Algorithms for partitioning a graph

The k-way graph partitioning problem is considered with two efficient heuristic procedures. Algorithms “local extreme exchange” (LEE) and “overall extreme exchange” (OEE) are presented by modifying Kernighan-Lin's two way uniform partitioning method. In algorithm LEE, a node which maximizes the reduced cost is selected and exchanged with a node in another cluster such that the gain from the exchange with the selected node is maximized. The computational time efficiency of LEE is verified to be excellent compared to Kernighan-Lin's method. Algorithm OEE which considers a node pair that maximizes the reduced exchange cost is illustrated to be superior to the Kernighan-Lin's method. The time requirement of the proposed algorithm is also shown to be smaller than that of Kernighan-Lin's procedure.     

Efficient algorithms for single- and two-layer linear placement of parallel graphs

This paper outlines an algorithm for optimum linear ordering (OLO) of a weighted parallel graph with O(n log k) worst-case time complexity, and O(n + k log(n/k) log k) expected-case time complexity, where n is the total number of nodes and k is the number of chains in the parallel graph. Next, the two-layer OLO problem is considered, where the goal is to place the nodes linearly in two routing layers minimizing the total wire length. The two-layer problem is shown to subsume the maxcut problem and a befitting heuristic algorithm is proposed. Experimental results on randomly generated samples show that the heuristic algorithm runs very fast and outputs optimum solutions in more than 90% instances.     

基于Ncut准则的图分割的新算法

针对有权图分割时不能很好解决子图内部耦合度不高的问题,使用可以同时优化子图内部顶点耦合度和子图之间顶点耦合度的Ncut准则,提出了一种新的基于迭代改善策略的RNK分割算法。算法通过不断交换可以改善Ncut值的顶点对优化现有分割。与传统分割算法相比,可以同时保证子图内最大耦合度和子图间最小的耦合度。并提出一种散列技术,提高查找最优交换顶点对的效率。当图为稠密矩阵时,改善效果尤为明显。通过对随机图分割的实验结果表明,该算法较传统的KL算法可以得到更理想的分割结果。     

On the k-path partition of graphs

The k-path partition problem is to partition a graph into the minimum number of paths, so that none of them has length more than k, for a given positive integer k. The problem is a generalization of the Hamiltonian path problem and the problem of partitioning a graph into the minimum number of paths. The k-path partition problem remains NP-complete on the class of chordal bipartite graphs if k is part of the input, and we show that it is NP-complete on the class of comparability graphs even for k=3. On the positive side, we present a polynomial-time solution for the problem, with any k, on bipartite permutation graphs, which form a subclass of chordal bipartite graphs.     

基于二元决策图的集群计算系统性能分析

针对节点计算能力相同但故障分布不同的集群系统的性能分析问题,基于k-to-l-out-of-n结构对集群系统的性能进行建模,并提出了一种基于二元决策图（BDD）的分析方法。针对k-to-l-out-of-n结构的BDD模型生成问题,分析了BDD的结构特征并设计自顶向下生成算法,克服了传统的自底向上生成算法必须生成大量中间冗余节点的缺陷;然后利用生成的BDD模型高效地计算出系统处于一个特定性能级别的概率;最后通过实例说明了BDD方法能够有效分析节点具有不同故障分布的集群系统性能。     

基于参考节点嵌入的图可达性查询

针对k步可达性查询算法无法解决带距离约束的图可达性查询问题,提出基于参考节点嵌入的图可达性查询算法。首先,从所有节点中选出极少数有代表性的全局参考节点,预先计算所有节点与全局参考节点之间的最短路径距离;然后,采用最短路径树和范围最小值查询技术求得局部参考节点;接着,利用三角不等式关系得到查询点对距离范围;最后,根据查询条件中的距离值与查询点对距离范围上、下限值的大小关系,可快速得出可达性结论。针对社会关系网络和公路网络数据,将所提算法与Dijkstra算法、K-Reach算法进行实验对比测试。相较于K-Reach算法,其索引建立时间小4个数量级,其索引规模小2个数量级;相较于Dijkstra算法,在公路网络和社会关系网络中,直接得出可达性结论的比例分别为92%和78.6%,其查询时间大大缩短,分别降低了95.5%和92%。实验结果表明：所提算法能够通过使用较小的索引开销,实现在线查询计算复杂度的降低,可很好地解决既适用于有权图又适用于无权图带距离约束的可达性查询问题。     

