A complete algebraic solvability test for the nonstrict Lyapunov inequality

For arbitrary equally sized square complex matrices A and Q (Q Hermitian), the paper provides a complete algebraic test for verifying the existence of a Hermitian solution X of the nonstrict Lyapunov inequality A^*X + XA + Q 0. If existing, we exhibit how to construct a solution. Our approach involves the validation problem for the linear matrix inequality Σ_j=1^k (A_j^*X_jB_j + B_j^*X_j^*A_j) + Q> 0 in X_j, for which we provide an algebraic solvability test and a construct solutions if the kernels of A_j or, dually, those of B_j form an isotonic sequence.     

Constant-time algorithm for computing the Euclidean distance maps of binary images on 2D meshes with reconfigurable buses

The computation of Euclidean distance maps (EDM), also called Euclidean distance transform, is a basic operation in computer vision, pattern recognition, and robotics. Fast computation of the EDM is needed since most of the applications using the EDM require real-time computation. It is shown in L. Chen and H.Y.H. Chuang [Information Processing Letters, 51, pp. 25–29 (1994)] that a lower bound Ω(n²) is required for any sequential EDM algorithm due to the fact that in any EDM algorithm each of the n² pixels has to be scanned at least once. Recently, many parallel EDM algorithms have been proposed to speedup its computation. Chen and Chuang proposed an algorithm for computing the EDM on an n×n mesh in O(n) time [L. Chen and H.Y.H. Chuang Parallel Computing, 21, pp. 841–852 (1995)]. Clearly, the VLSI complexities of both the sequential and the mesh algorithm described in L. Chen and H.Y.H. Chuang [Parallel Computing, 21, pp. 841–852 (1995)] are AT²=O(n⁴), where A is the VLSI layout area of the design and T is the computation time using area A when implemented in VLSI. In this paper, we propose a new and faster parallel algorithm for computing the EDM problem on the reconfigurable VLSI mesh model. For the same problem, our algorithm runs in O(1) time on a two-dimensional n²×n² reconfigurable mesh. We show that the VLSI complexity of our algorithm is the same as those of the above sequential algorithm and the mesh algorithm, while it uses much less time. To our best knowledge, this is the first constant-time EDM algorithm on any parallel computational model.     

Inferring evolutionary trees with strong combinatorial evidence

We consider the problem of inferring the evolutionary tree of a set of n species. We propose a quartet reconstruction method which specifically produces trees whose edges have strong combinatorial evidence. Let Q be a set of resolved quartets defined on the studied species, the method computes the unique maximum subset Q^* of Q which is equivalent to a tree and outputs the corresponding tree as an estimate of the species’ phylogeny. We use a characterization of the subset Q^* due to Bandelt and Dress (Adv. Appl. Math. 7 (1986) 309–343) to provide an O(n⁴) incremental algorithm for this variant of the NP-hard quartet consistency problem. Moreover, when chosing the resolution of the quartets by the four-point method (FPM) and considering the Cavender–Farris model of evolution, we show that the convergence rate of the Q^* method is at worst polynomial when the maximum evolutive distance between two species is bounded. We complete these theoretical results by an experimental study on real and simulated data sets. The results show that (i) as expected, the strong combinatorial constraints it imposes on each edge leads the Q^* method to propose very few incorrect edges; (ii) more surprisingly; the method infers trees with a relatively high degree of resolution.     

离散线性一致性算法噪声问题研究

多智能体一致性问题在传感网、社交网、协同控制等诸多领域有着广泛的实际应用背景, 本文对离散线性一致性算法的噪声问题进行了研究, 证明了离散线性 一致性算法的噪声不可控性; 提出基于抑噪算子ε(t)的噪声控制策略, 指出当ε(t)为t-0.5的高阶无穷小时, 抑噪后的一致性算法噪声可控; 分析了抑噪算子对一致性 算法收敛性的影响, 证明了在无噪声条件下, 当抑噪算子ε(t为t-1的低阶无穷小时, 抑噪后的一致性算法依然可以使Agent收敛至原收敛状态x*.在上述结论基础上进一步指出, 当t→∞ 时, 若抑噪算子ε(t)的阶在t-0.5~t-1之间, 所有Agent 的状态将以原收敛状态x* 为中心呈正态分布. 最后, 以DHA 为例对相应理论结果进行了验证和讨论. 本文为线性一致性算法的噪声控制提供了理论依据, 对抑噪算s子的确定有较强的指导意义.     

