P2-Packing问题参数算法的改进

P_2-Packing问题是一个典型的NP难问题.目前这个问题的最好结果是时间复杂度为O(2~(5.301k))的参数算法,其核的大小为15k.通过对P_2-packing问题的结构作进一步分析,提出了改进的核心化算法,得到大小为7k的核,并在此基础上提出了一种时间复杂度为O(2~(4.142k))的参数算法,大幅度改进了目前文献中的最好结果.     

超平面覆盖问题的参数化改进算法

超平面覆盖问题是计算几何领域中一类典型的NP难问题,在实际生活中有着广泛的应用.针对NP难问题的难解性,人们提出了一些传统的方法用来求解这些NP难问题.但由于这些方法具有各自的局限性,不能满足实际应用中的各种需求,人们从新的理论角度为固定参数可解的NP难问题设计参数算法.通过深入分析直线覆盖问题(超平面覆盖问题的一个特例)的结构特征,并利用深度有界搜索树的方法,提出了一个时间复杂度为O(k3(0.736k)k+nlogk)的确定性参数算法,极大地改进了当前最好的结果O((k/2.2)2k+nlogk).通过对上述算法在高维空间中的进一步扩展,提出了关于超平面覆盖问题时间复杂度为O(dkd+1(dk)!/((d!)kk!)+nd+1)确定性参数算法,对当前的最好结果O(kd(k+1)+nd+1)有较大改进.     

一类层次环网络的构造及路由算法

讨论了一类层次环网络 HRN的构造方法、拓扑性质和路由策略 .重点讨论了 HRN网络的一个子类 ,即RP(P,k1 ,k2 )网络 ,分析了其拓扑性质 ,并和 2 D Torus,3D Torus,Hypercube和 De Bruijn Graph等拓扑结构进行了分析比较 .结果表明 ,RP(P,k1 ,k2 )网络的拓扑结构简单 ,路由策略方便 ,是一种实用的互联网络 .接着 ,讨论了RP(P,k1 ,k2 )网络上的路由问题 ,给出了点点路由、Broadcast路由、All- to- all路由和置换路由算法 ,前 3个算法分别需要 k2 / 2 k1 / 2 2 ,k2 / 2 k1 / 2 2 ,10× k1 × k2 - 4个时间步 ,置换路由需要 4 min{ k2 ,k1 } (k2 - 1)×(k1 - 1)个路由时间步 .最后 ,提出了两个参数 ,即最优节点分组和最优网络划分 ,用于评价互联网络的效率 ,并据此分析了 RP(P,k1 ,k2 ) ,2 D Torus和 Hypercube网络的性能 .     

加权3D-Matching的改进算法

Matching问题构成了一类重要的NP难问题.此类问题在诸多领域中有着重要的应用,如调度、代码优化等领域.对于加权3D-matching问题,通过深入分析问题的结构特性,可以转化成加权3D-matching augmentation问题进行求解,即从一个最大加权的k-matching着手构造权值最大的(k+1)-matching.从问题的特殊结构特性出发,给出了加权3D-matching augmentation问题特有的性质: k- matching中存在2列使得该2列至少有2k/3元素被包含在(k+1)-matching中所对应的2列中.基于给出的性质,通过运用color-coding和动态规划技术,给出了一个时间复杂度为O* (4.823k)的参数算法,最终求解加权3D-matching问题.该算法较目前文献中的最好结果O* (5.473k)有了极大的改进.     