基于贪婪策略的紧密k核子图查询

k核查询是一种社团查询,由于其可以在线性时间内被有效计算,因此在社团检测中具有较广泛的应用。图中边的权值在很多场景下具有较强的语义关系,但现有研究较少考虑图中边的权值。为提升k核查询的效率,在k核的基础上定义加权图中的紧密k核子图查询（CRKSQ）问题,并使用归约方法证明该问题是NP-难的。基于贪婪策略设计启发式算法CRK-G,通过迭代删除节点为CRKSQ问题找到一个近似解。在此基础上,从降低图规模和减少迭代次数两方面研究CRK-G算法的优化策略,分别提出使用图压缩策略的算法CRK-C及使用单次多节点删除策略的算法CRK-F。在Bio-GRID、Email-Enron、DBLP 3个数据集上的实验结果表明,相对于CRK-G算法,CRK-C、CRK-F算法在查询速度上有较大的提升,且平均误差均在8%以内。     

Partitioning approach oriented to the decentralised predictive control of large-scale systems

In this paper, a partitioning approach for large-scale systems based on graph-theory is presented. The algorithm starts with the translation of the system model into a graph representation. Once the system graph is obtained, the problem of graph partitioning is then solved. The resultant partition consists in a set of non-overlapping subgraphs whose number of vertices is as similar as possible and the number of interconnecting edges between them is minimal. To achieve this goal, the proposed algorithm applies a set of procedures based on identifying the highly connected subgraphs with balanced number of internal and external connections. In order to illustrate the use and application of the proposed partitioning approach, it is used to decompose a dynamical model of the Barcelona drinking water network (DWN). Moreover, a hierarchical-like DMPC strategy is designed and applied over the resultant set of partitions in order to assess the closed-loop performance. Results obtained when used several simulation scenarios show the effectiveness of both the partitioning approach and the DMPC strategy in terms of the reduced computational burden and, at the same time, of the admissible loss of performance in contrast to a centralised MPC strategy.     

Automated sub-zoning of water distribution systems

Water distribution systems (WDS) are complex pipe networks with looped and branching topologies that often comprise thousands to tens of thousands of links and nodes. This work presents a generic framework for improved analysis and management of WDS by partitioning the system into smaller (almost) independent sub-systems with balanced loads and minimal number of interconnections. This paper compares the performance of three classes of unsupervised learning algorithms from graph theory for practical sub-zoning of WDS: (1) Global clustering – a bottom-up algorithm for clustering n objects with respect to a similarity function, (2) Community structure – a bottom-up algorithm based on the property of network modularity, which is a measure of the quality of network partition to clusters versus randomly generated graph with respect to the same nodal degree, and (3) Graph partitioning – a flat partitioning algorithm for dividing a network with n nodes into k clusters, such that the total weight of edges crossing between clusters is minimized and the loads of all the clusters are balanced. The algorithms are adapted to WDS to provide a practical decision support tool for water utilities. Visual qualitative and quantitative measures are proposed to evaluate models' performance. The three methods are applied for two large-scale water distribution systems serving heavily populated areas in Singapore.     

Minimum cost source location problem with vertex-connectivity requirements in digraphs

Given a digraph (or an undirected graph) G=(V,E) with a set V of vertices v with nonnegative real costs w(v), and a set E of edges and a positive integer k, we deal with the problem of finding a minimum cost subset SV such that, for each vertex vV−S, there are k vertex-disjoint paths from S to v. In this paper, we show that the problem can be solved by a greedy algorithm in time in a digraph (or in time in an undirected graph), where n=|V| and m=|E|. Based on this, given a digraph and two integers k and ℓ, we also give a polynomial time algorithm for finding a minimum cost subset SV such that for each vertex vV−S, there are k vertex-disjoint paths from S to v as well as ℓ vertex-disjoint paths from v to S.     