Nagging: A scalable fault-tolerant paradigm for distributed search

This paper describes nagging, a technique for parallelizing search in a heterogeneous distributed computing environment. Nagging exploits the speedup anomaly often observed when parallelizing problems by playing multiple reformulations of the problem or portions of the problem against each other. Nagging is both fault tolerant and robust to long message latencies. In this paper, we show how nagging can be used to parallelize several different algorithms drawn from the artificial intelligence literature, and describe how nagging can be combined with partitioning, the more traditional search parallelization strategy. We present a theoretical analysis of the advantage of nagging with respect to partitioning, and give empirical results obtained on a cluster of 64 processors that demonstrate nagging's effectiveness and scalability as applied to A^* search, β minimax game tree search, and the Davis–Putnam algorithm.     

Computation of generalised inverse of a special class of matrices

The computation of the generalised Inverse A⁺ of matrix A depends critically of the rank of A and involves several matrix multiplications. It is shown here that if A is of the form where A_i are now vectors, then A⁺ can be computed efficiently and accurately by a simple algebraic method.     

Efficient enumeration of all minimal separators in a graph

This paper presents an efficient algorithm for enumerating all minimal a-b separators separating given non-adjacent vertices a and b in an undirected connected simple graph G = (V, E), Our algorithm requires O(n³R_ab) time, which improves the known result of O(n⁴R_ab) time for solving this problem, where ¦V¦= n and R_ab is the number of minimal a-b separators. The algorithm can be generalized for enumerating all minimal A-B separators that separate non-adjacent vertex sets A, B < V, and it requires O(n²(n − n_A − n_b)R_AB) time in this case, where n_a = ¦A¦, n_B = ¦B¦ and r_AB is the number of all minimal A−B separators. Using the algorithm above as a routine, an efficient algorithm for enumerating all minimal separators of G separating G into at least two connected components is constructed. The algorithm runs in time O(n³R⁺_Σ + n⁴R_Σ), which improves the known result of O(n⁶R_Σ) time, where R_σ is the number of all minimal separators of G and R_ΣR⁺_Σ = ∑_1i, v_j) ER_vivj n − 1)/2 − m)R_Σ. Efficient parallelization of these algorithms is also discussed. It is shown that the first algorithm requires at most O((n/log n)R_ab) time and the second one runs in time O((n/log n)R⁺_Σ+n log nR_Σ) on a CREW PRAM with O(n³) processors.     

Regular Model Checking using Widening Techniques

In this paper, we consider symbolic model checking of safety properties of linear parametrized systems. Sets of configurations are represented by regular languages and actions by regular relations. Since the verification problem amounts to the computation of the reachability set, we focus on the computation of R^*(φ) for a regular relation R and a regular language φ. We present a technique called regular widening that allows, when it terminates, the computation of either the reachability set R^*(φ) of a system or the transitive closure R^* of a regular relation. We show that our method can be uniformly applied to several parametrized systems. Furthermore, we show that it is powerful enough to simulate some existing methods that compute either R^* or R^*(φ) for each R (resp. φ) belonging to a subclass of regular relations (resp. belonging to a subclass of regular languages).     

基于图勾勒的图链路预测方法

针对已有链路预测算法复杂度高,不适于在大规模图上进行链接预测的问题,本文基于图勾勒近似技术对已有链路预测方法进行优化,提出了基于图勾勒的链路预测方法。该方法将链路预测算法的计算复杂度由O（n³）降低至O（n²k²log²n）。为进一步提高链接预测效率,给出了基于Spark的并行化链路预测实现方法。在真实图数据集上进行测试,实验结果表明本文方法在保证链接预测精度的前提下,可有效提升算法效率。     

Shortest path and closure algorithms for banded matrices

A fast algorithm is given for the all-pairs shortest paths problem for banded matrices having band-width b. It solves the negative-cycle problem and calculates all path lengths within the band in O(nb²) time and calculates all other path lengths in O(n²b) time.     

