Most uniform path partitioning and its use in image processing

Lucertini, M.; Perl, Yehoshua; Simeone, Bruno

doi:10.1016/0166-218x(93)90048-s

Cited by 54 publications

(16 citation statements)

References 10 publications

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“…Therefore, it is very unlikely that the three partition problems can be solved in polynomial time even for series-parallel graphs, although the three problems can be solved in pseudo-polynomial time for graphs of bounded tree-width, including series-parallel graphs [8]. On the other hand, all the three partition problems can be solved in linear time for paths [10], and a related minimization problem can be solved in polynomial time for special classes of trees such as stars, worms, caterpillars and so on [4]. In this paper, we present the first polynomial-time algorithm to solve the three partition problems for arbitrary trees.…”

Section: Introductionmentioning

confidence: 96%

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Partitioning a Weighted Tree into Subtrees with Weights in a Given Range

et al. 2011

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Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are given integers such that 0 ≤ l ≤ u. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such a partition is called an (l, u)-partition. We deal with three problems to find an (l, u)-partition of a given graph: the minimum partition problem is to find an (l, u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l, u)-partition with a given number p of components. All these problems are NP-hard even for series-parallel graphs, but are solvable in linear time for paths. In this paper, we present the first polynomial-time algorithm to solve the three problems for arbitrary trees.

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Section: Introductionmentioning

confidence: 96%

“…1(b) is a solution for the 6-partition problem and for the maximum partition problem. The three partition problems often appear in many practical situations such as load balancing [1], image processing [6,10], paging systems of operation systems [12], and political districting [2,13].…”

Section: Introductionmentioning

confidence: 99%

Partitioning a Weighted Tree into Subtrees with Weights in a Given Range

et al. 2011

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“…An open problem in that paper is the most uniform vertex partitioning problem for trees, in which the objective is to minimize the difference between the maximum and the minimum weights of the vertex set in the partition. For a special case that the tree is a path, a solution was given in [13]. One can image that the problem is more difficult than the min-max or max-min problem since both the smallest and the largest parts are concerned.…”

Section: Introductionmentioning

confidence: 98%

On the uniform edge-partition of a tree

Wang

Kuan

et al. 2007

Discrete Applied Mathematics

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We study the problem of uniformly partitioning the edge set of a tree with n edges into k connected components, where k n. The objective is to minimize the ratio of the maximum to the minimum number of edges of the subgraphs in the partition. We show that, for any tree and k 4, there exists a k-split with ratio at most two. For general k, we propose a simple algorithm that finds a k-split with ratio at most three in O(n log k) time. Experimental results on random trees are also shown.

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“…The three partition problems often appear in many practical situations such as with image processing [5,8], paging systems of operation systems [11], and political districting [3,12]. Consider, for example, political districting.…”

Section: Introductionmentioning

confidence: 99%

Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size

Ito

Zhou

Nishizeki

2006

Journal of Discrete Algorithms

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Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an "almost uniform" partition is called an (l, u)-partition. We deal with three problems to find an (l, u)-partition of a given graph; the minimum partition problem is to find an (l, u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l, u)-partition with a fixed number p of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time O(u 4 n) and the p-partition problem can be solved in time O(p 2 u 4 n) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.

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Most uniform path partitioning and its use in image processing

Cited by 54 publications

References 10 publications

Partitioning a Weighted Tree into Subtrees with Weights in a Given Range

Partitioning a Weighted Tree into Subtrees with Weights in a Given Range

On the uniform edge-partition of a tree

Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size

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