Structured Permuting in Place on Parallel Disk Systems

The ability to perform permutations of large data sets in place reduces the amount of necessary available disk storage. The simplest way to perform a permutation often is to read the records of a data set from a source portion of data storage, permute them in memory, and write them to a separate target portion of the same size. It can be quite expensive, however, to provide disk storage that is twice the size of very large data sets. Permuting in place reduces the expense by using only a small amount of extra disk storage beyond the size of the data set. This paper features in-place algorithms for commonly used structured permutations. We have developed an asymptotically optimal algorithm for performing BMMC (bit-matrix-multiply/complement) permutations in place that requires at most \frac{2N}{BD}\left( 2\ceil{\frac{\rank{\gamma}}{\lg (M/B)}} + \frac{7}{2}\right) parallel disk accesses, as long as $M \geq 2BD$, where $N$ is the number of records in the data set, $M$ is the number of records that can fit in memory, $D$ is the number of disks, $B$ is the number of records in a block, and $\gamma$ is the lower left $\lg (N/B) \times \lg B$ submatrix of the characteristic matrix for the permutation. This algorithm uses $N+M$ records of disk storage and requires only a constant factor more parallel disk accesses and insignificant additional computation than a previously published asymptotically optimal algorithm that uses $2N$ records of disk storage. We also give algorithms to perform mesh and torus permutations on a $d$-dimensional mesh. The in-place algorithm for mesh permutations requires at most 3\ceil{N/BD} parallel I/Os and the in-place algorithm for torus permutations uses at most $4dN/BD$ parallel I/Os. The algorithms for mesh and torus permutations require no extra disk space as long as the memory size~$M$ is at least~$3BD$. The torus algorithm improves upon the previous best algorithm in terms of both time and space

Permuting operations on strings and their relation to prime numbers

Some length-preserving operations on strings only permute the symbol positions in strings; such an operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by $X_n$. We call an integer $n$ $X$-{\em prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). Then we show some properties of these $X$-primes, particularly, how $X$-primes are related to $X^\prime$-primes as well as to ordinary prime numbers. Here $X$ and $X^\prime$ range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on Archimedes spiral and on the Josephus problem

Idempotent permutations

Together with a characteristic function, idempotent permutations uniquely determine idempotent maps, as well as their linearly ordered arrangement simultaneously. Furthermore, in-place linear time transformations are possible between them. Hence, they may be important for succinct data structures, information storing, sorting and searching. In this study, their combinatorial interpretation is given and their application on sorting is examined. Given an array of n integer keys each in [1,n], if it is allowed to modify the keys in the range [-n,n], idempotent permutations make it possible to obtain linearly ordered arrangement of the keys in O(n) time using only 4log(n) bits, setting the theoretical lower bound of time and space complexity of sorting. If it is not allowed to modify the keys out of the range [1,n], then n+4log(n) bits are required where n of them is used to tag some of the keys.Comment: 32 page

Permuting operations on strings: Their permutations and their primes

We study some length-preserving operations on strings that permute the symbol positions in strings. These operations include some well-known examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on Archimedes spiral. Such a permuting operation $X$ gives rise to a family $\{p(X,n)\}_{n\geq2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by such a permutation $p(X,n)$. We call an integer $n$ $X$-prime if $p(X,n)$ consists of a single cycle of length $n$ ($n\geq2$). Then we show some properties of these $X$-primes, particularly, how $X$-primes are related to $X^\prime$-primes as well as to ordinary prime numbers

Six-qubit permutation-based decoherence-free orthogonal basis

There is a natural orthogonal basis of the 6-qubit decoherence-free (DF) space robust against collective noise. Interestingly, most of the basis states can be obtained from one another just permuting qubits. This property: (a) is useful for encoding qubits in DF subspaces, (b) allows the implementation of the Bennett-Brassard 1984 (BB84) protocol in DF subspaces just permuting qubits, which completes a the method for quantum key distribution using DF states proposed by Boileau et al. [Phys. Rev. Lett. 92, 017901 (2004)], and (c) points out that there is only one 6-qubit DF state which is essentially new (not obtained by permutations) and therefore constitutes an interesting experimental challenge.Comment: REVTeX4, 5 page

Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS

We study the realization of anyon-permuting symmetries of topological phases on the lattice using tensor networks. Working on the virtual level of a projected entangled pair state, we find matrix product operators (MPOs) that realize all unitary topological symmetries for the toric and color codes. These operators act as domain walls that enact the symmetry transformation on anyons as they cross. By considering open boundary conditions for these domain wall MPOs, we show how to introduce symmetry twists and defect lines into the state.Comment: 11 pages, 6 figures, 2 appendices, v2 published versio