一种在元素与颜色规模相近时的有效着色算法及其应用

着色算法(color-coding)是求解NP难问题的重要手段之一.而在应用着色算法时,着色算法所产生的着色方案的规模极大地影响着问题的求解性能,故构造一个尽可能小的着色方案是着色算法所寻求的目标.目前存在的着色算法均基于完全散列函数,并要求元素数目n远大于颜色数目k,且k比较小,这个限制条件使得这些着色算法在一些实际情况下无法应用.该文主要研究在元素与颜色规模相近时(n2k)的有效着色算法,并着重分析在n2k情况下着色算法的性能.该文提出了一种基于划分思想的着色方案构造算法PBCC,证明了由PBCC产生的着色方案确实可以覆盖到所有的子集,并具体给出了可应用于(l,d)-(20,16)Motif查找问题的403种着色的构造方法.文章进一步分析了PBCC产生的着色方案规模,并证明了在n2k且n-k2的情况下,任何着色算法所产生的着色方案的规模|S(n,k)|都不小于[n/2 n-k] [[n n-k]-n/2 n-k]2~(n-k)]/(2~(n-k)-2).此外,文中也采用了渐进分析技术,证明了PBCC算法生成着色方案规模为O(e2Rootof(ex-eμx 1)(n-k)),在n=2k的情况下结果是O(2.62n-k);同时,文中也证明了n2k情况下着色方案规模的下界为2n-k.     

个体单体型问题参数化算法研究

个体单体型问题指如何利用个体DNA测序片断数据,根据不同的优化准则确定该个体单体型的计算问题.因为技术上的限制,DNA测序实验中能直接测定的片断长度是有限的,一个片断所覆盖的最大SNP位点数k1通常小于10;出于时间和金钱的考虑,覆盖一个SNP位点的最大片断数k2也不是很大,通常约为10左右;与要测定的单体型SNP位点总数,n及所测序的DNA片断总数m相比,k1和k2均很小.在此基础上,文中对个体单体型问题最少SNP位点删除MSR和最少片段删除MFR模型进行了参数化,提出了时间复杂度分别为O(nk1k2+mlogm+mk1)和O(mk22+mlogm+nk2)求解无空隙MSR和MFR的精确算法.和Bafna等提出的时间复杂度为O(mn2)和O(m2n+m2)的精确算法相比,文中的算法效率提高了很多,具有较高的实用价值.     

带权无向图中反馈顶点集的固定参数枚举算法

反馈顶点集(FVS)问题是一个经典的NP-完全问题,在很多领域有重要的应用.人们对该问题进行了大量的研究,但目前还没有有效的算法枚举带权无向图的反馈顶点集.文中通过对带权无向图中反馈顶点集问题的结构的深入分析,给出了一个有效的基于分支搜索技术的固定参数枚举算法.算法将反馈顶点集问题转化为反馈边集问题,通过枚举z个权值最大的森林来枚举z个权值最小的含k条边的反馈边集,从而得到z个权值最小的含k个顶点的反馈顶点集,算法时间复杂度为O(5kn2(logn+k)+3kz(n2logn+z)).     

加权3-Set Packing 的改进算法

Packing问题构成了一类重要的NP难问题.对于加权3-SetPacking问题,把问题转化成加权3-SetPacking Augmentation问题进行求解,即主要讨论如何从一个已知的最大加权k-packing求得一个权值最大的(k+1)-packing.通过对问题结构的分析,结合Color-Coding技术,首先给出了一种时间复杂度为O*(10.63k)的参数算法,极大地改进了目前文献中的最好结果O*(12.83k).通过对(k+1)-packing结构的进一步分析,利用集合划分技术将上述结果降到O*(7.563k).     

Fast Parallel Molecular Algorithm for Solving the Discrete Logarithm Problem over Group on Zp+ DNA-based Computing

加群Zp+上离散对数问题在公钥密码系统分析中具有非常广泛的应用.研究一种加群Zp+上离散对数问题的DNA计算算法.算法主要由解空间生成器、并行乘法器、并行加法器、解转换器及解搜索器组成.其中解空间生成器借鉴传统计算机中3表算法的思想,将解空间的生成分为3个部分来完成,极大减少了非法解的搜索空间.本算法的生物操作时间复杂度为O(k2),需要O(1)个试管数、O(2k)条DNA链,最长DNA链长为O(k2)(其中k为加群上离散对数问题群阶p的二进制编码位数).最后,通过DNA计算通用的试验方法对算法进行了仿真,验证了算法的可行性和有效性.     

