How efficiently can room at the bottom be traded away for speed at the top?

Given exponential 2 ⁿ space, we know that an Adleman-Lipton computation can decide many hard problems – such as boolean formula and boolean circuit evaluation – in a number of steps that is linear in the problem size n. We wish to better understand the process of designing and comparing bio-molecular algorithms that trade away weakly exponential space to achieve as low a running time as possible, and to analyze the efficiency of their space and time utilization relative to those of their best extant classical/bio-molecular counterparts. We propose a randomized framework which augments that of the sticker model of Roweis et al. to provide an abstract setting for analyzing the space-time efficiency of both deterministic and randomized bio-molecular algorithms. We explore its power by developing and analyzing such algorithms for theCovering Code Creation (CCC) and k-SAT problems. In the process, we uncover new classical algorithms for CCC andk-SAT that, while exploiting the same space-time trade-off as the best previously known classical algorithms, are exponentially more efficient than them in terms of space-time product utilization. This work indicates that the proposed abstract bio-molecular setting for randomized algorithm design provides a logical tool of independent interest. This revised version was published online in June 2006 with corrections to the Cover Date.     

高速流环境下近似连续k代表轮廓查询算法

k代表轮廓查询是从传统轮廓查询中衍生出来的一类查询.给定多维数据集合D,轮廓查询从D中找到所有不被其他对象支配的对象,将其返回给用户,便于用户结合自身偏好选择高质量对象.然而,轮廓对象规模通常较大,用户需要从大量数据中进行选择,导致选择速度和质量无法得到保证.与传统轮廓查询相比,k代表轮廓查询从所有轮廓对象中选择“代表性”最强的k个对象返回给用户,有效地解决了传统轮廓查询存在的这一问题.给定滑动窗口W和连续查询q,q监听窗口中的数据.当窗口滑动时,查询q返回窗口中,组合支配面积最大的k个对象.现有算法的核心思想是:实时监测当前窗口中的轮廓对象集合,当轮廓对象集合更新时,算法更新k代表轮廓.然而,实时监测窗口中,轮廓集合的计算代价通常较大.此外,当轮廓集合规模较大时,从中选择k代表轮廓的计算代价是同样巨大的,导致已有算法无法在高速流环境下使用.针对上述问题,提出了ρ-近似k代表轮廓查询.为了支持该查询,提出了查询处理框架PAKRS(predict-basedapproximatekrepresentativeskyline).首先,PAKRS利用高速流的特性对当前窗口进行划分,根据划分结...     

Systolic parsing of context-free languages

We report on a new systolic context-free language parsing algorithm. We present both single parse and multiple parse versions of the algorithm. For an input of lengthn the single parse version has a time complexity ofO(n) and a space complexity ofO(n) ². This equals the performance of the most effcient known single parse algorithm. The space complexity of the multiple parse version is also ofO(n) ² while its time complexity is ofO(mn) wherem is the number of differenti parses for the input. This equals the time complexity of the most efficient known multiple parse algorithms and brings an improvenment of a factor of logn over its space complexity.This work was supported by Université de Montréal, the Natural Science Engineering Resarch Council of Canada and the Communittee of Vice-Chancellors and Principals of the Universities of the United Kingdom.     

Efficient randomized generation of optimal algorithms for multiplication in certain finite fields

LetN _max(q) denote the maximum number of points of an elliptic curve over F _q . Given a prime powerq=p ^f and an integern satisfying 1/2q+1<n(N _max(q)–2)/2, we present an algorithm which on inputq andn produces an optimal bilinear algorithm of length 2n for multiplication in F _q ⁿ /F _q . The algorithm takes roughlyO(q ⁴+n ⁴logq) F _q -operations or equivalentlyO((q ⁴+n ⁴logq)f ²log² p) bit-operations to compute the output data.     

