Efficient algorithms for single- and two-layer linear placement of parallel graphs

This paper outlines an algorithm for optimum linear ordering (OLO) of a weighted parallel graph with O(n log k) worst-case time complexity, and O(n + k log(n/k) log k) expected-case time complexity, where n is the total number of nodes and k is the number of chains in the parallel graph. Next, the two-layer OLO problem is considered, where the goal is to place the nodes linearly in two routing layers minimizing the total wire length. The two-layer problem is shown to subsume the maxcut problem and a befitting heuristic algorithm is proposed. Experimental results on randomly generated samples show that the heuristic algorithm runs very fast and outputs optimum solutions in more than 90% instances.     

On fast planning of suboptimal paths amidst polygonal obstacles in plane

The problem of planning a path for a point robot from a source point s to a destination point d so as to avoid a set of polygonal obstacles in plane is considered. Using well-known methods, a shortest path from s to d can be computed with a time complexity of O(n²) where n is the total number of obstacle vertices. The focus here is in

1. (a) planning paths faster at the expense of setting for suboptimal path lengths and

2. (b) performance analysis of simple and/or well-known suboptimal methods.

A method that enables a hierarchical implementation of any path planning algorithm with no increase in the worst-case time complexity, is presented; this implementation enables fast planning of simple paths. Then methods are presented based on the Voronoi diagrams, trapezoidal decomposition and triangulation, which compute (suboptimal) paths in O(n√log n) time with the preprocessing costs of O(n log n), O(n²) and O(n log n), respectively. Using existing navigational algorithms for unknown terrains, algorithms that run in O(n log n) time (after preprocessing) and yield suboptimal paths, are presented. For all these algorithms, upper bounds on the path lengths are estimated in terms of the shortest of the obstacles, etc.     

A new complexity bound for the least-squares problem

For a least-squares problem of m degree polynomial regression with n measured values (n ≥ m + 1), the traditional methods need O(n²m) arithmetic operations. We prove that the arithmetic complexity of this problem does not exceed O(nlog²₂m).     

Computational complexity of distributed detection problems with information constraints

For a system consisting of a set of sensors S = {S₁, S₂, …, S_m} and a set of objects O = {O₁, O₂, …, O_n}, there are information constraints given by a relation R S × O such that (S_i, O_j) R if and only if S_i is capable of detecting O_j. Each (S_i, O_j) R is assigned a confidence factor (a positive real number) which is either explicitly given or can be efficiently computed. Given that a subset of sensors have detected obstacles, the detection problem is to identify a subset H O with the maximum confidence value. The computational complexity of the detection problem, which depends on the nature of the confidence factor and the information constraints, is the main focus of this paper. This problem exhibits a myriad of complexity levels ranging from a worst-case exponential (in n) lower bound in a general case to an O(m + n) time solvability. We show that the following simple versions of a detection problem are computationally intractable: (a) deterministic formulation, where confidence factors are either 0 or 1; (b) uniform formulation where (S_i, O_j) R, for all S_i S, O_j O; (c) decomposable systems under multiplication operation. We then show that the following versions are solvable in polynomial (in n) time: (a) single object detection; (b) probabilistically independent detection; (c) decomposable systems under additive and nonfractional multiplicative measures; and (d) matroid systems.     

A method of inexact steepest descent for systems of linear equations

Our approach combines the method of inexact steepest descent with the method of contractor directions to obtain an algorithm for solving systems of linear equations. In order to enhance the scope of applicability, we consider an iterative method with variable step-size iterations. We prove the convergence and given an error estimate for our method.

The algorithm is well-suited for parallel computation. In fact, for systems with m equations and n unknowns, each iteration may be computed in parallel time O(log m + log n), on an EREW PRAM with O(mn) processors.     

On the power of alternation on reversal-bounded alternating Turing machines with a restriction

Whether or not there is a difference of the power among alternating Turing machines with a bounded number of alternations is one of the most important problems in the field of computer science. This paper presents the following result: Let R(n) be a space and reversal constructible function. Then, for any k 1, we obtain that the class of languages accepted by off-line 1-tape rσ_k machines running in reversal O(R(n)) is equal to the class of languages accepted by off-line 1-tape σ₁ machines running in reversal O(R(n)). An off-line 1-tape σ_k machine M is called an off-line 1-tape rσ_k machine if M always limits the non-blank part of the work-tape to at most O(R(n) log n) when making an alternation between universal and existential states during the computation.     