合取范式3可满足问题的局部搜索近似算法

合取范式最大可满足问题是理论计算机科学的核心问题.局部搜索被许多求解实践证明是解答合取范式最大可满足问题十分有效的方法,但未见关于局部搜索算法解答该问题性能分析的结果.文中讨论最大3可满足问题(Max-(3)-Sat)的局部搜索算法并分析算法性能.证明Max-(3)-Sat问题的一位跳变局部搜索算法的近似性能比为4/3;证明一位跳变局部搜索后跟有条件全体跳变算法,解答Max-(3)-Sat问题的近似性能比为5/4.设计一位跳变加全体跳变的新局部搜索算法,证明新算法解答Max-(3)-Sat问题的近似性能比为8/7.将8/7-近似局部搜索算法推广为解答Max-(k)-Sat问题的局部搜索算法,证明算法的近似性能比为(2k+2)/(2k+1),k≥4.设计解答Max-(k)-Sat问题的两位跳变局部搜索算法,证明两位跳变局部搜索算法的近似性能比为1+1/(2k+1+k(k-1)/(n-k)),k≥4.局部搜索算法经多次运行可进一步提高求解性能.文中结果显示,局部搜索算法在合取范式最大可满足问题求解实践中表现出高性能,有其必然性.     

A fast parallel algorithm for routing unicast assignments in Benesnetworks

This paper presents a new parallel algorithm for routing unicast (one-to-one) assignments in Benes networks. Parallel routing algorithms for such networks were reported earlier, but these algorithms were designed primarily to route permutation assignments. The routing algorithm presented in this paper removes this restriction without an increase in the order of routing cost or routing time. We realize this new routing algorithm on two different topologies. The algorithm routes a unicast assignment involving O(k) pairs of inputs and outputs in O(lg ² k+lg n) time on a completely connected network of n processors and in O(lg⁴ k+lg² k lg n) time on an extended shuffle-exchange network of n processors. Using O(n lg n) professors, the same algorithm can be pipelined to route α unicast assignments each involving O(k) pairs of inputs and outputs, in O(lg² k+lg n+(α-1) lg k) time on a completely connected network and in O(lg⁴ k+lg² k lg n+(α-1)(lg ³ k+lg k lg n)) time on the extended shuffle-exchange network. These yield an average routing time of O(lg k) in the first case, and O(lg³ k+1g k lg n) in the second case, for all α⩾lg n. These complexities indicate that the algorithm given in this paper is as fast as Nassimi and Sahni's algorithm for unicast assignments, and with pipelining, it is faster than the same algorithm at least by a factor of O(lg n) on both topologies. Furthermore, for sparse assignments, i.e., when k=O(1), it is the first algorithm which has an average routing time of O(1g n) on a topology with O(n) links     

On robust online scheduling algorithms

While standard parallel machine scheduling is concerned with good assignments of jobs to machines, we aim to understand how the quality of an assignment is affected if the jobs’ processing times are perturbed and therefore turn out to be longer (or shorter) than declared. We focus on online scheduling with perturbations occurring at any time, such as in railway systems when trains are late. For a variety of conditions on the severity of perturbations, we present bounds on the worst case ratio of two makespans. For the first makespan, we let the online algorithm assign jobs to machines, based on the non-perturbed processing times. We compute the makespan by replacing each job’s processing time with its perturbed version while still sticking to the computed assignment. The second is an optimal offline solution for the perturbed processing times. The deviation of this ratio from the competitive ratio of the online algorithm tells us about the “price of perturbations”. We analyze this setting for Graham’s algorithm, and among other bounds show a competitive ratio of 2 for perturbations decreasing the processing time of a job arbitrarily, and a competitive ratio of less than 2.5 for perturbations doubling the processing time of a job. We complement these results by providing lower bounds for any online algorithm in this setting. Finally, we propose a risk-aware online algorithm tailored for the possible bounded increase of the processing time of one job, and we show that this algorithm can be worse than Graham’s algorithm in some cases.     