面向移动对象连续k近邻查询的双层索引结构

移动对象连续k近邻（CKNN）查询是指给定一个连续移动的对象集合,对于任意一个k近邻查询q,实时计算查询q的k近邻并在查询有效时间内对查询结果进行实时更新.现实生活中,交通出行、社交网络、电子商务等领域许多基于位置的应用服务都涉及移动对象连续k近邻查询这一基础问题.已有研究工作解决连续k近邻查询问题时,大多需要通过多次迭代确定一个包含k近邻的查询范围,而每次迭代需要根据移动对象的位置计算当前查询范围内移动对象的数量,整个迭代过程的计算代价占查询代价的很大部分.为此,提出了一种基于网络索引和混合高斯函数移动对象分布密度的双重索引结构（grid GMM index,GGI）,并设计了移动对象连续k近邻增量查询算法（incremental search for continuous k nearest neighbors,IS-CKNN）.GGI索引结构的底层采用网格索引对海量移动对象进行维护,上层构建混合高斯模型模拟移动对象在二维空间中的分布.对于给定的k近邻查询q,IS-CKNN算法能够基于混合高斯模型直接确定一个包含q的k近邻的查询区域,减少了已有算法求解该区域的多次迭代过程;当移动对象和查询q位置发生变化时,进一步提出一种高效的增量查询策略,能够最大限度地利用已有查询结果减少当前查询的计算量.最后,在滴滴成都网约车数据集以及两个模拟数据集上进行大量实验,充分验证了算法的性能.     

RBF neural network based on q-Gaussian function in function approximation

To enhance the generalization performance of radial basis function (RBF) neural networks, an RBF neural network based on a q-Gaussian function is proposed. A q-Gaussian function is chosen as the radial basis function of the RBF neural network, and a particle swarm optimization algorithm is employed to select the parameters of the network. The non-extensive entropic index q is encoded in the particle and adjusted adaptively in the evolutionary process of population. Simulation results of the function approximation indicate that an RBF neural network based on q-Gaussian function achieves the best generalization performance.     

Efficiently Certifying Non-Integer Powers

We describe a randomized algorithm that, given an integer a, produces a certificate that the integer is not a pure power of an integer in expected (log a)^1+o(1) bit operations under the assumption of the Generalized Riemann Hypothesis. The certificate can then be verified in deterministic (log a)^1+o(1) time. The certificate constitutes for each possible prime exponent p a prime number q_p, such that a mod q_p is a p^th non-residue. We use an effective version of the Chebotarev density theorem to estimate the density of such prime numbers q_p.     

求解大规模谱聚类的近似加权核k-means算法

谱聚类将聚类问题转化成图划分问题,是一种基于代数图论的聚类方法.在求解图划分目标函数时,一般利用Rayleigh熵的性质,通过计算Laplacian矩阵的特征向量将原始数据点映射到一个低维的特征空间中,再进行聚类.然而在谱聚类过程中,存储相似矩阵的空间复杂度是O(n²),对Laplacian矩阵特征分解的时间复杂度一般为O(n³),这样的复杂度在处理大规模数据时是无法接受的.理论证明,Normalized Cut图聚类与加权核k-means都等价于矩阵迹的最大化问题.因此,可以用加权核k-means算法来优化Normalized Cut的目标函数,这就避免了对Laplacian矩阵特征分解.不过,加权核k-means算法需要计算核矩阵,其空间复杂度依然是O(n²).为了应对这一挑战,提出近似加权核k-means算法,仅使用核矩阵的一部分来求解大数据的谱聚类问题.理论分析和实验对比表明,近似加权核k-means的聚类表现与加权核k-means算法是相似的,但是极大地减小了时间和空间复杂性.     

Semantics of sub-probabilistic programs

The aim of this paper is to extend the probabilistic choice in probabilistic programs to sub-probabilistic choice, i.e., of the form (p)P ^⋈(q)Q where p + q ⩽ 1. It means that program P is executed with probability p and program Q is executed with probability q. Then, starting from an initial state, the execution of a sub-probabilistic program results in a sub-probability distribution. This paper presents two equivalent semantics for a sub-probabilistic while-programming language. One of these interprets programs as sub-probabilistic distributions on state spaces via denotational semantics. The other interprets programs as bounded expectation transformers via wp-semantics. This paper proposes an axiomatic systems for total logic, and proves its soundness and completeness in a classical pattern on the structure of programs.     

