基于回溯机制的互联网AS拓扑的Betweenness算法

Betweenness能够刻画节点或边在网络中的重要程度．在Internet中，Betweenness直接反应了特定网络拓扑结构下节点或链路可能承载的网络流量，能够对网络的动态行为进行预测．但传统的Betweenness计算复杂度较高，为O（n^3），但这些算法是为加权网络设计的，而很多实际的网络模型并没有考虑权重．另一方面，目前的算法都没有考虑边的语义，而互联网AS（autonomous system）拓扑中的边具有语义．针对简单无权网络提出一种基于回溯的时间复杂度为O（nm）的Betweenness计算方法．在进一步考虑Internet AS拓扑的特殊性，即任意两个相连的AS都具有某种商业关系的基础上提出了互联网AS层拓扑的Betweenness计算方法．     

Maximal Biclique Subgraphs and Closed Pattern Pairs of the Adjacency Matrix: A One-to-One Correspondence and Mining Algorithms

Maximal biclique (also known as complete bipartite) subgraphs can model many applications in Web mining, business, and bioinformatics. Enumerating maximal biclique subgraphs from a graph is a computationally challenging problem, as the size of the output can become exponentially large with respect to the vertex number when the graph grows. In this paper, we efficiently enumerate them through the use of closed patterns of the adjacency matrix of the graph. For an undirected graph G without self-loops, we prove that 1) the number of closed patterns in the adjacency matrix of G is even, 2) the number of the closed patterns is precisely double the number of maximal biclique subgraphs of G, and 3) for every maximal biclique subgraph, there always exists a unique pair of closed patterns that matches the two vertex sets of the subgraph. Therefore, the problem of enumerating maximal bicliques can be solved by using efficient algorithms for mining closed patterns, which are algorithms extensively studied in the data mining field. However, this direct use of existing algorithms causes a duplicated enumeration. To achieve high efficiency, we propose an O(mn) time delay algorithm for a nonduplicated enumeration, in particular, for enumerating those maximal bicliques with a large size, where m and n. are the number of edges and vertices of the graph, respectively. We evaluate the high efficiency of our algorithm by comparing it to state- of-the-art algorithms on three categories of graphs: randomly generated graphs, benchmarks, and a real-life protein interaction network. In this paper, we also prove that if self-loops are allowed in a graph, then the number of closed patterns in the adjacency matrix is not necessarily even, but the maximal bicliques are exactly the same as those of the graph after removing all the self-loops.     

Inserting an Edge into a Planar Graph

Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e where all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a combinatorial embedding of a planar graph G where the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, that is able to find a solution with the minimum number of crossings.     

Backtracking with multi-level dynamic search rearrangement

Summary The order in which the variables are tested in a backtrack program can have a major effect on its running time. The best search order usually varies among the branches of the backtrack tree, so the number of possible search orders can be astronomical. We present an algorithm that chooses a search order dynamically by investigating all possibilities for k levels below the current level, extending beyond k levels wherever possible by setting the variables that have unique forced values. The algorithm takes time O(n ^k+1) to process a node. For k=2 and binary variables the analysis for selecting the next variable to introduce into the backtrack tree makes complete use of the information contained in the two-level investigations. For larger k or variables of higher degree there is no polynomial-time algorithm that makes complete use of the k-level investigations to limit searching (unless P=NP). The search rearrangement algorithm is closely related to constraint propagation. Experimental studies on conjunctive normal form predicates confirm that 1-level search rearrangement saves a great deal of time compared to 0-level (ordinary backtracking), and show that 2-level saves time over 1-level on large problems. For such problems with 256 variables 2-level is better than 1-level by a factor of two.     