面向非平衡类问题的k近邻分类算法

针对k近邻（kNN）方法不能很好地解决非平衡类问题,提出一种新的面向非平衡类问题的k近邻分类算法。与传统k近邻方法不同,在学习阶段,该算法首先使用划分算法（如K-Means）将多数类数据集划分为多个簇,然后将每个簇与少数类数据集合并成一个新的训练集用于训练一个k近邻模型,即该算法构建了一个包含多个k近邻模型的分类器库。在预测阶段,使用划分算法（如K-Means）从分类器库中选择一个模型用于预测样本类别。通过这种方法,提出的算法有效地保证了k近邻模型既能有效发现数据局部特征,又能充分考虑数据的非平衡性对分类器性能的影响。另外,该算法也有效地提升了k近邻的预测效率。为了进一步提高该算法的性能,将合成少数类过抽样技术（SMOTE）应用到该算法中。KEEL数据集上的实验结果表明,即使对采用随机划分策略划分的多数类数据集,所提算法也能有效地提高k近邻方法在评价指标recall、g-mean、f-measure和AUC上的泛化性能;另外,过抽样技术能进一步提高该算法在非平衡类问题上的性能,并明显优于其他高级非平衡类处理方法。     

On parallelizing the multiprocessor scheduling problem

Existing heuristics for scheduling a node and edge weighted directed task graph to multiple processors can produce satisfactory solutions but incur high time complexities, which tend to exacerbate in more realistic environments with relaxed assumptions. Consequently, these heuristics do not scale well and cannot handle problems of moderate sizes. A natural approach to reducing complexity, while aiming for a similar or potentially better solution, is to parallelize the scheduling algorithm. This can be done by partitioning the task graphs and concurrently generating partial schedules for the partitioned parts, which are then concatenated to obtain the final schedule. The problem, however, is nontrivial as there exists dependencies among the nodes of a task graph which must be preserved for generating a valid schedule. Moreover, the time clock for scheduling is global for all the processors (that are executing the parallel scheduling algorithm), making the inherent parallelism invisible. In this paper, we introduce a parallel algorithm that is guided by a systematic partitioning of the task graph to perform scheduling using multiple processors. The algorithm schedules both the tasks and messages, and is suitable for graphs with arbitrary computation and communication costs, and is applicable to systems with arbitrary network topologies using homogeneous or heterogeneous processors. We have implemented the algorithm on the Intel Paragon and compared it with three closely related algorithms. The experimental results indicate that our algorithm yields higher quality solutions while using an order of magnitude smaller scheduling times. The algorithm also exhibits an interesting trade-off between the solution quality and speedup while scaling well with the problem size     

The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan

In this paper we consider the unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an O(n(n/m+1)^m) time dynamic programming algorithm and an O(mk^k+1P^2k−1) time dynamic programming algorithm, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the processing times of all families. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme for the problem.     

基于k-核过滤的社交网络影响最大化算法

针对现有社交网络影响最大化算法影响范围小和时间复杂度高的问题,提出一种基于独立级联模型的k-核过滤算法。首先,介绍了一种节点影响力排名不依赖于整个网络的现有影响力最大化算法;然后,通过预训练k,找到对现有算法具有最佳优化效果且与选择种子数无关的k值;最后,通过计算图的k-核过滤不属于k-核子图的节点和边,在k-核子图上执行现有影响最大化算法,达到降低计算复杂度的目的。为验证k-核过滤算法对不同算法有不同的优化效果,在不同规模数据集上进行了实验。结果显示,应用k-核过滤算法后：与原PMIA算法相比,影响范围最多扩大13.89%,执行时间最多缩短8.34%;与原核覆盖算法（CCA）相比,影响范围没有太大差异,但执行时间最多缩短28.5%;与OutDegree算法相比,影响范围最多扩大21.81%,执行时间最多缩短26.96%;与Random算法相比,影响范围最多扩大71.99%,执行时间最多缩短24.21%。进一步提出了一种新的影响最大化算法GIMS,它比PMIA和IRIE的影响范围更大,执行时间保持在秒级别,而且GIMS算法的k-核过滤算法与原GIMS算法的影响范围和执行时间差异不大。实验结果表明,k-核过滤算法能够增大现有算法选择种子节点集合的影响范围,并且减少执行时间;GIMS算法具有更好的影响范围效果和执行效率,并且更加鲁棒。     

