We find the minimal cutwidth and bisection width values for abelian Cayley graphs with up to 4 generators and present an algorithm for finding the corresponding optimal ordering. We also find minimal cuts of each orde...
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We find the minimal cutwidth and bisection width values for abelian Cayley graphs with up to 4 generators and present an algorithm for finding the corresponding optimal ordering. We also find minimal cuts of each order. (C) 2007 Elsevier B.V. All rights reserved.
Obstacle-avoiding Steiner routing has arisen as a fundamental problem in the physical design of modern VLSI chips. In this paper, we present EBOARST, an efficient four-step algorithm to construct a rectilinear obstacl...
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Obstacle-avoiding Steiner routing has arisen as a fundamental problem in the physical design of modern VLSI chips. In this paper, we present EBOARST, an efficient four-step algorithm to construct a rectilinear obstacle-avoiding Steiner tree for a given set of pins and a given set of rectilinear obstacles. Our contributions are fourfold. First, we propose a novel algorithm, which generates sparse obstacle-avoiding spanning graphs efficiently. Second, we present a fast algorithm for the minimum terminal spanning tree construction step, which dominates the running time of several existing approaches. Third, we present an edge-based heuristic, which enables us to perform both local and global refinements, leading to Steiner trees with small lengths. Finally, we discuss a refinement technique called segment translation to further enhance the quality of the trees. The time complexity of our algorithm is O(n log n). Experimental results on various benchmarks show that our algorithm achieves 16.56 times speedup on average, while the average length of the resulting obstacle-avoiding rectilinear Steiner trees is only 0.46% larger than the best existing solution.
Obstacle-avoiding Steiner routing has arisen as a fundamental problem in the physical design of modern VLSI chips. In this paper, we present EBOARST, an efficient four-step algorithm to construct a rectilinear obstacl...
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Obstacle-avoiding Steiner routing has arisen as a fundamental problem in the physical design of modern VLSI chips. In this paper, we present EBOARST, an efficient four-step algorithm to construct a rectilinear obstacle-avoiding Steiner tree for a given set of pins and a given set of rectilinear obstacles. Our contributions are fourfold. First, we propose a novel algorithm, which generates sparse obstacle-avoiding spanning graphs efficiently. Second, we present a fast algorithm for the minimum terminal spanning tree construction step, which dominates the running time of several existing approaches. Third, we present an edge-based heuristic, which enables us to perform both local and global refinements, leading to Steiner trees with small lengths. Finally, we discuss a refinement technique called segment translation to further enhance the quality of the trees. The time complexity of our algorithm is O(n log n). Experimental results on various benchmarks show that our algorithm achieves 16.56 times speedup on average, while the average length of the resulting obstacle-avoiding rectilinear Steiner trees is only 0.46% larger than the best existing solution.
Obtaining a matching in a graph satisfying a certain objective is an important class of graph problems. Matching algorithms have received attention for several decades. However, while there are efficient algorithms to...
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Obtaining a matching in a graph satisfying a certain objective is an important class of graph problems. Matching algorithms have received attention for several decades. However, while there are efficient algorithms to obtain a maximum weight matching, not much is known about the maximum weight maximum cardinality, and maximum cardinality maximum weight matching problems for general graphs. Our contribution in this work is to show that for bounded weight input graphs one can obtain an algorithm for both maximum weight maximum cardinality (for real weights), and maximum cardinality maximum weight matching (for integer weights) by modifying the input and running the existing maximum weight matching algorithm. Also, given the current state of the art in maximum weight matching algorithms, we show that, for bounded weight input graphs, both maximum weight maximum cardinality, and maximum cardinality maximum weight matching have algorithms of similar complexities to that of maximum weight matching. Subsequently, we also obtain approximation algorithms for maximum weight maximum cardinality, and maximum cardinality maximum weight matching.
Given a positive integer k and an edge-weighted undirected graph G = (V, E;w), the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected comp...
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ISBN:
(纸本)9783540921813
Given a positive integer k and an edge-weighted undirected graph G = (V, E;w), the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. This problem is a natural generalization of the classical minimum cut problem and has been well-studied in the literature. A simple and natural method to solve the minimum k-way cut problem is the divide-and-conquer method: getting a minimum k-way cut by properly separating the graph into two small graphs and then finding minimum k'-way cut and k ''-way cut respectively in the two small graphs, where k' + k '' = k. In this paper, we present the first algorithm for the tight case of k' = [k/2]. Our algorithm runs in O(n(4k-lgk)) time and can enumerate all minimum k-way cuts, which improves all the previously known divide-and-conquer algorithms for this problem.
