Near-Linear Time Approximation Algorithms for Curve Simplification

We consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P’ whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P’ while ensuring that the error between P’ and P is below a certain threshold. We consider two different error measures: Hausdorff and Frechet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ε > 0, compute a simplified polygonal curve P’ whose error is less than ε and size at most the size of an optimal simplified polygonal curve with error ε/2. We consider monotone curves in ℝ² in the case of the Hausdorff error measure under the uniform distance metric and arbitrary curves in any dimension for the Frechet error measure under L_p metrics. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice.     

A Fast Approximation Algorithm for the Subset-sum Problem

The subset-sum problem (SSP) is defined as follows: given a positive integer bound and a set of n positive integers find a subset whose sum is closest to, but not greater than, the bound. We present a randomized approximation algorithm for this problem with linear space complexity and time complexity of O( n log n ). Experiments with random uniformly-distributed instances of SSP show that our algorithm outperforms, both in running time and average error, Martello and Toth's (1984) quadratic greedy search, whose time complexity is O( n ²). We propose conjectures on the expected error of our algorithm for uniformly-distributed instances of SSP and provide some analytical arguments justifying these conjectures. We present also results of numerous tests.     

Optimal polygonal approximation of digitized curves using the sum of square deviations criterion

We address the problem of finding an optimal polygonal approximation of digitized curves. Several optimal algorithms have already been proposed to determine the minimum number of points that define a polygonal approximation of the curve, with respect to a criterion error. We present a new algorithm with reasonable complexity to determine the optimal polygonal approximation using the mean square error norm. The idea is to estimate the remaining number of segments and to integrate the cost in the A^* algorithm. The solution is optimal in the minimum number of segments.     

A systolic-based parallel bin packing algorithm

A systolic based parallel approximation algorithm that obtains solutions to the I-D bin packing problem is presented. The algorithm has an asymptotic error bound of 1.5 and time complexity O(n). An experimental study demonstrates that the heuristic offers improved packing and execution performance over parallelizations of two well-known serial algorithms     

RCC8的一致分割及其算法

区域连接演算（Region Connection Calculus,RCC）是一种用于空间定性表示和推理的形式化模型,如RCC5,RCC8等,其一致性检查被证明是一个NP问题。幸运的是,在其可处理子集上,路径一致性和一致性等价,即便这样也有[O(n3)]的时间复杂度和[O(n2)]的空间复杂度。为了提高一致性检查的效率,提出了一致分割的概念,给出了其定义和成立的充分必要条件,用来将RCC8的约束图在保持一致性的前提下分割成若干个子图,分而求解各个子图的一致性;并随后给出了几种一致分割的充分条件,和相应的高效分割算法。在随机生成的大型、稀疏约束图上的实验表明了一致分割的有效性。     

New Results on Path Approximation

In this paper we give bounds on the complexity of some algorithms for approximating 2-D and 3-D polygonal paths with the infinite beam measure of error. While the time/space complexities of the algorithms known for other error measures are well understood, path approximation with infinite beam measure seems to be harder due to the complexity of some geometric structures that arise in the known approaches. Our results answer some open problems left in previous work. We also present a more careful analysis of the time complexity of the general iterative algorithm for infinite beam measure and show that it could perform much better in practice than in the worst case. We then propose a new approach for path approximation that consists of a breadth first traversal of the path approximation graph, without constructing the graph. This approach can be applied to any error criterion in any constant dimension. The running time of the new algorithm depends on the size of the output: if the optimal approximating path has m vertices, the algorithm performs F(m) iterations rather than n–1 in the previous approaches, where F(m) \le n–1 is the number of vertices of the path approximation graph that can be reached with at most m–2 edges. This is the first output sensitive path approximation algorithm.     

Design and Performance Evaluation of Sequence Partition Algorithms

Tradeoffs between time complexities and solution optimalities are important when selecting algorithms for an NP-hard problem in different applications. Also, the distinction between theoretical upper bound and actual solution optimality for realistic instances of an NP-hard problem is a factor in selecting algorithms in practice. We consider the problem of partitioning a sequence of n distinct numbers into minimum number of monotone （increasing or decreasing） subsequences. This problem is NP-hard and the number of monotone subsequences can reach [√2n＋1/1-1/2]in the worst case. We introduce a new algorithm, the modified version of the Yehuda-Fogel algorithm, that computes a solution of no more than [√2n＋1/1-1/2]monotone subsequences in O（n^1.5） time. Then we perform a comparative experimental study on three algorithms, a known approximation algorithm of approximation ratio 1.71 and time complexity O（n^3）, a known greedy algorithm of time complexity O（n^1.5 log n）, and our new modified Yehuda-Fogel algorithm. Our results show that the solutions computed by the greedy algorithm and the modified Yehuda-Fogel algorithm are close to that computed by the approximation algorithm even though the theoretical worst-case error bounds of these two algorithms are not proved to be within a constant time of the optimal solution. Our study indicates that for practical use the greedy algorithm and the modified Yehuda-Fogel algorithm can be good choices if the running time is a major concern.     