Solving quantified constraint satisfaction problems with value selection rules

Solving a quantified constraint satisfaction problem (QCSP) is usually a hard task due to its computational complexity. Exact algorithms play an important role in solving this problem, among which backtrack algorithms are effective. In a backtrack algorithm, an important step is assigning a variable by a chosen value when exploiting a branch, and thus a good value selection rule may speed up greatly. In this paper, we propose two value selection rules for existentially and universally quantified variables, respectively, to avoid unnecessary searching. The rule for universally quantified variables is prior to trying failure values in previous branches, and the rule for existentially quantified variables selects the promising values first. Two rules are integrated into the state-of-the-art QCSP solver, i.e., QCSPSolve, which is an exact solver based on backtracking. We perform a number of experiments to evaluate improvements brought by our rules. From computational results, we can conclude that the new value selection rules speed up the solver by 5 times on average and 30 times at most. We also show both rules perform well particularly on instances with existentially and universally quantified variables occurring alternatively.     

Fair circulation of a token

Suppose that a distributed system is modeled by an undirected graph G = (V, E), where V and E, respectively, are the sets of processes and communication links. Israeli and Jalfon (1990) proposed a simple self-stabilizing mutual exclusion algorithm: a token is circulated among the processes (i.e., vertices) and a process can access the critical section only when it holds the token. In order to guarantee equal access chance to all processes, the token circulation needs to be fair in the sense that all processes have the same probability of holding the token. However, the Israeli-Jalfon token circulation scheme does not meet the requirement. This paper proposes a new scheme for making it fair. We evaluate the average of the longest waiting times in terms of the cover time and show an O(deg(G)n²) upper bound on the cover time for our scheme, where n and deg(G) are the number of processes and the maximum degree of G, respectively. The same (tight) upper bound is known for the Israeli-Jalfon scheme     

Drawing Planar Graphs Symmetrically, III: Oneconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries.     

The Linear Arrangement Problem Parameterized Above Guaranteed Value

A linear arrangement (LA) is an assignment of distinct integers to the vertices of a graph. The cost of an LA is the sum of lengths of the edges of the graph, where the length of an edge is defined as the absolute value of the difference of the integers assigned to its ends. For many application one hopes to find an LA with small cost. However, it is a classical NP-complete problem to decide whether a given graph G admits an LA of cost bounded by a given integer. Since every edge of G contributes at least one to the cost of any LA, the problem becomes trivially fixed-parameter tractable (FPT) if parameterized by the upper bound of the cost. Fernau asked whether the problem remains FPT if parameterized by the upper bound of the cost minus the number of edges of the given graph; thus whether the problem is FPT "parameterized above guaranteed value." We answer this question positively by deriving an algorithm which decides in time O(m + n + 5.88^k) whether a given graph with m edges and n vertices admits an LA of cost at most m + k (the algorithm computes such an LA if it exists). Our algorithm is based on a procedure which generates a problem kernel of linear size in linear time for a connected graph G. We also prove that more general parameterized LA problems stated by Serna and Thilikos are not FPT, unless P = NP.     

A linear-time algorithm to compute a MAD tree of an interval graph

The average distance of a connected graph G is the average of the distances between all pairs of vertices of G. We present a linear time algorithm that determines, for a given interval graph G, a spanning tree of G with minimum average distance (MAD tree). Such a tree is sometimes referred to as a minimum routing cost spanning tree.     

采用超节点协同的多智能体系统一致性算法

针对如何提高多智能体系统达到一致性的收敛速度的问题,提出了一种采用超节点协同的多智能体系统一致性算法.新算法对多智能体系统建立图信号模型,在图中选出超节点进行协同,有效提高一致性收敛速度.首先利用单跳采样算法对图进行超节点的选取和局部集的划分,并对局部集内的节点进行一次协同.然后超节点之间进行边的连接得到粗化图,用粗化图的拉普拉斯矩阵特征值设计图滤波器的系数.最后超节点的信号经过图滤波器迭代达到平均值后,传输给其一阶邻居节点,使所有节点达到平均一致.仿真结果表明所提算法能够最终实现平均一致性,与现有方法相比,可以显著提高收敛速度,并减少计算量.     

