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Safety in s-t Paths, Trails and Walks

algorithmICA 2022年第3期84卷 719-741页

作者： Cairo, Massimo Khan, Shahbaz Rizzi, Romeo Schmidt, Sebastian Tomescu, Alexandru, I Univ Helsinki Dept Comp Sci Helsinki Finland Univ Verona Dept Comp Sci Verona Italy

Given a directed graph G and a pair of nodes s and t, an s-t bridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of "safety" from bioinformatics. We say that a walk W is safe with respect to a set W of s-t walks, if W is a subwalk of all walks in W. We start by considering the maximal safe walks when W consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-t articulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural "safety"-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.

关键词： Directed graph Connectivity problem graph algorithm Strong bridge Strong articulation point Safety Genome assembly

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Computing bounded-degree phylogenetic roots of disconnected graphs

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JOURNAL OF algorithmS-COGNITION INFORMATICS AND LOGIC 2006年第2期59卷 125-148页

作者： Chen, Zhi-Zhong Tsukiji, Tatsuie Tokyo Denki Univ Dept Math Sci Hatoyama Saitama 3500394 Japan Tokyo Denki Univ Dept Informat Sci Hatoyama Saitama 3500394 Japan

The Phylogenetic kth Root Problem (PRk) is the problem of finding a (phylogenetic) tree T from a given graph G = (V, E) such that (1) T has no degree-2 internal nodes, (2) the external nodes (i.e., leaves) of T are exactly the elements of V, and (3) (u, v) is an element of E if and only if the distance between it and v in tree T is at most k, where k is some fixed threshold k. Such a tree T, if exists. is called a phylogenetic kth root of graph G. The computational complexity of PRk is open, except for k <= 4. Recently, Chen et al. investigated PRk under a natural restriction that the maximum degree of the phylogenetic root is bounded from above by a constant. They presented a linear-time algorithm that determines if a given connected G has such a phylogenetic kth root, and if so. demonstrates one. In this paper, we supplement their work by presenting a linear-time algorithm for disconnected graphs. (c) 2005 Elsevier Inc. All rights reserved.

关键词： phylogeny phylogenetic root computational biology graph power tree power graph algorithm

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Multicast Routing Using Delay Intervals for Collaborative and Competitive Applications

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IEEE TRANSACTIONS ON COMMUNICATIONS 2018年第12期66卷 6329-6338页

作者： Andrus, James Banik, Shankar M. Swart, Breeanne Baker Verdicchio, Michael The Citadel Dept Math & Comp Sci Charleston SC 29409 USA

Collaborative and competitive applications require that participants receive messages almost simultaneously and before a specified time. These requirements have been addressed by the delay variation-bounded multicasting tree (DVBMT) problem. In this paper, we propose the interval multicast subgraph (IMS) problem to address these requirements. IMS addresses these constraints with an interval of acceptable delay values for paths as user input from a source to a destination, eliminating the need to optimize delays for a variation value. By solving IMS rather than DVBMT and other variants of DVBMT, we are able to find solutions for larger graphs more efficiently. Our proposed interval multicast algorithm (IMA) accounts for an interval of acceptable delay as user input and guarantees the weight of each path from the source to a distinct destination is within the given interval if that path exists. We provide proofs of correctness and complexity of IMA, as well as simulation experiments, to illustrate the effects of various parameters on our algorithm. Simulations show that IMS is significantly less costly than finding the minimum variation for the average and best case. By remodeling the DVBMT problem to IMS, we have created a new problem that addresses the quality of service requirements of multicasting and is able to be solved efficiently for the average case for relatively large graphs.

