半动态矩形交查询算法

本文讨论了动态矩形交查询算法.文中介绍了两个半动态矩形查询的新算法，它们分别基于一维数据结构和二维数据结构.一维查询算法的查询时间复杂度是O（logM＋k′），更新时间复杂度是O（logMlogn），空间复杂度是O（nlogM/）.二维查询算法的查询时间复杂度是O（log²M＋k），更新时间复杂度是O（log²Mlogn），空间复杂度是O（nlog²M）.本文分别实现了这两个算法，通过对它们的性能进行比较，发现一维查询算法是一种高效、实用的算法.     

确定任意多边形凸凹顶点的算法

本文提出一种确定任意多边形凸凹顶点的算法．该算法的时间复杂性为O(n²logn)次乘法和O(n²)次比较．     

虫孔路由Mesh上的连通分量算法及其应用

用倍增技术在带有Wormhole路由技术的n×n二维网孔机器上提出了时间复杂度为O(log²n)的连通分量和传递闭包并行算法,并在此基础上提出了一个时间复杂度为O(log³n)的最小生成树并行算法.这些都改进了Store-and-Forward路由技术下的时间复杂度下界O(n).同其他运行在非总线连接分布式存储并行计算机上的算法相比,此连通分量和传递闭包算法的时间复杂度是最优的.     

一种高效频繁子图挖掘算法

由于在频繁项集和频繁序列上取得的成功,数据挖掘技术正在着手解决结构化模式挖掘问题--频繁子图挖掘.诸如化学、生物学、计算机网络和WWW等应用技术都需要挖掘此类模式.提出了一种频繁子图挖掘的新算法.该算法通过对频繁子树的扩展,避免了图挖掘过程中高代价的计算过程.目前最好的频繁子图挖掘算法的时间复杂性是O(n³·2ⁿ),其中,n是图集中的频繁边数.提出算法的时间复杂性是O〔2ⁿ·n^2.5／logn〕,性能提高了O(√n·logn)倍.实验结果也证实了这一理论分析.     

模糊聚类计算的最佳算法

给出模糊关系传递闭包在对应模糊图上的几何意义,并提出一个基于图连通分支计算的模糊聚类最佳算法.对任给的n个样本,新算法最坏情况下的时间复杂性函数T(n)满足O(n)≤T(n)≤O(n²).与经典的基于模糊传递闭包计算的模糊聚类算法的O(n³logn)计算时间相比,新算法至少降低了O(n相似文献   

PRAM和LARPBS模型上有向序列翻转距离并行算法

分别在两种重要并行计算模型中给出计算有向基因组排列的反转距离新的并行算法.基于Hannenhalli和Pevzner理论,分3个主要部分设计并行算法:构建断点图、计算断点图中圈数、计算断点图中障碍的数目.在CREW-PRAM模型上,算法使用O(n²)处理器,时间复杂度为O(log²n);在基于流水光总线的可重构线性阵列系统(linear array with a reconfigurable pipelined bus system, LARPBS)模型上,算法使用O(n³)处理器,计算时间复杂度为O(logn).     

无向图的边极大匹配并行算法及其应用*

在EREW PRAM(exclusive-read and exclusive-write parallel random access machine)并行计算模型上,对范围很广的一类无向图的边极大匹配问题,给出时间复杂性为O(logn),使用O((n+m)/logn)处理器的最佳、高速并行算法.     

矩阵条件数及高斯算法平滑分析的进一步研究

算法的复杂度平滑分析是对许多算法在实际应用中很有效但其最坏情况复杂度却很糟这一矛盾给出的更合理的解释.高性能计算机被广泛用于求解大规模线性系统及大规模矩阵的分解.求解线性系统的最简单且容易实现的算法是高斯消元算法(高斯算法).用高斯算法求解n个方程n个变量的线性系统所需要的算术运算次数为O(n³).如果这些方程中的系数用m位表示,则最坏情况下需要机器位数mn位来运行高斯算法.这是因为在消元过程中可能产生异常大的中间项.但大量的数值实验表明,在实际应用中,需要如此高的精度是罕见的.异常大的矩阵条件数和增长因子是导致矩阵A病态,继而导致解的误差偏大的主要根源.设-A为任意矩阵,A是-A受到微小幅度的高斯随机扰动所得到的随机矩阵,方差σ²≤1.Sankar等人对矩阵A的条件数及增长因子进行平滑分析,证明了Pr[K(A)≥α]≤(3.64n(1+4√log(α)))/ασ.在此基础上证明了运行高斯算法输出具有m位精度的解所需机器位数的平滑复杂度为m+71og₂(n)+3log₂(1/σ)+log₂log₂n+7.在上述结果的证明过程中存在错误,将其纠正后得到以下结果:m+71og₂n+3log₂(1/σ)+4√2+log₂n+log₂(1/σ)+7.367.通过构造两个分别关于矩阵范数和随机变量乘积的不等式,将关于矩阵条件数的平滑分析结果简化到Pr[K(A)≥α]≤(6√2n²)/α·σ.部分地解决了Sankar等人提出的猜想:Pr[K(A)≥α]≤O(n/α·σ).并将运行高斯算法输出具有m位精度的解所需机器位数的平滑复杂度降低到m+81og₂n+3log₂(1/σ)+7.实验结果表明,所得到的平滑复杂度更好.     