遗传算法在曲线多边形近似中的应用

在平面数字曲线的多边形近似中,为克服顶点的检测只依靠部区域,缺 乏全局信息的弱点,文中把多边形近似问题作了寻找在满足一定的近似误差下使顶点数最少,或者使顶点数和近似误差都尽可能少的最优化问题来处理。     

Efficient EREW PRAM algorithms for parentheses-matching

We present four polylog-time parallel algorithms for matching parentheses on an exclusive-read and exclusive-write (EREW) parallel random-access machine (PRAM) model. These algorithms provide new insights into the parentheses-matching problem. The first algorithm has a time complexity of O(log² n) employing O(n/(log n)) processors for an input string containing n parentheses. Although this algorithm is not cost-optimal, it is extremely simple to implement. The remaining three algorithms, which are based on a different approach, achieve O(log n) time complexity in each case, and represent successive improvements. The second algorithm requires O(n) processors and working space, and it is comparable to the first algorithm in its ease of implementation. The third algorithm uses O(n/(log n)) processors and O(n log n) space. Thus, it is cost-optimal, but uses extra space compared to the standard stack-based sequential algorithm. The last algorithm reduces the space complexity to O(n) while maintaining the same processor and time complexities. Compared to other existing time-optimal algorithms for the parentheses-matching problem that either employ extensive pipelining or use linked lists and comparable data structures, and employ sorting or a linked list ranking algorithm as subroutines, the last two algorithms have two distinct advantages. First, these algorithms employ arrays as their basic data structures, and second, they do not use any pipelining, sorting, or linked list ranking algorithms     

基因组Translocation排序问题的改进多项式算法

该文给出基因组Transhocation排序问题的一个改进多项式算法，原算法所有存储空间O（n)，时间复杂度为O(n^3)，文中改进算法仍采用O(n)存储空间，时间复杂度为O(n^2logn),具体地，将计算Translocation距离的时间复杂度由O(n^3)改进为O(n^2),将计算Translocation序列的时间复杂度由O(n^3)改进为O(n^2logn).     

一种基于离散微粒群优化的数字曲线的多边形近似算法

数字曲线的多边形近似是图像分析研究领域的一个热点问题.获取数字曲线的优化多边形近似是一个复杂的问题,其计算复杂度非常高.微粒群算法是近些年来提出的一种新的优化方法,已经被广泛应用于各种优化问题的求解.提出了一种求解数字曲线的多边形近似问题的基于整数编码的离散微粒群算法(IPSO).IPSO通过重新定义标准微粒群算法的速度和位置更新公式中的加法、乘法和减法运算,使得算法能运行在离散的解空间.IPSO的位置向量修复机制保证了解的可行性,而局部优化器提高了算法的搜索精度.实验结果表明,IPSO求解的质量和求解的效率均优于遗传算法和0-1编码的微粒群算法.     

On signal representations within the Bayes decision framework

This work presents new results in the context of minimum probability of error signal representation (MPE-SR) within the Bayes decision framework. These results justify addressing the MPE-SR criterion as a complexity-regularized optimization problem, demonstrating the empirically well understood trade-off between signal representation quality and learning complexity. Contributions are presented in three folds. First, the stipulation of conditions that guarantee a formal tradeoff between approximation and estimation errors under sequence of embedded transformations are provided. Second, the use of this tradeoff to formulate the MPE-SR as a complexity regularized optimization problem, and an approach to address this oracle criterion in practice is given. Finally, formal connections are provided between the MPE-SR criterion and two emblematic feature transformation techniques used in pattern recognition: the optimal quantization problem of classification trees (CART tree pruning algorithms), and some versions of Fisher linear discriminant analysis (LDA).     

汉诺塔非递归算法研究

汉诺塔(Tower of Hanoi)问题是求在三个柱子之间移动圆盘的方法,它是递归程序设计的经典例子,已经证明其时间复杂度下限是O(2n),空间复杂度是O(n),实际使用时很容易溢出.给出汉诺塔问题的两个非递归算法:解集递推法和解集树法.解集递推法的时间复杂度和空间复杂度都是O(2n),该算法空间复杂度很大,无法实际使用,提出该算法的目的是为了引出解集树法.解集树法可以计算出指定的任意一步移动方法,时间复杂度和空间复杂度分别是O(n*2n)和O(1).并证明了汉诺塔问题的空间复杂度下限是O(1).     

命题μ-演算全局模型检测的高效算法设计

在Long, Browne, Jha 和 Marrero等人工作的基础上,详细分析了用Tarski不动点定理计算不动点交替嵌套深度为4的命题μ-演算公式的计算过程,找到了计算中间结果间具有的两组偏序关系,利用这两组偏序关系设计了一个高效的命题μ-演算全局模型检测算法,该算法与Long等人提出的算法有相似的时间复杂度(O((2n+1)\\+{[d/2]+1})相对于O(n\\+[d/2]+1})),但空间复杂度有很大的改进(O(dn)相对于O(n\\+{[d/2]+1})),其中n是变迁系统的状态规模,d是命题μ-演算公式中不动点算子的嵌套深度.算法性能的改进对于命题μ-演算模型检测技术的理论研究与实际推广应用都意义重大.     