LESS-mapping: Online environment segmentation based on spectral mapping

Given the features obtained from a sequence of consecutively acquired sensor readings, this paper proposes an on-line algorithm for unsupervisedly detecting a transition on this sequence, i.e. the frame that divides the sequence into two tightly related parts that are dissimilar between them. Contrary to recently proposed approaches that address this partitioning problem dealing with a sequence of robot’s poses, our proposal considers each individual feature as a node of an incrementally built graph whose edges link two nodes if their associated features were simultaneously observed. These graph edges carry non-negative weights according to the locality of the features. Given a feature, its locality defines the set of features that has been observed simultaneously with it at least once. At each execution of the algorithm, the feature-based graph is split into two subgraphs using a normalized spectral clustering algorithm. The obtained partitions correspond to those parts in the environment that share the minimum amount of information. If this graph partition is validated, the algorithm determines that there is a significant change on the perceived scenario, assuming that a transition area has been traversed. In a map partitioning framework, we have tested the proposed approach in real environments where features are obtained using 2D laser sensors or vision (stereo and monocular cameras). The statistical evaluation of the experimental results demonstrates the performance of the proposal.     

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.     

多编码树GPU并行轴心子图匹配

子图同构问题是非确定多项式(NP)完全问题,而轴心子图同构是一种特殊的子图同构问题.针对现在已经有许多高效的子图同构算法,然而对于轴心子图同构问题目前并没有基于GPU的搜索算法,且通过改造已有的子图同构算法来解决轴心子图匹配问题会产生大量不必要的中间结果这一问题,提出了一种基于GPU的轴心子图同构算法.首先,通过一种新...     

On a multimode test sequencing problem

Test sequencing is a binary identification problem wherein one needs to develop a minimal expected cost test procedure to determine which one of a finite number of possible failure states, if any, is present. In this paper, we consider a multimode test sequencing (MMTS) problem, in which tests are distributed among multiple modes and additional transition costs will be incurred if a test sequence involves mode changes. The multimode test sequencing problem can be solved optimally via dynamic programming or AND/OR graph search methods. However, for large systems, the associated computation with dynamic programming or AND/OR graph search methods is substantial due to the rapidly increasing number of OR nodes (denoting ambiguity states and current modes) and AND nodes (denoting next modes and tests) in the search graph. In order to overcome the computational explosion, we propose to apply three heuristic algorithms based on information gain: information gain heuristic (IG), mode capability evaluation (MC), and mode capability evaluation with limited exploration of depth and degree of mode Isolation (MCLEI). We also propose to apply rollout strategies, which are guaranteed to improve the performance of heuristics, as long as the heuristics are sequentially improving. We show computational results, which suggest that the information-heuristic based rollout policies are significantly better than traditional information gain heuristic. We also show that among the three information heuristics proposed, MCLEI achieves the best tradeoff between optimality and computational complexity.     

k-Anonymization with Minimal Loss of Information

The technique of k-anonymization allows the releasing of databases that contain personal information while ensuring some degree of individual privacy. Anonymization is usually performed by generalizing database entries. We formally study the concept of generalization, and propose three information-theoretic measures for capturing the amount of information that is lost during the anonymization process. The proposed measures are more general and more accurate than those that were proposed by Meyerson and Williams and Aggarwal et al. We study the problem of achieving k-anonymity with minimal loss of information. We prove that it is NP-hard and study polynomial approximations for the optimal solution. Our first algorithm gives an approximation guarantee of O(ln k) for two of our measures as well as for the previously studied measures. This improves the best-known O(k)-approximation in. While the previous approximation algorithms relied on the graph representation framework, our algorithm relies on a novel hypergraph representation that enables the improvement in the approximation ratio from O(k) to O(ln k). As the running time of the algorithm is O(n^2k}), we also show how to adapt the algorithm in in order to obtain an O(k)-approximation algorithm that is polynomial in both n and k.     

基于语义位置保护的轨迹隐私保护的k-CS算法

针对轨迹数据隐私保护算法数据可用性低及易受语义位置攻击和最大运行速度攻击等问题,提出了一种在路网环境中基于语义轨迹的隐私保护算法——k-CS算法。首先,提出了两种路网环境中针对轨迹数据的攻击模型;然后,将路网环境中基于语义轨迹的隐私问题定义为k-CS匿名问题,并证明了该问题是一个NP难问题;最后,提出了一种基于图上顶点聚类的近似算法将图上的顶点进行匿名,将语义位置由相应的匿名区域取代。实验对所提算法和轨迹隐私保护经典算法（k,δ）-anonymity进行了对比,实验结果表明：k-CS算法在数据可用性、查询误差率、运行时间等方面优于（k,δ）-anonymity算法;平均信息丢失率比（k,δ）-anonymity算法降低了20%左右;算法运行时间比（k,δ）-anonymity算法减少近10%。