有中断时间代价的一致并行机抢先调度问题

提出了一种具有中断时间代价的抢先调度问题(P|ptmn(δ)|C_max):在抢先调度中,一个任务发生一次中断,其总的执行时间会增加一个δ.该问题在工程任务分配、分布式计算和网络通信等实际问题中有着广泛的应用背景.证明了这是一个NP-hard问题,给出了一个时间复杂度为O(nlogn+m)的脱线近似算法LPT-Wrap,其近似比小于等于1.40825,并分析了P|ptmn(δ)|C_max的在线特性,给出一个线性时间复杂度的在线近似算法,其竞争比为2.     

一类实际网络中的最小截算法

讨论了节点和边都有容量限制的无向平面网络中的两点间的最小截问题.传统方法是把节点和边都有容量的网络中的最小截问题转化为只有边有容量的问题,但该方法用在平面网络时不能保持网络的平面性,因此网络的平面性不能得到利用.使用传统方法的计算时间为O(n²logn)(其中n为网络的节点数).给出了可以充分利用网络平面性的方法.对源和汇共面的s-t平面网络,把最小截问题转化为平面图上两点间的最短路径问题,从而可以得到O(n)时间的算法;对一般的平面网络,给出了新的将节点和边都有容量的问题转化为仅边有容量问题的方法,这种转化方法不破坏网络的平面性,从而可以利用平面网络中仅边有容量问题的计算方法,使原问题在O(nlogn)时间内获得解决.     

Towards optimal range medians

We consider the following problem: Given an unsorted array of n elements, and a sequence of intervals in the array, compute the median in each of the subarrays defined by the intervals. We describe a simple algorithm which needs O(nlogk+klogn) time to answer k such median queries. This improves previous algorithms by a logarithmic factor and matches a comparison lower bound for k=O(n). The space complexity of our simple algorithm is O(nlogn) in the pointer machine model, and O(n) in the RAM model. In the latter model, a more involved O(n) space data structure can be constructed in O(nlogn) time where the time per query is reduced to O(logn/loglogn). We also give efficient dynamic variants of both data structures, achieving O(log²n) query time using O(nlogn) space in the comparison model and O((logn/loglogn)²) query time using O(nlogn/loglogn) space in the RAM model, and show that in the cell-probe model, any data structure which supports updates in O(log^O(1)n) time must have Ω(logn/loglogn) query time.Our approach naturally generalizes to higher-dimensional range median problems, where element positions and query ranges are multidimensional—it reduces a range median query to a logarithmic number of range counting queries.     

Quasi-optimal upper bounds for simplex range searching and new zone theorems

This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in ?^d so that, given any query simplexq, the points inP ∩q can be counted or reported efficiently. Ifm units of storage are available (n <m <n ^d ), then we show that it is possible to answer any query inO(n ^1+?/m ^1/d ) query time afterO(m ^1+?) preprocessing. This bound, which holds on a RAM or a pointer machine, is almost tight. We also show how to achieveO(logn) query time at the expense ofO(n ^d+?) storage for any fixed ? > 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.     

Efficient temporal counting with bounded error

This paper studies aggregate search in transaction time databases. Specifically, each object in such a database can be modeled as a horizontal segment, whose y-projection is its search key, and its x-projection represents the period when the key was valid in history. Given a query timestamp q _t and a key range , a count-query retrieves the number of objects that are alive at q _t , and their keys fall in . We provide a method that accurately answers such queries, with error less than , where N _alive(q _t ) is the number of objects alive at time q _t , and ɛ is any constant in (0, 1]. Denoting the disk page size as B, and n =  N / B, our technique requires O(n) space, processes any query in O(log _B n) time, and supports each update in O(log _B n) amortized I/Os. As demonstrated by extensive experiments, the proposed solutions guarantee query results with extremely high precision (median relative error below 5%), while consuming only a fraction of the space occupied by the existing approaches that promise precise results.     

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.     

Lower bounds for set intersection queries

We consider the followingset intersection reporting problem. We have a collection of initially empty sets and would like to process an intermixed sequence ofn updates (insertions into and deletions from individual sets) andq queries (reporting the intersection of two sets). We cast this problem in thearithmetic model of computation of Fredman [F1] and Yao [Ya2] and show that any algorithm that fits in this model must take time (q+nq) to process a sequence ofn updates andq queries, ignoring factors that are polynomial in logn. We also show that this bound is tight in this model of computation, again to within a polynomial in logn factor, improving upon a result of Yellin [Ye]. Furthermore, we consider the caseq=O(n) with an additional space restriction. We only allow the use ofm memory locations, wherem n^3/2. We show a tight bound of (n²/m^1/3) for a sequence ofn operations, again ignoring the polynomial in logn factors.     

