One of the major disadvantages of using Branch and Bound algorithm to solve permutation bases problems by computer is the size of main memory required. The single Array Branch and Bound (SABB) method introduction here aims to overcome this deficiency, using only N memory locations as a vector of size N, and a single cell C with N bits.

The Discrete Memory Machine (DMM) is a theoretical parallel computing model that captures the essence of the shared memory access of GPUs. The bank conflicts should be avoided for maximizing the bandwidth of the shared memory access. Offline permutation of an array is a task to copy all elements in array a into array b along a permutation given in advance. The main contribution of this paper is...

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 th...

A Costas array of order n is an n× n permutation matrix such that all vectors between pairs of ones are distinct. Thus, a permutation matrix fails to be a Costas array if and only if it contains ones that form a (possibly degenerate) parallelogram. In this paper, we enumerate parallelograms in an n × n permutation matrix. We use our new formulas to improve Davies’s O(n−1) result for the density...

An (n, d, k)-mapping f is a mapping from binary vectors of length n to permutations of length n + k such that for all x, y ∈ {0, 1}n , dH ( f (x), f (y)) ≥ dH (x, y) + d , if dH (x, y) ≤ (n+k)−d and dH ( f (x), f (y)) = n+k, if dH (x, y) > (n+k)−d . In this paper, we construct an (n, 3, 2)-mapping for any positive integer n ≥ 6. An (n, r)-permutation array is a permutation array of length n and...

and a positive integer m. The resulted array has ( |PA(p, k)| · p(m−1)(p−k) )m rows. Compared to the other constructions, the new construction gives a permutation array of far bigger size with a large minimum distance, for example, when k ≥ 2p/3. Moreover the proposed construction provides an algorithm to find the i-th row of PA ( pm, pm−1k ) for a given index i very simply. key words: permutat...

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 pe...

Recently, a new parameter of a code, referred to as the remoteness, has been introduced.This parameter can be viewed as a dual to the covering radius. It is exactly determined for the cyclic and dihedral groups. In this paper, we consider the group $U_{6n}$ as a subgroup of $S_{2n+3}$ and obtain its remoteness.  We show that the remoteness of the permutation code $U_{6n}$ is $2n+2$.  Moreover, ...

Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. In particular, the number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is bas...

This paper introduces and studies the notion of an equidistant frequency permutation array (EFPA). An EFPA of length n = mλ, distance d and size v is defined to be a v×n array T such that 1) each row is a multipermutation on a multiset of m symbols, each repeated with frequency λ, and 2) the Hamming distance between any two distinct rows of T is precisely d. Such an array generalizes the well-s...