Bounded size dictionary compression: SC-completeness and NC algorithms

We study the parallel complexity of a bounded size dictionary version (LRU deletion heuristic) of the LZ2 compression algorithm. The unbounded version was shown to be P-complete. When the size of the dictionary is O(log^kn), the problem of computing the LZ2 compression is shown to be hard for the class of problems solvable simultaneously in polynomial time and O(log^kn) space (that is, SC^k). We also introduce a variation of this heuristic that turns out to be an SC^k-complete problem (the original heuristic belongs to SC^k+1). In virtue of these results, we argue that there are no practical parallel algorithms for LZ2 compression with LRU deletion heuristic or any other heuristic deleting dictionary elements in a continuous way. For simpler heuristics (SWAP, RESTART, FREEZE), practical parallel algorithms are given.     

Inductive inference of regular languages based on model inference

This paper is concerned with an algorithm for identifying an unknown regular language from examples of its members and non-members. The algorithm is based on the model inference algorithm given by Shapiro. In our setting, however, a given first order language for describing a target logic program has countably many unary predicate symbols: q ₀,q ₁,q ₂…. On the other hand, the oracle which gives information about the unknown regular language to the inference algorithm has no interpretation for predicates other than the predicate q ₀. In such a setting,we cannot directly take advantage of the contradiction backtracing algorithm which is one of the most important parts for the efficiency of the model inference algorithm. In order to overcome this disadvantage, we develop a method for giving an interpretation for predicates other than the predicate q ₀ indirectly, which is based on the idea of using the oracle and a one to one mapping from a set of predicates to a set of strings. Furthermore, we propose a model inference algorithm for regular languages using the method, then argue the correctness and the time complexity of the algorithm     

A parallel algorithm for the integer knapsack problem

A sequential algorithm with complexity O(M²+n) for the integer knapsack problem is presented. M is the capacity of the knapsack, and n the number of objects. The algorithm admits an efficient parallelization on a p-processor ring machine. The corresponding parallel algorithm is O(M²/p+n). The parallel algorithm is compared with a version of the well-known Lee algorithm adapted to the integer knapsack problem. Computational results on both a local area network and a transputer are reported.     

A Polynomial-Time Approximation Algorithm for One Problem Simulating the Search in a Time Series for the Largest Subsequence of Similar Elements

We analyze the mathematical aspects of the data analysis problem consisting in the search (selection) for a subset of similar elements in a group of objects. The problem arises, in particular, in connection with the analysis of data in the form of time series (discrete signals). One of the problems in modeling this challenge is considered, namely, the problem of searching in a finite sequence of points from the Euclidean space for the subsequence with the greatest number of terms such that the quadratic spread of the elements of this subsequence with respect to its unknown centroid does not exceed a given percentage of the quadratic spread of elements of the input sequence with respect to its centroid. It is shown that the problem is strongly NP-hard. A polynomial-time approximation algorithm is proposed. This algorithm either establishes that the problem has no solution or finds a 1/2-approximate solution if the length M^* of the optimal subsequence is even, or it yields a \(\frac{1}{2}\left(\begin{array}{c}1-\frac{1}{M^*}\\ \end{array}\right)\)-approximate solution if M* is odd. The time complexity of the algorithm is O(N³(N²+q)), where N is the number of points in the input sequence and q is the space dimension. The results of numerical simulation that demonstrate the effectiveness of the algorithm are presented.     

Online scheduling on batching machines to minimise the total weighted completion time of jobs with precedence constraints and identical processing times

This article studies online scheduling of equal length jobs with precedence constraints on m parallel batching machines. The jobs arrive over time. The objective is to minimise the total weighted completion time of jobs. Denote the size of each batch by b with b?=?∞ in the unbounded batching and b? m , where ρ _m is the positive solution of ρ ^m+1???ρ?=?1. The algorithm is also best possible when the jobs have identical weights. For the bounded batching version with identical weights of jobs, we provide an online algorithm with a competitive ratio of 2.     

A new scheme for the deterministic simulation of PRAMs in VLSI

A deterministic scheme for the simulation of (n, m)-PRAM computation is devised. Each PRAM step is simulated on a bounded degree network consisting of a mesh-of-trees (MT) of siden. The memory is subdivided inn modules, each local to a PRAM processor. The roots of the MT contain these processors and the memory modules, while the otherO(n ²) nodes have the mere capabilities of packet switchers and one-bit comparators. The simulation algorithm makes a crucial use of pipelining on the MT, and attains a time complexity ofO(log² n/log logn). The best previous time bound wasO(log² n) on a different interconnection network withn processors. While the previous simulation schemes use an intermediate MPC model, which is in turn simulated on a bounded degree network, our method performs the simulation directly with a simple algorithm.This work has been supported in part by Ministero della Pubblica Istruzione of Italy under a research grant.     

