two elements in the array is at least . FPA generalizes the notion of permutation array. In this paper, under the Chebyshev distance, we first prove lower and upper bounds on the size of FPA. Then we give several constructions of FPAs, and some of them come with efficient encoding and decoding capabilities. Moreover, we show one of our designs is locally decodable, i.e., we can decode a message bit by reading at most + 1 symbols, which has an interesting application to private information retrieval.

IEEE Transactions on Information Theory, 2000

Permutation arrays have found applications in powerline communication. One construction method for permutation arrays is to map good codes to permutations using a distance-preserving mappings (DPM). DPMs are mappings from the set of all q-ary vectors of a fixed length to the set of permutations of some fixed length (the same or longer) such that every two distinct vectors are mapped to permutations with the same or larger Hamming distance than that of the vectors. A DPM is called distance increasing (DIM) if the distances are strictly increased (except when the two vectors are equal). In this correspondence, we propose constructions of DPMs and DIMs from ternary vectors. The constructed DPMs and DIMs improve many lower bounds on the maximal size of permutation arrays.

The technique of in-situ associative permuting is introduced which is an association of in-situ permuting and in-situ inverting. It is suitable for associatively permutable permutations of {1,2,...,n} where the elements that will be inverted are negative and stored in order relative to each other according to their absolute values. Let K[1...n] be an array of n integer keys each in the range [1,n], and it is allowed to modify the keys in the range [-n,n]. If the integer keys are rearranged such that one of each distinct key having the value i is moved to the i'th position of K, then the resulting arrangement (will be denoted by K^P) can be transformed in-situ into associatively permutable permutation pi^P using only logn additional bits. The associatively permutable permutation pi^P not only stores the ranks of the keys of K^P but also uniquely represents K^P. Restoring the keys from pi^P is not considered. However, in-situ associative permuting pi^P in O(n) time using logn addi...

IEEE Transactions on Information Theory, 2000

We give a simple construction of distance-preserving mappings from ternary vectors to permutations (3-DPM). Our result gives a lower bound for permutation arrays, i.e., P (n; d) A (n; d), which significantly improves previous lower bounds for d .

Designs, Codes and Cryptography, 2011

Permutation codes (or permutation arrays) have received considerable interest in recent years, partly motivated by a potential application to powerline communication. Powerline communication is the transmission of data over the electricity distribution system. This environment is rather hostile to communication and the requirements are such that permutation codes may be suitable. The problem addressed in this study is the construction of permutation codes with a specified length and minimum Hamming distance, and with as many codewords (permutations) as possible. A number of techniques are used including construction by automorphism group and several variations of clique search based on vertex degrees. Many significant improvements are obtained to the size of the best known codes.

Theoretical Computer Science, 2018

• Amortized time complexity is O (n3). • With sufficient memory, for transposition, we should be able compute diam((Sn)) for n = 14. • Instead of SSSP from In we compute distances only from Sn(0) utilizing properties of Sn.

2012

Permutation is the different arrangements that can be made with a given number of things taking some or all of them at a time. The notation P(n,r) is used to denote the number of permutations of n things taken r at a time. Permutation is used in various fields such as mathematics, group theory, statistics, and computing, to solve several combinatorial problems such as the job assignment problem and the traveling salesman problem. In effect, permutation algorithms have been studied and experimented for many years now. Bottom-Up, Lexicography, and Johnson-Trotter are three of the most popular permutation algorithms that emerged during the past decades. In this paper, we are implementing three of the most eminent permutation algorithms, they are respectively: Bottom-Up, Lexicography, and Johnson-Trotter algorithms. The implementation of each algorithm will be carried out using two different approaches: brute-force and divide and conquer. The algorithms codes will be tested using a computer simulation tool to measure and evaluate the execution time between the different implementations.

2008

This paper presents two new algorithms for inline transforming an integer array 'a' into its own sorting permutation - that is: after performing either of these algorithms, a(i) is the index in the unsorted input array 'a' of its i'th largest element (i=0,1..n-1). The difference between the two IPS (Inline Permutation Substitution) algorithms is that the first and fastest generates an unstable permutation while the second generates the unique, stable, permutation array. The extra space needed in both algorithms is O(log n) - no extra array of length n is needed! The motivation for using these algorithms is given along with their pseudo code. To evaluate their efficiency, they are tested relative to sorting the same array with Quicksort on 4 different machines and for 14 different distributions of the numbers in the input array, with n=10, 50, 250.. 97M. This evaluation shows that both IPS algorithms are generally faster than Quicksort for values of n less th...

Theoretical Computer Science, 2013

Previous compact representations of permutations have focused on adding a small index on top of the plain data π(1), π(2),. .. π(n) , in order to efficiently support the application of the inverse or the iterated permutation. In this paper we initiate the study of techniques that exploit the compressibility of the data itself, while retaining efficient computation of π(i) and its inverse. In particular, we focus on exploiting runs, which are subsets (contiguous or not) of the domain where the permutation is monotonic. Several variants of those types of runs arise in real applications such as inverted indexes and suffix arrays. Furthermore, our improved results on compressed data structures for permutations also yield better adaptive sorting algorithms.

Acta Universitatis Sapientiae, Mathematica

A permutation p of [k] = {1, 2, 3, …, k} is called Layman permutation iff i + p(i) is a Fibonacci number for 1 ≤ i ≤ k. This concept is introduced by Layman in the A097082 entry of the Encyclopedia of Integers Sequences, that is the number of Layman permutations of [n]. In this paper, we will study Layman permutations. We introduce the notion of the Fibonacci complement of a natural number, that plays a crucial role in our investigation. Using this notion we prove some results on the number of Layman permutations, related to a conjecture of Layman that is implicit in the A097083 entry of OEIS.

IEEE Transactions on Information Theory, 2003

Mappings of the set of binary vectors of a fixed length to the set of permutations of the same length are useful for the construction of permutation codes. In this correspondence, several explicit constructions of such mappings preserving or increasing the Hamming distance are given. Some applications are given to illustrate the usefulness of the construction. In particular, a new lower bound on the maximal size of permutation arrays (PAs) is given.