Given a graph G = (V, E) with n vertices and m edges, and a subset T of 1 vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of k edges (non-terminal vertices), whose...
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ISBN:
(纸本)9783540797081
Given a graph G = (V, E) with n vertices and m edges, and a subset T of 1 vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of k edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for l >= 3 but well-known to be polynomial-time solvable for 1 = 2 by the flow technique. In this paper, we show that Edge Multiterminal Cut is polynomial-time solvable for k = O(log n) by presenting an O(2(k)lT(n, m)) algorithm, where T(n, m) O(min(n(2/3), m(1/2))m) is the running time of finding a minimum (s,t) cut in an unweighted graph. We also give two algorithms for Vertex Multiterminal Cut that run in O(l(k)T(n, m)) time and O((k!)T-2(n, m)) time respectively. The former one indicates that Vertex Multiterminal Cut is solvable in polynomial time for I being a constant and k = O(log n), and the latter one improves the best known algorithm of running time O(4(k3) n(O(1))). When l = 3, we show that the running times can be improved to O(1.415(k) T(n, m)) for Edge Multiterminal Cut and O(2.059(k)T(n, m)) for Vertex Multiterminal. Cut. Furthermore, we present a simple idea to solve another important problem Multicut by finding minimum multiterminal. cuts. Our algorithms for Multicuts are also faster than the previously best algorithm. Based on a notion farthest minimum isolating cut, we present some properties for Multiterminal Cuts, which help shed light on the structure of optimal cut problems, and enables us to design efficient algorithms for Multiterminal Cuts, as well as some other related cut problems.
Recent researches show that the benefits of image segmentation have been exploited in object categorization and recognition approaches. In most of these works, objects are segmented from the background around to incre...
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ISBN:
(纸本)9781424420209
Recent researches show that the benefits of image segmentation have been exploited in object categorization and recognition approaches. In most of these works, objects are segmented from the background around to increase recognition accuracy. However, it is generally hard to find a segmentation that captures all correct object boundaries in images of real world scene. So some researches begin to choose several segmentations for representing the objects and performing object categorization. In this paper, we take advantage of an efficient graph-based algorithm for image segmentation, and combine a visual attention model to locate the salient and effective segmentations in a real world image. We propose a model which extends the Bag-of-features method for modeling the semantic objects. We evaluate our approach on two experiments: multiclass categorization in Caltech 101 datasets and high-level features extraction in video datasets of TRECVID2007. The results show that combining segmentation and visual attention makes our model achieve competitive performance.
In many applications it is an important algorithmic task to find a densest subgraph in an input graph. The complexity of this task depends on how density is defined. If density means the ratio of the number of edges a...
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In many applications it is an important algorithmic task to find a densest subgraph in an input graph. The complexity of this task depends on how density is defined. If density means the ratio of the number of edges and the number of vertices in the subgraph, then the algorithmic problem has long been known efficiently solvable. On the other hand, the task becomes NP-hard with closely related but somewhat modified concepts of density. To capture many possible tractable density concepts of interest in a common model, we define and analyze a general concept of density, called F-density. Here F is a family of graphs and we are looking for a subgraph of the input graph, such that this subgraph is the densest in terms of containing the highest number of graphs from F relative to the size of the subgraph. We show that for any fixed finite family F, a subgraph of maximum F-density can be found in polynomial time. As our main tool we develop an algorithm, that may be of independent interest, which can find an independent set of maximum independence ratio in a certain class of weighted graphs. The independence ratio is the weight of the independent set divided by the weight of its neighborhood.
Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i-th class have pairwise distance greater than i. We solve an open problem of Goddard et al. and show that the dec...
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ISBN:
(纸本)9783540922476
Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i-th class have pairwise distance greater than i. We solve an open problem of Goddard et al. and show that the decision whether a tree allows a packing coloring with at most k classes is NP-complete. We accompany this NP-hardness result by a polynomial time algorithm for trees for closely related variant of the packing coloring problem where the lower bounds on the distances between vertices inside color classes are determined by an infinite nondecreasing sequence of bounded integers.
Two graph algorithms which derive from bread-first search were implemented by using C language in this paper. One algorithm is a replacement method for finding out a graph's all spanning tree, the other is the Pat...
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Two graph algorithms which derive from bread-first search were implemented by using C language in this paper. One algorithm is a replacement method for finding out a graph's all spanning tree, the other is the Paton algorithm for finding out all essential circuit of a graph. This paper mainly introduced the flow of the two graph algorithms. An illustrative example was presented. Also, a practical operational platform of graph algorithm was designed and realized in this paper.
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