Monotonicity of non-deterministic graph searching

In graph searching, a team of searchers are aiming at capturing a fugitive moving in a graph. In the initial variant, called invisible graph searching, the searchers do not know the position of the fugitive until they catch it. In another variant, the searchers permanently know the position of the fugitive, i.e. the fugitive is visible. This latter variant is called visible graph searching. A search strategy that catches any fugitive in such a way that the part of the graph reachable by the fugitive never grows is called monotone. A priori, monotone strategies may require more searchers than general strategies to catch any fugitive. This is however not the case for visible and invisible graph searching. Two important consequences of the monotonicity of visible and invisible graph searching are: (1) the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in NP, and (2) computing optimal search strategies is simplified by taking into account that there exist some that never backtrack.

Fomin et al. [F.V. Fomin, P. Fraigniaud, N. Nisse, Nondeterministic graph searching: From pathwidth to treewidth, in: Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, MFCS’05, 2005, pp. 364–375] introduced an important graph searching variant, called non-deterministic graph searching, that unifies visible and invisible graph searching. In this variant, the fugitive is invisible, and the searchers can query an oracle that permanently knows the current position of the fugitive. The question of the monotonicity of non-deterministic graph searching was however left open.

In this paper, we prove that non-deterministic graph searching is monotone. In particular, this result is a unified proof of monotonicity for visible and invisible graph searching. As a consequence, the decision problem corresponding to non-deterministic graph searching belongs to NP. Moreover, the exact algorithms designed by Fomin et al. do compute optimal non-deterministic search strategies.     

Nondeterministic Graph Searching: From Pathwidth to Treewidth

We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this game-theoretic approach and graph decompositions called q -branched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard) tree decomposition are two extreme cases of q-branched tree decompositions. The equivalence between nondeterministic graph searching and q-branched tree decomposition enables us to design an exact (exponential time) algorithm computing q-branched treewidth for all q≥0, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required to search a graph with the minimum number of searchers. Additional support of F.V. Fomin by the Research Council of Norway. Additional supports of P. Fraigniaud from the INRIA Project “Grand Large”, and from the Project PairAPair of the ACI “Masse de Données”. Additional supports of N. Nisse from the Project Fragile of the ACI “Sécurité & Informatique”.     

Edge and node searching problems on trees

In this paper, we consider the edge searching and node searching problems on trees. Given a tree, we show a transformation from an optimal node-search strategy to an optimal edge-search strategy. Using our transformation, we simplify a previous linear-time algorithm for determining the edge-search number of a tree, and improve the running time of a previous algorithm for constructing an optimal edge-search strategy of an n-vertex tree from O(nlogn) to O(n). We also improve the running time of a previous algorithm for constructing an optimal min-cut linear layout of an n-vertex tree with the maximum degree 3 from O(nlogn) to O(n).     

An annotated bibliography on guaranteed graph searching

Graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers. In this annotated bibliography, we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field.     

Time constrained graph searching

We consider time constraints for four models of searching graphs for intruders. One model is the standard cops and robber vertex-searching model with complete visibility. The second model differs from the preceding one only in that none of the searchers can see the intruder. The third model is a vertex-searching model in which searchers and an intruder move simultaneously and none of the searchers can see the intruder. The fourth model is simultaneous edge searching with an arbitrarily fast intruder.     

Connected searching of weighted trees

In this paper we consider the problem of connected edge searching of weighted trees. Barrière et al. claim in [L. Barrière, P. Flocchini, P. Fraigniaud, N. Santoro, Capture of an intruder by mobile agents, in: SPAA’02: Proceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures, ACM, New York, NY, USA, 2002, pp. 200-209] that there exists a polynomial-time algorithm for finding an optimal search strategy, that is, a strategy that minimizes the number of used searchers. However, due to some flaws in their algorithm, the problem turns out to be open. It is proven in this paper that the considered problem is strongly NP-complete even for node-weighted trees (the weight of each edge is 1) with one vertex of degree greater than 2. It is also shown that there exists a polynomial-time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices. This is an FPT algorithm with respect to the maximum degree of a tree.     