一种支持多维资源描述的高效P2P路由算法

在分析现有P2P(peer to peer)路由算法的基础上,提出了一种基于二阶矩定位、支持多维资源数据描述的高效资源路由算法--FAN(flabellate addressable network)路由算法.FAN算法将节点映射到统一的多维笛卡尔空间,并以节点相对空间原点的二阶矩作为子空间管理和资源搜索的依据.FAN路由算法具有O(log(N/k))的高路由效率,在节点加入和退出FAN网络时,更新路由信息的代价为O(klog(N/k)).实验结果表明,FAN路由算法具有路由效率高、维护代价小的优点,是一种P2P环境中支持多维资源数据描述的高效结构化资源路由算法.而且,目前部分基于CAN(content-addressable network)网络的改进算法也可以在FAN网络中适用,并获得更好的路由效率和更低的维护代价.     

Two-Dimensional Packet Classification and Filter Conflict Resolution in the Internet

In this paper, we study the packet classification problem and the filter conflict resolution problem, both motivated by applications in the routing of packets in an IP network. For the first problem, we focus on the static 2-dimensional conflict-free (i.e., nested) filters. We design a linear space data structure with O(T _w (n)+(log log n)²) query time on a RAM with word size O(w) bits where n is the number of filters in the router, w is the number of bits in an IP address and

Higher Order May Be Better or May Not Be Better: Investigations with the FDEM (Finite Difference Element Method)

The FDEM program package is a black-box solver for nonlinear systems of elliptic and parabolic PDEs. A FDM of arbitrary consistency order q on an unstructured basic FEM mesh is used. With the difference formula of order q+2 an error estimate is computed. A sophisticated algorithm has been developed to generate reliable difference formulae of order q on an unstructured grid. Completely new is the usage of the error estimate to determine under such general conditions for each node i its individual optimal order q(i) out of the orders 2, 4, 6. This order control tells us that higher order may be better or may not be better, depending on the character of the solution and the used mesh.     

Analytical Fully-Recursive Sensitivity Analysis for Multibody Dynamic Chain Systems

This paper presents a novel fully recursive method, a direct differentiation based approach, which facilitates first-order sensitivityanalysis in optimal design problems involving multibody dynamic systems. A state space O(n) dynamic analysis algorithm based on avelocity space projection method, as promoted by Kane [18], forms the foundation of the underlying formulation. This algorithm cansignificantly reduce the massive number of mathematical and associated computational operations involved in explicitly generatingand solving the sensitivity equations. This benefit is particularly evident for systems involving a combination of many state variablesand design parameters. The development presented in this paper focuses on chain systems to illustrate the recursive nature of thealgorithm. The computational efficiency and solution accuracy of the presented algorithm are investigated through the proceduresapplication to the simulation and design sensitivity determination of spatial chain systems involving 2, 4, 6, ..., 24 degrees offreedom, as well as a simple planar double pendulum.     

Fast computation of sample entropy and approximate entropy in biomedicine

Both sample entropy and approximate entropy are measurements of complexity. The two methods have received a great deal of attention in the last few years, and have been successfully verified and applied to biomedical applications and many others. However, the algorithms proposed in the literature require O(N²) execution time, which is not fast enough for online applications and for applications with long data sets. To accelerate computation, the authors of the present paper have developed a new algorithm that reduces the computational time to O(N^3/2)) using O(N) storage. As biomedical data are often measured with integer-type data, the computation time can be further reduced to O(N) using O(N) storage. The execution times of the experimental results with ECG, EEG, RR, and DNA signals show a significant improvement of more than 100 times when compared with the conventional O(N²) method for N = 80,000 (N = length of the signal). Furthermore, an adaptive version of the new algorithm has been developed to speed up the computation for short data length. Experimental results show an improvement of more than 10 times when compared with the conventional method for N > 4000.