Improved time and space bounds for Boolean matrix multiplication

Summary Using modular arithmetic we obtain the following improved bounds on the time and space complexities for n × n Boolean matrix multiplication: O(n ^log ₂ ⁷ lognlogloglognloglogloglogn) bit operations and O(n ²loglog n) bits of storage on a logarithmic cost RAM having no multiply or divide instruction; O(n ^log ₂ ⁷(logn)^2–1/2log ₂ ⁷(loglog n)^1/2log ₂ ^7–1) bit operations and O(n ²log n) bits of storage on a RAM which can use indirect addressing for table lookups. The first algorithm can be realized as a Boolean circuit with O(n ^log ₂ ⁷lognlogloglognloglogloglogn) gates. Whenever n×n arithmetic matrix multiplication can be performed in less than O(n ^log ₂ ⁷) arithmetic operations, our results have corresponding improvements.This work was supported in part by the Office of Naval Research under contract N00014-67-0204-0063, by the National Research Council of Canada under grant A4307, and by the National Science Foundation under grants MCS76-17321 and GJ-43332     

Algorithms for dense graphs and networks on the random access computer

We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size = (logn) a maximal matching in ann-vertex bipartite graph in timeO(n ²+n ^2.5/)=O(n ^2.5/logn), how to compute the transitive closure of a digraph withn vertices andm edges in timeO(n ²+nm/), how to solve the uncapacitated transportation problem with integer costs in the range [O.C] and integer demands in the range [–U.U] in timeO ((n ³ (log log/logn)^1/2+n² logU) lognC), and how to solve the assignment problem with integer costs in the range [O.C] in timeO(n ^2.5 lognC/(logn/loglogn)^1/4).Assuming a suitably compressed input, we also show how to do depth-first and breadth-first search and how to compute strongly connected components and biconnected components in timeO(n+n ²/), and how to solve the single source shortest-path problem with integer costs in the range [O.C] in time0 (n ²(logC)/logn). For the transitive closure algorithm we also report on the experiences with an implementation.Most of this research was carried out while both authors worked at the Fachbereich Informatik, Universität des Saarlandes, Saarbrücken, Germany. The research was supported in part by ESPRIT Project No. 3075 ALCOM. The first author acknowledges support also from NSERC Grant No. OGPIN007.     

Computing largest empty circles with location constraints

LetQ = {q₁, q₂,..., q_n} be a set ofn points on the plane. The largest empty circle (LEG) problem consists in finding the largest circleC with center in the convex hull ofQ such that no pointq _i εQ lies in the interior ofC. Shamos recently outlined anO(n logn) algorithm for solving this problem.⁽⁹⁾ In this paper it is shown that this algorithm does not always work correctly. A different approach is proposed here and shown to also result in anO(n logn) algorithm. The new approach has the advantage that it can also solve more general problems. In particular, it is shown that if the center ofC is constrained to lie in an arbitrary convexn-gon, an0(n logn) algorithm can still be obtained. Finally, an0(n logn +k logn) algorithm is given for solving this problem when the center ofC is constrained to lie in an arbitrary simplen-gonP. wherek denotes the number of intersections occurring between edges ofP and edges of the Voronoi diagram ofQ andk ?O(n ²).     

Efficient worst-case data structures for range searching

In this paper we investigate the worst-case complexity of range searching: preprocess N points in k-space such that range queries can be answered quickly. A range query asks for all points with each coordinate in some range of values, and arises in many problems in statistics and data bases. We develop three different structures for range searching in this paper. The first structure has absolutely optimal query time (which we prove), but has very high preprocessing and storage costs. The second structure we present has logarithmic query time and O(N ¹⁺²) preprocessing and storage costs, for any fixed >0. Finally we give a structure with linear storage, O(N ln N) preprocessing and O(N ) query time.Research in this paper has been supported partially under Office of Naval Research contract N000014-76-C-0373, USA, and by the Austrian Federal Ministry for Science and Research     

Improved Constructions for Non-adaptive Threshold Group Testing

The basic goal in combinatorial group testing is to identify a set of up to d defective items within a large population of size n?d using a pooling strategy. Namely, the items can be grouped together in pools, and a single measurement would reveal whether there are one or more defectives in the pool. The threshold model is a generalization of this idea where a measurement returns positive if the number of defectives in the pool reaches a fixed threshold u>0, negative if this number is no more than a fixed lower threshold ?<u, and may behave arbitrarily otherwise. We study non-adaptive threshold group testing (in a possibly noisy setting) and show that, for this problem, O(d ^g+2(logd)log(n/d)) measurements (where g:=u???1 and u is any fixed constant) suffice to identify the defectives, and also present almost matching lower bounds. This significantly improves the previously known (non-constructive) upper bound O(d ^u+1log(n/d)). Moreover, we obtain a framework for explicit construction of measurement schemes using lossless condensers. The number of measurements resulting from this scheme is ideally bounded by O(d ^g+3(logd)logn). Using state-of-the-art constructions of lossless condensers, however, we obtain explicit testing schemes with O(d ^g+3(logd)quasipoly(logn)) and O(d ^g+3+β poly(logn)) measurements, for arbitrary constant β>0.