An augmented Voronoi roadmap for 3D translational motion planning for a convex polyhedron moving amidst convex polyhedral obstacles

This paper concerns the development of a piecewise linear Voronoi roadmap for translating a convex polyhedron in a three-dimensional (3-D) polyhedral world. In general the Voronoi roadmap is incomplete for motion planning, i.e., it can have several disjoint components in one connected component of free space. An analysis of the roadmap shows that incompleteness is caused by the occurrence of the following simple geometric structure: a polygon in the Voronoi surface containing one or more polygons inside it. We formally bring out the details of this geometric structure and give an efficient augmentation of the roadmap that makes it complete. We show that the roadmap has size e = O(n²Q²l²), where n is the total number of faces on the obstacles, Q is the total number of obstacles and l is the number of faces on the moving object. We also present an algorithm to construct the roadmap in O((n + Ql)e + Q²log Q) time.     

Contrast-optimal k out of n secret sharing schemes in visual cryptography

Visual cryptography and (k,n)-visual secret sharing schemes were introduced by Naor and Shamir (Advances in Cryptology — Eurocrypt 94, Springer, Berlin, 1995, pp. 1–12). A sender wishing to transmit a secret message distributes n transparencies amongst n recipients, where the transparencies contain seemingly random pictures. A (k,n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k−1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message. The important parameters of a scheme are its contrast, i.e., the clarity with which the message becomes visible, and the number of subpixels needed to encode one pixel of the original picture. Naor and Shamir constructed (k,k)-schemes with contrast 2^−(k−1). By an intricate result from Linial (Combinatorica 10 (1990) 349–365), they were also able to prove the optimality of these schemes. They also proved that for all fixed kn, there are (k,n)-schemes with contrast . For k=2,3,4 the contrast is approximately and . In this paper, we show that by solving a simple linear program, one is able to compute exactly the best contrast achievable in any (k,n)-scheme. The solution of the linear program also provides a representation of a corresponding scheme. For small k as well as for k=n, we are able to analytically solve the linear program. For k=2,3,4, we obtain that the optimal contrast is at least and . For k=n, we obtain a very simple proof of the optimality of Naor's and Shamir's (k,k)-schemes. In the case k=2, we are able to use a different approach via coding theory which allows us to prove an optimal tradeoff between the contrast and the number of subpixels.     

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.     

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.     

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。     

Experimental numerical studies on a supercomputer of natural convection in an enclosure with localized heating

We present particle simulations of natural convection of a symmetrical, nonlinear, three-dimensional cavity flow problem. Qualitative studies are made in an enclosure with localized heating. The assumption is that particles interact locally by means of a compensating Lennard-Jones type force F, whose magnitude is given by −G/r^p + H/r^q.

In this formula, the parameters G, H, p, q depend upon the nature of the interacting particles and r is the distance between two particles. We also consider the system to be under the influence of gravity. Assuming that there are n particles, the equations relating position, velocity and acceleration at time t_k = kΔt, K = 0, 1, 2, …, are solved simultaneously using the “leap-frog” formulas. The basic formulas relating force and acceleration are Newton's dynamical equations F_i,k = m_ia_i,k, I = 1, 2, 3, …, n, where m_i is the mass of the ith particle.

Extensive and varied computations on a CRAY X - MP/24 are described and discussed, and comparisons are made with the results of others.     

Efficient algorithms for automatic viewer orientation

The calculation of a perspective image of an object involves the position and orientation of the viewer. In many graphics applications the viewpoint and object are fixed, and an orientation is sought in which the object is “centered” in the field of view. Previous work has proposed that the viewing direction be the axis of the narrowest circular cone emanating from the viewpoint and containing the object, and has shown how this direction can be calculated based on the viewpoint and a set of n object points which includes the object's extreme points. This paper presents algorithms which efficiently accomplish this task, in O(n) average and O(n log(n)) worst-case time. The orientation problem is converted into a problem in spherical geometry, and the proposed algorithms are based on existing algorithms for the analogous plane geometry problems.     

