State-complexity hierarchies of uniform languages of alphabet-size length

We study the state complexity of certain simple languages. If A$A$ is an alphabet of k$k$ letters, then a k$k$-language   is a nonempty set of words of length k$k$, that is, a uniform language of length k$k$. We show that the minimal state complexity of a k$k$-language is k+2$k+2$, and the maximal, (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. We prove constructively that, for every i$i$ between the minimal and maximal bounds, there is a language of state complexity i$i$. We introduce a class of automata accepting sets of words that are permutations of A$A$; these languages define a complete hierarchy of complexities between k²−k+3$k^2-k+3$ and 2^k+1$2^k+1$. The languages of another class of automata, based on k$k$-ary trees, define a complete hierarchy of complexities between 2^k+1$2^k+1$ and (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. This provides new examples of uniform languages of maximal complexity.     

On the Berlekamp/Massey algorithm and counting singular Hankel matrices over a finite field

We derive an explicit count for the number of singular n×n$n\times n$ Hankel (Toeplitz) matrices whose entries range over a finite field with q$q$ elements by observing the execution of the Berlekamp/Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n×n$n\times n$ Toeplitz matrices with 0’s on the diagonal is q²ⁿ⁻³+qⁿ⁻¹−qⁿ⁻²$q^{2n-3}+q^{n-1}-q^{n-2}$.     

An efficient construction of one-to-many node-disjoint paths in folded hypercubes

A folded hypercube is basically a hypercube with additional links augmented, where the additional links connect all pairs of nodes with longest distance in the hypercube. In an n$n$-dimensional folded hypercube, it has been shown that n+1$n+1$ node-disjoint paths from one source node to other n+1$n+1$ (mutually) distinct destination nodes, respectively, can be constructed in O(n⁴)$O\left(n^4\right)$ time so that their maximal length is not greater than ⌈n/2⌉+1$\lceil n/2\rceil +1$, where n+1$n+1$ is the connectivity and ⌈n/2⌉$\lceil n/2\rceil$ is the diameter. Besides, their maximal length is minimized in the worst case. In this paper, we further show that by minimizing the computations of minimal routing functions, these node-disjoint paths can be constructed in O(n³)$O\left(n^3\right)$ time, which is more efficient, and is hard to be reduced because it must take O(n³)$O\left(n^3\right)$ time to compute a minimal routing function by solving a corresponding maximum weighted bipartite matching problem with the best known algorithm.     

Column Subset Selection Problem is UG-hard

We address two problems related to selecting an optimal subset of columns from a matrix. In one of these problems, we are given a matrix A∈R^m×n$A\in {\mathbb{R}}^{m\times n}$ and a positive integer k, and we want to select a sub-matrix C of k   columns to minimize _{‖A−ΠCA‖F}${\Vert A-{\Pi }_{C}A\Vert }_F$, where _ΠC=CC⁺${\Pi }_C=C{C}^{+}$ denotes the matrix of projection onto the space spanned by C  . In the other problem, we are given A∈R^m×n$A\in {\mathbb{R}}^{m\times n}$, positive integers c and r, and we want to select sub-matrices C and R of c columns and r rows of A  , respectively, to minimize _{‖A−CUR‖F}${\Vert A-CUR\Vert }_F$, where U∈R^c×r$U\in {\mathbb{R}}^{c\times r}$ is the pseudo-inverse of the intersection between C and R. Although there is a plethora of algorithmic results, the complexity of these problems has not been investigated thus far. We show that these two problems are NP-hard assuming UGC.     

Atomic routing games on maximum congestion

We study atomic routing games on networks in which players choose a path with the objective of minimizing the maximum congestion along the edges of their path. The social cost is the global maximum congestion over all edges in the network. We show that the price of stability is 1. The price of anarchy  , PoA$PoA$, is determined by topological properties of the network. In particular, PoA=O(?+logn)$PoA=O(?+logn)$, where ?$?$ is the length of the longest path in the player strategy sets, and n$n$ is the size of the network. Further, κ−1≤PoA≤c(κ²+log²n)$\kappa -1\le PoA\le c({\kappa }^2+{log}^{2}n)$, where κ$\kappa$ is the length of the longest cycle in the network, and c$c$ is a constant.     

OptQC: An optimized parallel quantum compiler

The software package Qcompiler (Chen and Wang 2013) provides a general quantum compilation framework, which maps any given unitary operation into a quantum circuit consisting of a sequential set of elementary quantum gates. In this paper, we present an extended software OptQC  , which finds permutation matrices P$P$ and Q$Q$ for a given unitary matrix U$U$ such that the number of gates in the quantum circuit of U=Q^TP^TU^′PQ$U=Q^T{P}^T{U}^{\prime }PQ$ is significantly reduced, where U^′$U^{\prime }$ is equivalent to U$U$ up to a permutation and the quantum circuit implementation of each matrix component is considered separately. We extend further this software package to make use of high-performance computers with a multiprocessor architecture using MPI. We demonstrate its effectiveness in reducing the total number of quantum gates required for various unitary operators.     

Approximately optimal trees for group key management with batch updates

We investigate the group key management problem for broadcasting applications. Previous work showed that, in handling key updates, batch rekeying can be more cost effective than individual rekeying. One model for batch rekeying is to assume that every user has probability p$p$ of being replaced by a new user during a batch period with the total number of users unchanged. Under this model, it was recently shown that an optimal key tree can be constructed in linear time when p$p$ is a constant and in O(n⁴)$O\left(n^4\right)$ time when p→0$p\to 0$. In this paper, we investigate more efficient algorithms for the case p→0$p\to 0$, i.e., when membership changes are sparse. We design an O(n)$O\left(n\right)$ heuristic algorithm for the sparse case and show that it produces a nearly 2-approximation to the optimal key tree. Simulation results show that its performance is even better in practice. We also design a refined heuristic algorithm and show that it achieves an approximation ratio of 1+?$1+?$ for any fixed ?>0$?>0$ and n$n$, as p→0$p\to 0$. Finally, we give another approximation algorithm for any p∈(0,0.693)$p\in (0,0.693)$ which is shown to be quite good by our simulations.     

On a compact encoding of the swap automaton

Given a string P of length m over an alphabet Σ of size σ, a swapped version of P is a string derived from P   by a series of local swaps, i.e., swaps of adjacent symbols, such that each symbol can participate in at most one swap. We present a theoretical analysis of the nondeterministic finite automaton for the language _?P′∈ΠPΣ^?P^′${?}_{P^{\prime }\in {\Pi }_P}{\Sigma }^{?}{P}^{\prime }$ (swap automaton, for short), where _ΠP${\Pi }_P$ is the set of swapped versions of P  . Our study is based on the bit-parallel simulation of the same automaton due to Fredriksson, and reveals an interesting combinatorial property that links the automaton to the one for the language Σ^?P${\Sigma }^{?}P$. By exploiting this property and the method presented by Cantone et al. (2012), we obtain a bit-parallel encoding of the swap automaton which takes O(σ²⌈k/w⌉)$O({\sigma }^2\lceil k/w\rceil )$ space and allows one to simulate the automaton on a string of length n   in time O(n⌈k/w⌉)$O(n\lceil k/w\rceil )$, where ⌈m/σ⌉?k?m$\lceil m/\sigma \rceil ?k?m$ is the size of a specific factorization of P defined by Cantone et al. (2012) and w is the word size in bits.