关键词： QoS graph algorithm multicast routing delay interval collaborative applications

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Enumerating All Rooted Trees Including k Leaves

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IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS 2012年第3期E95D卷 763-768页

作者： Ishikawa, Masanobu Yamanaka, Katsuhisa Otachi, Yota Nakano, Shin-ichi Gunma Univ Dept Comp Sci Kiryu Gunma 3768515 Japan Iwate Univ Dept Elect Engn & Comp Sci Morioka Iwate 0208551 Japan Tohoku Univ Grad Sch Informat Sci Sendai Miyagi 9808579 Japan

This paper presents an efficient algorithm to generate all (unordered) rooted trees with exactly vertices including exactly k leaves. There are known results on efficient enumerations of some classes of graphs embedded on a plane, for instance, biconnected and triconnected triangulations [3], [6], and floorplans [4]. On the other hand, it is difficult to enumerate a class of graphs without a fixed embedding. The paper is on enumeration of rooted trees without a fixed embedding. We already proposed an algorithm to generate all "ordered" trees with 17 vertices including k leaves [11], while the algorithm cannot seem to efficiently generate all (unordered) rooted trees with it vertices including k leaves. We design a simple tree structure among such trees, then by traversing the tree structure we generate all such trees in constant time per tree in the worst case. By repeatedly applying the algorithm for each k = 1, 2, . . . , n - 1, we can also generate all rooted trees with exactly n vertices.

关键词： graph algorithm enumeration rooted tree family tree

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Efficient semi-external depth-first search

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INFORMATION SCIENCES 2022年 599卷 170-191页

作者： Wan, Xiaolong Wang, Hongzhi Harbin Inst Technol Sch Comp Sci & Technol Harbin Peoples R China

As graphs grow in size, many real-world graphs are difficult to load into the primary memory of a computer. Thus, computing depth-first search (DFS) results (i.e., depth-first order or DFS-Tree) on the semi-external memory model is important to investigate. Semi external algorithms assume that the primary memory can at least hold a spanning tree T of a graph G and gradually restructure T into a DFS-Tree, which is nontrivial. In this paper, we present a comprehensive study for the semi-external DFS problem. Based on a theoretical analysis of this problem, we introduce a new semi-external DFS algorithm called EPDFS with a lightweight index N thorn -index. Unlike traditional algorithms, we focus on addressing such a complex problem efficiently with fewer I/Os, simpler CPU calculations (implementation-friendly), and less random I/O accesses (key-to-efficiency). Extensive experimental evaluations are performed on both synthetic and real graphs, and experimental results confirm that the proposed EP-DFS algorithm markedly outperforms existing algorithms. (c) 2022 Elsevier Inc. All rights reserved.

关键词： Depth-first search Semi-external memory graph algorithm

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Complexity and algorithms for Semipaired Domination in graphs

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THEORY OF COMPUTING SYSTEMS 2020年第7期64卷 1225-1241页

作者： Henning, Michael A. Pandey, Arti Tripathi, Vikash Univ Johannesburg Dept Math & Appl Math ZA-2006 Auckland Pk South Africa Indian Inst Technol Ropar Dept Math Nangal Rd Rupnagar 140001 Punjab India

For a graph G = (V, E) with no isolated vertices, a set D subset of V is called a semipaired dominating set of G if (i) D is a dominating set of G, and (ii) D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted by gamma(pr2)(G). TheMINIMUM SEMIPAIRED DOMINATION problem is to find a semipaired dominating set of G of cardinality gamma(pr2)(G). In this paper, we initiate the algorithmic study of the MINIMUM SEMIPAIRED DOMINATION problem. We show that the decision version of the MINIMUM SEMIPAIRED DOMINATION problem is NP-complete for bipartite graphs and chordal graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs. We also propose a 1 + ln(2 Delta + 2)-approximation algorithm for the MINIMUM SEMIPAIRED DOMINATION problem, where Delta denotes the maximum degree of the graph and show that the MINIMUM SEMIPAIRED DOMINATION problem cannot be approximated within (1 - epsilon) ln vertical bar V vertical bar for any epsilon > 0 unless P=NP.