An algorithmic view of gene teams

Comparative genomics is a growing field in computational biology, and one of its typical problem is the identification of sets of orthologous genes that have virtually the same function in several genomes. Many different bioinformatics approaches have been proposed to define these groups, often based on the detection of sets of genes that are “not too far” in all genomes. In this paper, we propose a unifying concept, called gene teams, which can be adapted to various notions of distance. We present two algorithms for identifying gene teams formed by n genes placed on m linear chromosomes. The first one runs in O(mn log² n) and uses a divide and conquer approach based on the formal properties of gene teams. We next propose an optimization of the original algorithm, and, in order to better understand the complexity bound of the algorithms, we recast the problem in the Hopcroft's partition refinement framework. This allows us to analyze the complexity of the algorithms with elegant amortized techniques. Both algorithms require linear space. We also discuss extensions to circular chromosomes that achieve the same complexity.     

A new routing algorithm for multirate rearrangeable Clos networks

In this paper, we study the problem of finding routing algorithms on the multirate rearrangeable Clos networks which use as few number of middle-stage switches as possible. We propose a new routing algorithm called the “grouping algorithm”. This is a simple algorithm which uses fewer middle-stage switches than all known strategies, given that the number of input-stage switches and output-stage switches are relatively small compared to the size of input and output switches. In particular, the grouping algorithm implies that m = 2n+(n−1)/2^k is a sufficient number of middle-stage switches for the symmetric three-stage Clos network C(n,m,r) to be multirate rearrangeable, where k is any positive integer and rn/(2^k−1).     

On the number of occurrences of all short factors in almost all words

We previously proved that almost all words of length n over a finite alphabet A with m letters contain as factors all words of length k(n) over A as n→∞, provided limsup_n→∞ k(n)/log n<1/log m.

In this note it is shown that if this condition holds, then the number of occurrences of any word of length k(n) as a factor into almost all words of length n is at least s(n), where lim_n→∞ log s(n)/log n=0. In particular, this number of occurrences is bounded below by C log n as n→∞, for any absolute constant C>0.     

A parallel two-list algorithm for the knapsack problem

An n-element knapsack problem has 2ⁿ possible solutions to search over, so a task which can be accomplished in 2″ trials if an exhaustive search is used. Due to the exponential time in solving the knapsack problem, the problem is considered to be very hard. In the past decade, much effort has been done in order to find techniques which could lead to practical algorithms with reasonable running time. In 1994, Chang et al. proposed a brilliant parallel algorithm, which needs O(2^n/8) processors to solve the knapsack problem in O(2^n/2) time; that is, the cost of Chang et al.'s parallel algorithm is O(2^5n/8). In this paper, we propose a parallel algorithm to improve Chang et al.'s parallel algorithm by reducing the time complexity to be O(2^3n/8) under the same O(2^n/8) processors available. Thus, the proposed parallel algorithm has a cost of O(2^n/2). It is an improvement over previous literature. We believe that the proposed parallel algorithm is pragmatically feasible at the moment when multiprocessor systems become more and more popular.     

A new way of counting n

In this paper, we shall give a combinatorial proof of the following equation:

,

where m and n are positive integers, m ≤ n, and k₁, k₂, …, k_n-1 are nonnegative integers.     

Build shape-shifting tries for fast IP lookup in O(n) time

Most data structures for packet forwarding are optimized for IPv4, and less capable of handling IPv6 routing tables. Shape-Shifting Trie (SST) is specifically designed to be scalable to IPv6. To reduce the worst-case lookup time that is proportional to the height of SST, it is desirable to construct a minimum-height SST. Breadth-First Pruning (BFP) algorithm takes O(n²) time to construct a minimum-height SST, where n is the number of nodes in the binary trie corresponding to the routing table. In this paper, we propose a Post-Order Minimum-Height Pruning (POMHP) algorithm that takes only O(n) time to construct a minimum-height SST. We further propose nodes merging algorithm to cut down SST size without affecting SST height.     

O(log n) bimodality analysis

The bimodality of a population P can be measured by dividing its range into two intervals so as to maximize the Fisher distance between the resulting two subpopulations P₁ and P₂. If P is a mixture of two (approximately) Gaussian subpopulations, then P₁ and P₂ are good approximations to the original Gaussians, if their Fisher distance is great enough. Moreover, good approximations to P₁ and P₂ can be obtained by dividing P into small parts; finding the maximum-distance (MD) subdivision of each part; combining small groups of these subdivisions into (approximate) MD subdivisions of larger parts; and so on. This divide-and-conquer approach yields an approximate MD subdivision of P in O(log n) computational steps using O(n) processors, where n is the size of P.