Minimizing Makespan in Batch Machine Scheduling

We study the scheduling of a set of n jobs, each characterized by a release (arrival) time and a processing time, for a batch processing machine capable of running at most B jobs at a time. We obtain an O(n log n)-time algorithm when B is unbounded. When there are only m distinct release times and the inputs are integers, we obtain an O(n(BR_max)^m-1(2/m)^m-3)-time algorithm where R_max is the difference between the maximum and minimum release times. When there are k distinct processing times and m release times, we obtain an O(n log m + k^k+2 B^k+1 m² log m)-time algorithm. We obtain even better algorithms for m=2 and for k=1. These algorithms improve most of the corresponding previous algorithms for the respective special cases and lead to improved approximation schemes for the general problem.     

The online Prize-Collecting Traveling Salesman Problem

We study the online version of the Prize-Collecting Traveling Salesman Problem (PCTSP), a generalization of the Traveling Salesman Problem (TSP). In the TSP, the salesman has to visit a set of cities while minimizing the length of the overall tour. In the PCTSP, each city has a given weight and penalty, and the goal is to collect a given quota of the weights of the cities while minimizing the length of the tour plus the penalties of the cities not in the tour. In the online version, cities are disclosed over time. We give a 7/3-competitive algorithm for the problem, which compares with a lower bound of 2 on the competitive ratio of any deterministic algorithm. We also show how our approach can be combined with an approximation algorithm in order to obtain an O(1)-competitive algorithm that runs in polynomial time.     

High-order RKDG Methods for Computational Electromagnetics

In this paper we introduce a new RKDG method for problems of wave propagation that achieves full high-order convergence in time and space. The novelty of the method resides in the way in which it marches in time. It uses an mth-order m-stage, low storage SSP-RK scheme which is an extension to a class of non-autonomous linear systems of a recently designed method for autonomous linear systems. This extension allows for a high-order accurate treatment of the inhomogeneous, time-dependent terms that enter the semi-discrete problem on account of the physical boundary conditions. Thus, if polynomials of degree k are used in the space discretization, the RKDG method is of overall order m = k + 1, for any k > 0. Moreover, we also show that the attainment of high-order space–time accuracy allows for an efficient implementation of post-processing techniques that can double the convergence order. We explore this issue in a one-dimensional setting and show that the superconvergence of fluxes previously observed in full space–time DG formulations is also attained in our new RKDG scheme. This allows for the construction of higher-order solutions via local interpolating polynomials. Indeed, if polynomials of degree k are used in the space discretization together with a time-marching method of order 2k + 1, a post-processed approximation of order 2k + 1 is obtained. Numerical results in one and two space dimensions are presented that confirm the predicted convergence properties.Mathematics Subject Classification 1991: Primary 65N30; Secondary 65M60.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Approximating Node Connectivity Problems via Set Covers

Given a graph (directed or undirected) with costs on the edges, and an integer $k$, we consider the problem of finding a $k$-node connected spanning subgraph of minimum cost. For the general instance of the problem (directed or undirected), there is a simple $2k$-approximation algorithm. Better algorithms are known for various ranges of $n,k$. For undirected graphs with metric costs Khuller and Raghavachari gave a $( 2+{2(k-1)}/{n})$-approximation algorithm. We obtain the following results: (i) For arbitrary costs, a $k$-approximation algorithm for undirected graphs and a $(k+1)$-approximation algorithm for directed graphs. (ii) For metric costs, a $(2+({k-1})/{n})$-approximation algorithm for undirected graphs and a $(2+{k}/{n})$-approximation algorithm for directed graphs. For undirected graphs and $k=6,7$, we further improve the approximation ratio from $k$ to $\lceil (k+1)/2 \rceil=4$; previously, $\lceil (k+1)/2 \rceil$-approximation algorithms were known only for $k \leq 5$. We also give a fast $3$-approximation algorithm for $k=4$. The multiroot problem generalizes the min-cost $k$-connected subgraph problem. In the multiroot problem, requirements $k_u$ for every node $u$ are given, and the aim is to find a minimum-cost subgraph that contains $\max\{k_u,k_v\}$ internally disjoint paths between every pair of nodes $u,v$. For the general instance of the problem, the best known algorithm has approximation ratio $2k$, where $k=\max k_u$. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve for $k \leq 7$ the approximation guarantee from $3$ to $2+{\lfloor (k-1)/2 \rfloor}/{k} < 2.5$.     