A knowledge-based approach to multiple query processing

The collective processing of multiple queries in a database system has recently received renewed attention due to its capability of improving the overall performance of a database system and its applicability to the design of knowledge-based expert systems and extensible database systems. A new multiple query processing strategy is presented which utilizes semantic knowledge on data integrity and information on predicate conditions of the access paths (plans) of queries. The processing of multiple queries is accomplished by the utilization of subset relationships between intermediate results of query executions, which are inferred employing both semantic and logical information. Given a set of fixed order access plans, the A^* algorithm is used to find the set of reformulated access plans which is optimal for a given collection of semantic knowledge.     

Spacetime-minimal systolic arrays for Gaussian elimination and the Algebraic path problem

In this paper, we derive time-minimal systolic arrays for Gaussian elimination and the Algebraic Path Problem (APP) that use a minimal number of processors. For a problem of size n, we obtain an execution time T(n) = 3n −1 using A(n) = n²/4+O(n) processors for Gaussian elimination, and T(n) = 5n −2 and A(n) = n³/+O(n) for the APP.     

Finding nondegenerate Hopf bifurcation points for four-dimensional two-parametric systems

We establish a numerically feasible algorithm to find a simplicial approximation A to a certain part of the boundary of the set of stable (or Hurwitz) polynomials of degree 4. Moreover, we have that . Using this, we build an algorithm to find a piecewise-linear approximation to the intersection curve of a given surface contained in ⁴ with . We have also devised an efficient computer program to perform all these operations. The main motivation is to find the curve of nondegenerate bifurcation points in parameter space for a given 2-parametric Hopf bifurcation problem of dimension 4.     

Enhanced A algorithms for multiple alignments: optimal alignments for several sequences and k-opt approximate alignments for large cases

The multiple alignment of the sequences of DNA and proteins is applicable to various important fields in molecular biology. Although the approach based on Dynamic Programming is well-known for this problem, it requires enormous time and space to obtain the optimal alignment. On the other hand, this problem corresponds to the shortest path problem and the A^* algorithm, which can efficiently find the shortest path with an estimator, is usable.

First, this paper directly applies the A^* algorithm to multiple sequence alignment problem with more powerful estimator in more than two-dimensional case and discusses the extensions of this approach utilizing an upper bound of the shortest path length and of modification of network structure. The algorithm to provide the upper bound is also proposed in this paper. The basic part of these results was originally shown in Ikeda and Imai [11]. This part is similar to the branch-and-bound techniques implemented in MSA program in Gupta et al. [6]. Our framework is based on the edge length transformation to reduce the problem to the shortest path problem, which is more suitable to generalizations to enumerating suboptimal alignments and parametric analysis as done in Shibuya and Imai [15–17]. By this enhanced A^* algorithm, optimal multiple alignments of several long sequences can be computed in practice, which is shown by computational results.

Second, this paper proposes a k-group alignment algorithm for multiple alignment as a practical method for much larger-size problem of, say multiple alignments of 50–100 sequences. A basic part of these results were originally presented in Imai and Ikeda [13]. In existing iterative improvement methods for multiple alignment, the so-called group-to-group two-dimensional dynamic programming has been used, and in this respect our proposal is to extend the ordinary two-group dynamic programming to a k-group alignment programming. This extension is conceptually straightforward, and here our contribution is to demonstrate that the k-group alignment can be implemented so as to run in a reasonable time and space under standard computing environments. This is established by generalizing the above A^* search approach. The k-group alignment method can be directly incorporated in existing methods such as iterative improvement algorithms [2, 5] and tree-based (iterative) algorithms [9]. This paper performs computational experiments by applying the k-group method to iterative improvement algorithms, and shows that our approach can find better alignments in reasonable time. For example, through larger-scale computational experiments here, 34 protein sequences with very high homology can be optimally 10-group aligned, and 64 sequences with high homology can be optimally 5-group aligned.     