New Results on Path Approximation

In this paper we give bounds on the complexity of some algorithms for approximating 2-D and 3-D polygonal paths with the infinite beam measure of error. While the time/space complexities of the algorithms known for other error measures are well understood, path approximation with infinite beam measure seems to be harder due to the complexity of some geometric structures that arise in the known approaches. Our results answer some open problems left in previous work. We also present a more careful analysis of the time complexity of the general iterative algorithm for infinite beam measure and show that it could perform much better in practice than in the worst case. We then propose a new approach for path approximation that consists of a breadth first traversal of the path approximation graph, without constructing the graph. This approach can be applied to any error criterion in any constant dimension. The running time of the new algorithm depends on the size of the output: if the optimal approximating path has m vertices, the algorithm performs F(m) iterations rather than n–1 in the previous approaches, where F(m) \le n–1 is the number of vertices of the path approximation graph that can be reached with at most m–2 edges. This is the first output sensitive path approximation algorithm.     

一种节省空间的排序算法

目前报道的一些排序算法，空间复杂度都比较大．提出了一种改进其空间复杂度的方法，其特点是算法简单、稳定，时间复杂度为O(n^2)，空间复杂度为2n，达到下界．与传统的排序算法用变量与变量比较的思路不同，本文提出的是一种用变量与其分布区间进行比较的新思路．本算法特别适合那些范围确定且分布基本均匀的待排数据，也适合一般数据对象的排序．     

多序列的近似LCS改进算法

提出2种针对3条源序列的近似LCS算法,近似因子均为1/|?|。其中,线性近似LCS算法的时空复杂度均为 , 为最长源序列的长度,适于解决大规模问题。递归近似LCS算法时空复杂度均为O(nlogn),适于要求高精度问题。同时,这2种算法都能用于解决多序列的LCS和CLCS问题。实验验证了这2种算法的有效性。     

Parallel consistent labeling algorithms

Mackworth and Freuder have analyzed the time complexity of several constraint satisfaction algorithms.⁽¹⁾ Mohr and Henderson have given new algorithms, AC-4 and PC-3, for arc and path consistency, respectively, and have shown that the arc consistency algorithm is optimal in time complexity and of the same order space complexity as the earlier algorithms.⁽²⁾ In this paper, we give parallel algorithms for solving node and arc consistency. We show that any parallel algorithm for enforcing are consistency in the worst case must have O(na) sequential steps, wheren is number of nodes, anda is the number of labels per node. We give several parallel algorithms to do arc consistency. It is also shown that they all have optimal time complexity. The results of running the parallel algorithms on a BBN Butterfly multiprocessor are also presented.This work was partially supported by NSF Grants MCS-8221750, DCR-8506393, and DMC-8502115.     

Movement Planning in the Presence of Flows

This paper investigates the problem of time-optimum movement planning in two and three dimensions for a point robot which has bounded control velocity through a set of n polygonal regions of given translational flow velocities. This intriguing geometric problem has immediate applications to macro-scale motion planning for ships, submarines, and airplanes in the presence of significant flows of water or air. Also, it is a central motion planning problem for many of the meso-scale and micro-scale robots that have been constructed recently, that have environments with significant flows that affect their movement. In spite of these applications, there is very little literature on this problem, and prior work provided neither an upper bound on its computational complexity nor even a decision algorithm. It can easily be seen that an optimum path for the 2D version of this problem can consist of at least an exponential number of distinct segments through flow regions. We provide the first known computational complexity hardness result for the 3D version of this problem; we show the problem is PSPACE hard. We give the first known decision algorithm for the 2D flow path problem, but this decision algorithm has very high computational complexity. We also give the first known efficient approximation algorithms with bounded error.     

Dynamic Programming on the Word RAM

Dynamic programming is one of the fundamental techniques for solving optimization problems. In this paper we propose a general framework which can be used to decrease the time and space complexity of these algorithms with a logarithmic factor. The framework is based on word encoding, i.e. by representing subsolutions as bits in an integer. In this way word parallelism can be used in the evaluation of the dynamic programming recursion. Using this encoding the subset-sum problem can be solved in O( n b/ log b) time and O(b/ log b) space, where n is the number of integers given and b is the target sum. The knapsack problem can be solved in O( n m/ log m) time and O(m/ log m) space, where n is the number of items and m = max{b,z} is the maximum of the capacity b and the optimal solution value z . The problem of finding a path of a given length b in a directed acyclic graph G=(V,E) can be solved in O(|E|b/ log b) time and O(|V|b/ log b) space. Several other examples are given showing the generality of the achieved technique. Extensive computational experiments are provided to demonstrate that the achieved results are not only of theoretical interest but actually lead to algorithms which are up to two orders of magnitude faster than their predecessors. This is a surprising observation as the increase in speed is larger than the word size of the processor.