Scheduling parallel jobs to minimize the makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a prespecified, job-dependent number of machines when being processed. We prove that the makespan of any nonpreemptive list-schedule is within a factor of 2 of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-scheduling algorithm can achieve a better performance guarantee than 2 for the nonpreemptive problem, no matter which priority list is chosen. List-scheduling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. We show that no list-scheduling algorithm has a constant competitive ratio. Still, we present the first online algorithm for scheduling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bound of 2.25 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bound for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over time.     

Approximating Rooted Connectivity Augmentation Problems

A graph is called {\em $\el$-connected from $U$ to $r$} if there are $\el$ internally disjoint paths from every node $u \in U$ to $r$. The {\em Rooted Subset Connectivity Augmentation Problem} ({\em RSCAP}) is as follows: given a graph $G=(V+r,E)$, a node subset $U \subseteq V$, and an integer $k$, find a smallest set $F$ of new edges such that $G+F$ is $k$-connected from $U$ to $r$. In this paper we consider mainly a restricted version of RSCAP in which the input graph $G$ is already $(k-1)$-connected from $U$ to $r$. For this version we give an $O(\ln\! |U|)$-approximation algorithm, and show that the problem cannot achieve a better approximation guarantee than the Set Cover Problem (SCP) on $|U|$ elements and with $|V|-|U|$ sets. For the general version of RSCAP we give an $O(\ln k \ln\!|U|)$-approximation algorithm. For $U=V$ we get the {\em Rooted Connectivity Augmentation Problem} ({\em RCAP}). For directed graphs RCAP is polynomially solvable, but for undirected graphs its complexity status is not known: no polynomial algorithm is known, and it is also not known to be NP-hard. For undirected graphs with the input graph $G$ being $(k-1)$-connected from $V$ to $r$, we give an algorithm that computes a solution of size at most ${\it opt}+\min\{opt,k\}/2$, where {\it opt} denotes the optimal solution size.     

Fast Algorithms for Computing the Smallest k-Enclosing Circle

For a set $P$ of $n$ points in the plane and an integer $k \leq n$, consider the problem of finding the smallest circle enclosing at least $k$ points of $P$. We present a randomized algorithm that computes in $O( n k )$ expected time such a circle, improving over previously known algorithms. Further, we present a linear time $\delta$-approximation algorithm that outputs a circle that contains at least $k$ points of $P$ and has radius less than $(1+\delta)r_{opt}(P,k)$, where $r_{opt}(P,k)$ is the radius of the minimum circle containing at least $k$ points of $P$. The expected running time of this approximation algorithm is $O(n + n \cdot\min((1/k\delta^3) \log^2 (1/\delta), k))$.     

一种结合完全连接的改进Apriori算法

基于Apriori算法原理，提出一种有效的完全连接条件，在频繁2k-项集的集合L2k进行自身Apriori连接得频繁(2k+1)-项集的同时，自身完全连接产生未剪枝的候选4k-项集；对频繁(2k+1)-项集的集合L2k+1，直接对其项集进行完全连接产生未剪枝的候选(4k+2)-项集。改进的算法减少了连接的比较次数、迭代运算次数。实验表明该算法在保证无遗漏的情况下有效地提高了Apriori算法的挖掘速度。