EvIA - Evidential Interval Algebra and Heuristic Backtrack-Free Algorithm

The Interval Algebra (IA) framework for temporal reasoning encodes indefinite knowledge in terms of disjunctions of relations. Many problems arising in practice can have evidences from past or from other external sources to indicate that some relations in a disjunction may be more probable than others. IA framework is inadequate to encode this information. The aim of the present study is two fold. First, to extend IA framework by associating numeric weights to the relations for capturing additional information and provide a reasoning methodology for the extended framework. Second, to apply the extended framework for developing a heuristic algorithm which finds a solution of the conventional IA network problem without backtrack. We make use of well-known evidential reasoning techniques to develop the new framework, Evidential Interval Algebra (EvIA). EvIA is an augmentation of interval algebra with evidential techniques. The constraint, constraint operators namely converse, composition and intersection, and path consistency algorithm of interval algebra are overlayed by evidential function and evidential operations to get enhanced expressiveness and efficient reasoning capability. The efficiency of the EvIA framework is demonstrated in the form of a heuristic which finds a solution of the interval algebra network without backtrack. Experimental results of the heuristic algorithm reveal that the algorithm is sound and for some specific types of the problems, the success of finding a solution is more than 90 percent. The results also show that the algorithm is efficient in terms of runtime when compared with a backtrack search algorithm.     

GenMax: An Efficient Algorithm for Mining Maximal Frequent Itemsets

We present GenMax, a backtrack search based algorithm for mining maximal frequent itemsets. GenMax uses a number of optimizations to prune the search space. It uses a novel technique called progressive focusing to perform maximality checking, and diffset propagation to perform fast frequency computation. Systematic experimental comparison with previous work indicates that different methods have varying strengths and weaknesses based on dataset characteristics. We found GenMax to be a highly efficient method to mine the exact set of maximal patterns.     

图的有损摘要问题的两阶段算法

问题如下:给定图G=(V, E)和正整数k,要求将图G中所有节点合并成为k个超节点,满足由这些超节点组成的摘要图能够在一定误差范围内表示原图G.这是一个基于图划分的组合优化问题,一个主要求解思路是逐次地随机抽取节点对集并用启发式方法从中选取节点对进行合并.本文提出一个有效的两阶段求解算法TS＿LGS.算法根据图G的平均点度特征设置阶段阈值:当前超节点数大于阶段阈值为第1阶段,期间算法在采样节点对中基于当前最佳合并分数批量选择节点对合并,旨在有效减少迭代次数;否则为第2阶段,期间算法在加权采样的基础上优先挑选相邻的节点对,旨在找到重构误差增量较小的节点对合并,直至超节点的个数为k.在典型的真实网络实例图上与现有最好算法SAA进行了实验对比,结果表明,算法TS＿LGS以较低时间复杂度提取到的图摘要具有更低的重构误差和查询误差.     

应用搜索原理解n皇后问题算法分析及优化

n皇后问题是非结构化的问题,人工智能中的搜索策略——回溯法是解决这类问题的有效方法。本文介绍了利用回溯法求解n皇后问题的基本思想以及实现方法,并对算法提出了优化的方法,使得算法的运行效率更高。     

An Effective Test Generation Algorithm for Combinational Circuits

In this paper,an analysis of backtrack behavior in PODEM(the test generation algorithmfor combinational circuits presented by P.Goel)is given.It is pointed out that there are stillmany unnecessary backtracks in PODEM on some occasions.A new test generation algorithmnamed IPODEM is therefore proposed in this paper.IPODEM is an improvement over PODEMwith emphasis on backtrack of decision tree.A new backtrack approach is developed in thisalgorithm.It is shown that only O(j)of backtrack consumption is needed in IPODEMcompared with O(2~j)in PODEM on certain occasions.Experiments pointed out that theseoccasions appear in not small proportion.Several other techniques are applied in IPODEM toaccelerate test generation process in other aspects.Experimental results demonstrated thatIPODEM is faster than PODEM for both hard-testing and easy-testing single stuck fault,andthat the former has higher test coverage than the latter.     