Graph searching with advice

Fraigniaud et al. [L. Blin, P. Fraigniaud, N. Nisse, S. Vial, Distributing chasing of network intruders, in: 13th Colloquium on Structural Information and Communication Complexity, SIROCCO, in: LNCS, vol. 4056, Springer-Verlag, 2006, pp. 70–84] introduced a new measure of difficulty for a distributed task in a network. The smallest number of bits of advice   of a distributed problem is the smallest number of bits of information that has to be available to nodes in order to accomplish the task efficiently. Our paper deals with the number of bits of advice required to perform efficiently the graph searching problem in a distributed setting. In this variant of the problem, all searchers are initially placed at a particular node of the network. The aim of the team of searchers is to clear a contaminated graph in a monotone connected way, i.e., the cleared part of the graph is permanently connected, and never decreases while the search strategy is executed. Moreover, the clearing of the graph must be performed using the optimal number of searchers, i.e. the minimum number of searchers sufficient to clear the graph in a monotone connected way in a centralized setting. We show that the minimum number of bits of advice permitting the monotone connected and optimal clearing of a network in a distributed setting is Θ(nlogn)$\Theta (nlogn)$, where n$n$ is the number of nodes of the network. More precisely, we first provide a labelling of the vertices of any graph G$G$, using a total of O(nlogn)$O(nlogn)$ bits, and a protocol using this labelling that enables the optimal number of searchers to clear G$G$ in a monotone connected distributed way. Then, we show that this number of bits of advice is optimal: any distributed protocol requires Ω(nlogn)$\Omega (nlogn)$ bits of advice to clear a network in a monotone connected way, using an optimal number of searchers.     

Blocking for external graph searching

In this paper we consider the problem of using disk blocks efficiently in searching graphs that are too large to fit in internal memory. Our model allows a vertex to be represented any number of times on the disk in order to take advantage of redundancy. We give matching upper and lower bounds for completed-ary trees andd-dimensional grid graphs, as well as for classes of general graphs that intuitively speaking have a close to uniform number of neighbors around each vertex. We also show that, for the special case of grid graphs blocked with isothetic hypercubes, there is a provably better speed-up if even a small amount of redundancy is permitted.Support was provided in part by an IBM Graduate Fellowship, by NSF Research Grants CCR-9007851 and IRI-9116451, and by Army Research Office Grant DAAL03-91-G-0035.Support was provided in part by NSF Grants CCR-9003299, CCR-9300079, and IRI-9116843, and by NSF/DARPA Grant CCR-8908092.Support was provided in part by a National Science Foundation Presidential Young Investigator Award CCR-9047466 with matching funds from IBM, by NSF Research Grant CCR-9007851, and by Army Research Office Grant DAAL03-91-G-0035.     

A spectral lower bound for the treewidth of a graph and its consequences

We give a lower bound for the treewidth of a graph in terms of the second smallest eigenvalue of its Laplacian matrix. We use this lower bound to show that the treewidth of a d-dimensional hypercube is at least ⌊3·2^d/(2(d+4))⌋−1. The currently known upper bound is . We generalize this result to Hamming graphs. We also observe that every graph G on n vertices, with maximum degree Δ

(1)

contains an induced cycle (chordless cycle) of length at least 1+log_Δ(μn/8) (provided G is not acyclic),

(2)

has a clique minor K_h for some ,

where μ is the second smallest eigenvalue of the Laplacian matrix of G.     

Pathwidth of cubic graphs and exact algorithms

We prove that for any ?>0 there exists an integer n? such that the pathwidth of every cubic (or 3-regular) graph on n>n? vertices is at most (1/6+?)n. Based on this bound we improve the worst case time analysis for a number of exact exponential algorithms on graphs of maximum vertex degree three.     