The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan

In this paper we consider the unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an O(n(n/m+1)^m) time dynamic programming algorithm and an O(mk^k+1P^2k−1) time dynamic programming algorithm, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the processing times of all families. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme for the problem.     

A simple solution to the l optimization problem

It is pointed out in this brief paper that the l¹ optimization problem min_{Q ε lqp¹} | H − U * Q * V |₁, H ε l_mn¹, U ε l_mq¹, V ε l_pn¹ can be solved in one step rather than two. The solution of the dual problem is obviated by the direct solution of the primal problem via linear programming. The method here is applicable to finite-dimensional problems or approximating finite-dimensional problems, in the general case.     

A fast vector algorithm for solving tridiagonal linear equations

We present a fast vector algorithm which solves tridiagonal linear equations by an optimum synthesis of the inherently recursive Gaussian elimination and the parallel though complex cyclic reduction. The idea is to perform an incomplete cyclic reduction to bring the dimension of the tridiagonal system efficiently below a characteristic size n^* and then to solve the remaining system by Gaussian elimination. Extensive numerical experiments on the CYBER 205 and the CRAY X-MP computers reveal a maximum vector speedup of 13 and prove n^* to reflect the architecture of the vector computer. The performance is further enhanced when a feq right-hand sides are treated simultaneously.     

Ensuring compatibility of G-continuous surface patches around a nodepoint

The problem of ensuring compatibility of mixed partial derivative vectors of surface patches joining G²-continuously around a common nodepoint is essential in modelling G²-continuous n-sided surfaces. Although the compatibility constraints can be removed by using C² Gregory patches, these patches have singularities at their corner points. This paper presents conditions for ensuring the compatibility of the mixed partial derivative vectors of surface patches joining G²-continuously around a common nodepoint. After investigating the solvability of these compatibility conditions, a new solution method exploiting G³-continuity of surface patches at a common nodepoint is given. Example surfaces based on this solution method are also provided.     

Maximal Common Subsequences and Minimal Common Supersequences

The problems of finding a longest common subsequence and a shortest common supersequence of a set of strings are well known. They can be solved in polynomial time for two strings (in fact the problems are dual in this case), or for any fixed number of strings, by dynamic programming. But both problems are NP-hard in general for an arbitrary numberkof strings. Here we study the related problems of finding a shortest maximal common subsequence and a longest minimal common supersequence. We describe dynamic programming algorithms for the case of two strings (for which case the problems are no longer dual), which can be extended to any fixed number of strings. We also show that both problems are NP-hard in general forkstrings, although the latter problem, unlike shortest common supersequence, is solvable in polynomial time for strings of length 2. Finally, we prove a strong negative approximability result for the shortest maximal common subsequence problem.     

多粒度粗糙集粒度约简的高效算法

针对已有多粒度粗糙集粒度约简算法效率较低的问题,提出一种多粒度粗糙集粒度约简的高效算法（EAGRMRS）。首先,以决策信息系统为对象,定义决策类下近似布尔矩阵,该矩阵能够将粒度约简过程中过多且有重复的集合运算转换为布尔运算,基于该矩阵给出计算决策类下近似算法和计算粒度重要度算法。然后,针对计算粒度重要度时存在冗余计算的问题,提出粒度动态增加时快速计算粒度重要度的算法,并在此基础上,提出EAGRMRS,该算法的时间复杂度为O（|A|·|U|²+|A|²·|U|）,其中|A|表示粒度集合大小,|U|表示决策信息系统中实例数。在UCI数据集上的实验结果验证了所提算法的有效性和高效性,并且随着数据集的增大,EAGRMRS相较于多粒度粗糙集粒度约简的启发式算法（HAGSS）效率优势更加明显。