Information Processing Letters最新文献

Recently, Jecker has introduced and studied the regular $D$-length of a monoid, as the length of its longest chain of regular $D$-classes. We use this parameter in order to improve the construction, originally proposed by Colcombet, of a deterministic automaton that allows to map a word to one of its forward Ramsey splits: these are a relaxation of factorisation forests that enjoy prefix stability, thus allowing a compositional construction. For certain monoids that have a small regular $D$-length, our construction produces an exponentially more succinct deterministic automaton. Finally, we apply it to obtain better complexity result for the problem of fast infix evaluation.

Let A, B, and C be three $n\times n$ matrices. We investigate the problem of verifying whether $AB=C$ over the ring of integers and finding the correct product AB. Given that C is different from AB by at most k entries, we propose an algorithm that uses $O(\sqrt{k}{n}^2+k^{2}n)$ operations. Let α be the largest absolute value of an entry in A, B, and C. The integers involved in the computation are of $O\left(n^3{\alpha }^2\right)$.

We study the problem of computing the center of cycle graphs whose vertices are weighted. The distance from a vertex to a point of the graph is defined as the weight of the vertex times the length of the shortest path between the vertex and the point. The weighted center of the graph is a point of the graph such that the maximum distance of the vertices of the graph to the point is minimum among all points of the graph. We present an $O\left(n\right)$-time algorithm for the discrete and continuous weighted center problem on cycle graphs with n vertices. Our algorithm improves upon the best known algorithm that takes $O(n\log n)$ time. Moreover, it is optimal for the weighted center problem of cycle graphs.

Recently, a class of quaternary sequences with period pq, where p and q are two distinct odd primes introduced by Zhang et al. were proved to possess high linear complexity and 4-adic complexity. In this paper, we determine the autocorrelation distribution of this class of quaternary sequence. Our results indicate that the studied quaternary sequence are weak with respect to the correlation property.

The qCCS model proposed by Feng et al. provides a powerful framework to describe and reason about quantum communication systems that could be entangled with the environment. However, they only studied weak bisimulation semantics. In this paper we propose a new branching bisimilarity for qCCS and show that it is a congruence. The new bisimilarity is based on the concept of ϵ-tree and preserves the branching structure of concurrent processes where both quantum and classical components are allowed. Furthermore, we present a polynomial time equivalence checking algorithm for the ground version of our branching bisimilarity.

In this paper we consider two vertex deletion problems. In the Block Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a block graph (a graph in which every biconnected component is a clique). In the Pathwidth One Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a graph with pathwidth at most one. We give a kernel for Block Vertex Deletion with $O\left(k^3\right)$ vertices and a kernel for Pathwidth One Vertex Deletion with $O\left(k^2\right)$ vertices. Our results improve the previous $O\left(k^4\right)$-vertex kernel for Block Vertex Deletion (Agrawal et al., 2016 [1]) and the $O\left(k^3\right)$-vertex kernel for Pathwidth One Vertex Deletion (Cygan et al., 2012 [3]).

In the Longest $(s,t)$-Detour problem, we look for an $(s,t)$-path that is at least k vertices longer than a shortest one. We study the parameterized complexity of Longest $(s,t)$-Detour when parameterized by k: this falls into the research paradigm of ‘parameterization above guarantee’. Whereas the problem is known to be fixed-parameter tractable (FPT) on undirected graphs, the status of Longest $(s,t)$-Detour on directed graphs remains highly unclear: it is not even known to be solvable in polynomial time for $k=1$. Recently, Fomin et al. made progress in this direction by showing that the problem is FPT on every class of directed graphs where the 3-Disjoint Paths problem is solvable in polynomial time. We improve upon their result by weakening this assumption: we show that only a polynomial-time algorithm for 2-Disjoint Paths is required.

The seminal work of Charikar et al. [1] called Count-Sketch suggests a sketching algorithm for real-valued vectors that has been used in frequency estimation for data streams and pairwise inner product estimation for real-valued vectors etc. One of the major advantages of Count-Sketch over other similar sketching algorithms, such as random projection, is that its running time, as well as the sparsity of sketch, depends on the sparsity of the input. Therefore, sparse datasets enjoy space-efficient (sparse sketches) and faster running time. However, on dense datasets, these advantages of Count-Sketch might be negligible over other baselines. In this work, we address this challenge by suggesting a simple and effective approach that outputs (asymptotically) a sparser sketch than that obtained via Count-Sketch, and as a by-product, we also achieve a faster running time. Simultaneously, the quality of our estimate is closely approximate to that of Count-Sketch. For frequency estimation and pairwise inner product estimation problems, our proposal Sparse-Count-Sketch provides unbiased estimates. These estimations, however, have slightly higher variances than their respective estimates obtained via Count-Sketch. To address this issue, we present improved estimators for these problems based on maximum likelihood estimation (MLE) that offer smaller variances even w.r.t. Count-Sketch. We suggest a rigorous theoretical analysis of our proposal for frequency estimation for data streams and pairwise inner product estimation for real-valued vectors.

Given a universe $U=R\cup B$ of a finite set of red elements R, and a finite set of blue elements B and a family $F$ of subsets of $U$, the Red Blue Set Cover problem is to find a subset $F^{\prime }$ of $F$ that covers all blue elements of B and minimum number of red elements from R.

We prove that the Red Blue Set Cover problem is NP-hard even when R and B respectively are sets of red and blue points in ${\mathrm{IR}}^2$ and the sets in $F$ are defined by axis−parallel lines i.e., every set is a maximal set of points with the same x or y coordinate.

We then study the parameterized complexity of a generalization of this problem, where $U$ is a set of points in ${\mathrm{IR}}^d$ and $F$ is a collection of set of axis−parallel hyperplanes in ${\mathrm{IR}}^d$ under different parameterizations, where d is a constant. For every parameter, we show that the problem is fixed-parameter tractable and also show the existence of a polynomial kernel. We further consider the Red Blue Set Cover problem for some special types of rectangles in ${\mathrm{IR}}^2$.

With the development of Lie theory, Lie groups have profound significance in many branches of mathematics and physics. In Lie theory, matrix exponential plays a crucial role between Lie groups and Lie algebras. Meanwhile, as finite analogues of Lie groups, finite groups of Lie type also have wide application scenarios in mathematics and physics owning to their unique mathematical structures. In this context, it is meaningful to explore the potential applications of finite groups of Lie type in cryptography. In this paper, we firstly built the relationship between matrix exponential and discrete logarithmic problem (DLP) in finite groups of Lie type. Afterwards, we proved that the complexity of solving non-abelian factorization (NAF) problem is polynomial with the rank n of the finite group of Lie type. Furthermore, combining with the Algebraic Span, we proposed an efficient algorithm for solving group factorization problem (GFP) in finite groups of Lie type. Therefore, it's still an open problem to devise secure cryptosystems based on Lie theory.