关键词： Domination Semipaired domination Interval graphs graph algorithm NP-complete Approximation algorithm

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algorithms and obstructions for linear-width and related search parameters

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DISCRETE APPLIED MATHEMATICS 2000年第1-3期105卷 239-271页

作者： Thilikos, DM Univ Politecn Catalunya Dept Llenguatges & Sistemes Informat E-08034 Barcelona Spain

The linear-width of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e(1),...,e(r)) in such a way that for every i = 1,..., r - 1, there are at most k vertices incident to edges that belong both to {e(1),...,e(i)} and to {e(i+1),...,e(r)}. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden miners for the class of graphs with linear-width at most two. Our proof also gives a linear time algorithm that either reports that a given graph has linear-width more than two or outputs an edge ordering of minimum linear-width. We further prove a structural connection between linear-width and the mixed search number which enables us to determine, for any k greater than or equal to 1, the set of acyclic forbidden miners for the class of graphs with linear-width less than or equal to k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixed search or edge search number at most two and, if so, construct the corresponding sequences of search moves. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： graph algorithm linear width graph minor obstruction set graph searching

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Efficient community detection with additive constrains on large networks

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KNOWLEDGE-BASED SYSTEMS 2013年 52卷 268-278页

作者： Li, Yakun Wang, Hongzhi Li, Jianzhong Gao, Hong Harbin Inst Technol Dept Comp Sci & Technol Harbin 150001 Peoples R China

The community structure is one of the most important patterns in network. Since finding the communities in the network can significantly improve our understanding of the complex relations, lots of work has been done in recent years. Yet it still lies vacant on the exact definition and practical algorithms for community detection. This paper proposes a novel definition for community which overcomes the drawbacks of existing methods. With the new definition, efficient community detection algorithms are developed, which take advantage of additive topological and other constrains to discover communities in arbitrary shape based on the feedback. The algorithm has a linear run time with the size of graph. Experimental results demonstrate that the community definition in this paper is effective and the algorithm is scalable for large graphs. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Community detection Feedback control Additive constrains Large networks Scale-free network graph algorithm

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AN O(LOG N) algorithm FOR PARALLEL UPDATE OF MINIMUM SPANNING-TREES

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INFORMATION PROCESSING LETTERS 1986年第5期22卷 223-229页

作者： PAWAGI, S RAMAKRISHNAN, IV Department of Computer Science University of Maryland College Park MD 20742 U.S.A.

Incremental graph algorithms deal with recomputing properties of a graph after an incremental change is made to that graph, such as adding and deleting vertices and edges. Such recomputations are ''updating'' graph properties. Efficient parallel algorithms are presented for edge and vertex insertion updating in a minimum spanning tree (MST) due to changes in edge costs or in vertex insertion, when a new node is inserted in the underlying graph. The algorithms are derived from a computational model of an unbounded parallel random access machine where simultaneous reads, but not simultaneous writes, are allowed into the same memory location. They are shown to be more efficient than previously proposed algorithms for the MST updating problem, requiring only O(log n) time and certain processors. Main features of the algorithms are that they: 1. solve MST updating problems based on the use of an inverted tree, and 2. exploit a certain MST property that allows novel solution for vertex updating.

关键词： Spanning tree graph algorithm

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Robustness of First- and Second-Order Consensus algorithms for a Noisy Scale-Free Small-World Koch Network

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2017年第1期25卷 342-350页

作者： Yi, Yuhao Zhang, Zhongzhi Shan, Liren Chen, Guanrong Fudan Univ Sch Comp Sci Shanghai Key Lab Intelligent Informat Proc Shanghai 200433 Peoples R China City Univ Hong Kong Dept Elect Engn Hong Kong Hong Kong Peoples R China

In this brief, we study first- and second-order consensus algorithms for the scale-free small-world Koch network, where vertices are subject to white noise. We focus on three cases of consensus schemes: 1) first-order leaderless algorithm;2) first-order algorithm with a single leader;and 3) second-order leaderless algorithm. We are concerned with the coherence of the Koch network in the H-2 norm, which captures the level of agreement of vertices in face of stochastic disturbances. Based on the particular network construction, we derive explicit expressions of the coherence for all the three consensus algorithms, as well as their dependence on the network size. Particularly, for the first-order leader-follower model, we show that coherence relies on the shortest-path distance between the leader and the largest-degree vertices, as well as the degree of the leader. The asymptotic behaviors for coherence of the three consensus algorithms in Koch network behave differently from those associated with other networks lacking scale-free small-world features, indicating significant influences of the scale-free small-world topology on the performance of the consensus algorithms in noisy environments.

关键词： Distributed average consensus graph algorithm noise scale-free network small-world network

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