Matching Algorithms Are Fast in Sparse Random Graphs

We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every non-maximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like the Hopcroft-Karp algorithm for bipartite graphs and the Micali-Vazirani algorithm for general graphs, which have a worst case running time of O(m√n), run in time O(m log n) with high probability, where m is the number of edges in the graph. Motwani proved these results for random graphs when the average degree is at least ln (n) [Average Case Analysis of Algorithms for Matchings and Related Problems, Journal of the ACM, 41(6):1329-1356, 1994]. Our results hold if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.     

Maximum gap labelings of graphs

Given a graph, we define a base set to be a set of integers of size equal to the number of vertices in the graph. Given a graph and a base set, a labeling of the graph from the base set is an assignment of distinct integers from the base set to the vertices of the graph. The gap of an edge in a labeled graph is the absolute value of the difference between the labels of its endpoints. The gap of a labeled graph is the sum of the gaps of its edges.The maximum gap graph labeling problem takes as input a graph and a base set and maximizes the gap of the graph over all possible labelings from the base set. We show that this problem is NP-complete even when the base set is restricted to consecutive integers. We also show that this restricted case has polynomial time approximations that achieve a factor of 2/3 for trees, of 1/2 for bipartite graphs, and of 1/4 for general graphs, with a deterministic algorithm, while an expected factor of 1/3 for general graphs is achieved with a randomized algorithm. The case of general base sets is approximated within an expected factor of 1/16 for general graphs with a randomized polynomial time algorithm. We finally give a polynomial time algorithm that solves the maximum gap graph labeling problem for a graph that has bounded degree and bounded treewidth. The maximum graph labeling problem shows connections with the graceful tree conjecture.     

Variable ordering and constraint propagation for constrained CP-nets

A Conditional Preferences network (CP-net) is a known graphical model for representing qualitative preferences. In many real world applications we are often required to manage both constraints and preferences in an efficient way. The goal here is to select one or more scenarios that are feasible according to the constraints while maximizing a given utility function. This problem has been modelled as a CP-net where some variables share a set of constraints. This latter framework is called a Constrained CP-net. Solving the constrained CP-net has been proposed in the past using a variant of the branch and bound algorithm called Search CP. In this paper, we experimentally study the effect of variable ordering heuristics and constraint propagation when solving a constrained CP-net using a backtrack search algorithm. More precisely, we investigate several look ahead strategies as well as the most constrained heuristic for variable ordering during search. The results of the experiments conducted on random Constrained CP-net instances generated through the RB model, clearly show a significant improvement when adopting these techniques for specific graph structures as well as the case where a large number of variables are sharing constraints.     

Efficiency of Randomized Parallel Backtrack Search

This paper presents an improved analysis of a randomized parallel backtrack search algorithm (RPBS). Our analysis uses the single-node-donation model that each donation contains a single tree node. It is shown that with high probability the total number of messages generated by RPBS is O(phd) where p is the number of processors, and h and d are the height and degree of the backtrack search tree. Under the assumption of unit-time message delivery, it is shown that with high probability the execution time of RPBS is n/p + O(hd) where n is the number of nodes of the backtrack search tree and the leading term n/p has no constant factor. As the result of limited communication requirement, RPBS can be efficiently implemented in message-passing or shared-memory multiprocessor systems. A general analysis of network implementation of RPBS is presented. The concept of total routing time, the sum of routing times of all messages, is introduced as a measure of communication cost. It is shown that the overall effect of message delay to the execution time of RPBS is small if the total routing time is small. Some experimental data on a shared-memory machine are reported. Received November 23, 1996; revised February 15, 1998.     

Combinatorial search versus genetic algorithms: A case study based on the generalized median graph problem

The computation of generalized median graphs (the graph with the smallest average edit distance to all graphs in a given set of graphs) is highly computationally complex. As a matter of fact, it is exponential in the number of nodes of the union of all graphs under consideration. Thus, the generalized median graph computation problem seems to be a suitable and challenging testbed for a comparison of combinatorial search and genetic algorithms. Two solutions are described in this paper. The first is an exact algorithm based on combinatorial search, while the second is a genetic algorithm. Both approaches are compared to each other in a series of experiments.