类属图密集近邻搜索的视觉跟踪算法研究

提出一种基于密集近邻搜索的视觉跟踪算法,能够有效应对目标跟踪过程中出现的形变和遮挡问题.基于马尔科夫随机场建立图像分割模型,提取出目标部件,建立目标部件的类属图矩阵;通过搜索类属图矩阵中的密集近邻,得到相邻帧之间目标部件的匹配关系;通过匹配关系得到跟踪目标位置概率图,确定目标跟踪位置.实验结果表明:本文提出的方法相比其他同类方法效果更好.     

面向大规模图计算的连通分量算法分析与优化

近年来,图计算在诸多领域发挥着越来越重要的作用。连通分量算法是图计算的重要基础算法,可以应用于可达性查询、一致性检测等众多场景。面向大规模图遍历Graph500标准测试,对连通分量算法进行了算法和数据结构优化。主要有以下创新:(1)对并查集提出了捷径向量算法,并测试了算法和数据结构的配合程度;(2)利用多线程迭代轮转对算法实现并行加速;(3)从多个维度比较了不同实现方法的优缺点。基于优化方法,对性能进行了评估分析,当scale=25(包含2²⁵个节点)时,捷径向量算法对基于二维向量和链表的按秩合并算法的加速比分别是1.38倍和1.40倍,对BFS和DFS的加速比分别为4.76倍和4.70倍,且空间占用为该2个算法的4.1%～4.6%,此外,并行对串行的加速比为1.57倍。     

Improved string searching

We show that it is possible to improve the average time of the Boyer-Moore string matching algorithm using more space. This is accomplished by applying a transformation that virtually increases the size of the alphabet in use. The improvement is such that for long patterns it is possible to obtain an algorithm more than 50 per cent faster than the original one. We include experimental results on random and English text. Some improvements for searching on English text are also discussed.     

Digraph decompositions and monotonicity in digraph searching

Graph decompositions such as tree-decompositions and associated width measures have been the focus of much attention in structural and algorithmic graph theory. In particular, it has been found that many otherwise intractable problems become tractable on graph classes of bounded tree-width.More recently, proposals have been made to define a similar notion to tree-width for directed graphs. Several proposals have appeared so far, supported by algorithmic applications.In this paper we explore the limits of algorithmic applicability of digraph decompositions and show that various natural candidates for problems, which potentially could benefit from digraphs having small “directed width”, remain NP-complete even on almost acyclic graphs.Closely related to graph and digraph decompositions are graph searching games. An important property of graph searching games is monotonicity and a large number of papers addresses the question whether particular variants of these games are monotone. However, so far for two natural types of graph searching games-underlying DAG- and Kelly-decompositions-the question whether they are monotone was still open.We settle this issue by showing that both variants, the visible and the inert invisible graph searching games on directed graphs, are non-monotone.     

Treewidth computations II. Lower bounds

For several applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast. Good lower bounds on the treewidth of a graph can, amongst others, help to speed up branch and bound algorithms that compute the treewidth of a graph exactly. A high lower bound for a specific graph instance can tell that a dynamic programming approach for solving a problem is infeasible for this instance. This paper gives an overview of several recent methods that give lower bounds on the treewidth of graphs.     

论计算机搜查中的隐私权保护

本文对计算机所包含的内容的特点进行分析,进而说明实际计算机搜查中启动条件的随意性、搜查对象的不确定性和司法审查的欠缺性,为计算机搜查的研究提出一些问题。     

Parameterized complexity of connected even/odd subgraph problems

In 2011, Cai an Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs: for a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about:

•

a connected k-edge subgraph with all vertices of odd degrees, the problem known as k-Edge Connected Odd Subgraph; and     

Fast string searching

Since the Boyer-Moore algorithm was described in 1977, it has been the standard benchmark for the practical string search literature. Yet this yardstick compares badly with current practice. We describe two algorithms that perform 47% fewer comparisons and are about 4.5 times faster across a wide range of architectures and compilers. These new variants are members of a family of algorithms based on the skip loop structure of the preferred, but often neglected, fast form of Boyer-Moore. We present a taxonomy for this family, and describe a toolkit of components that can be used to design an algorithm most appropriate for a given set of requirements.