Finding the ℓ-core of a tree

Becker, Ronald I.; Chang, Yen I.; Lari, Isabella; Scozzari, Andrea; Storchi, Giovanni

doi:10.1016/s0166-218x(01)00254-2

Cited by 23 publications

(21 citation statements)

References 16 publications

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“…Notice that if k ϭ 2 the problem reduces to finding an l-core of T, that is, a path P in T that minimizes DISTSUM, with length at most l [2].…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…As in the papers of Peng and Lo [9] and Becker et al [2], we use Remark 1 in a recursive algorithm that, given a "central" vertex v, finds a (k, l )-core containing v in T v . If it is not the (k, l )-core of the tree T, then the (k, l )-core must lie entirely in one of the subtrees rooted at the adjacent vertices of the "central" vertex.…”

Section: Definition 3 Given a Weighted Tree T A Central Vertex V Ofmentioning

confidence: 99%

“…Linear time algorithms for finding a core of a tree were presented in Morgan and Slater [8] and in Becker [1]. A number of authors later extended the problem to finding a core of specified size l on tree networks [2,7,9].…”

Section: Introductionmentioning

confidence: 99%

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Efficient algorithms for finding the (k, l)‐core of tree networks

et al. 2002

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Given a tree T ‫؍‬ (V, E), with ͦVͦ ‫؍‬ n, we consider the problem of selecting a subtree with at most k leaves and with a diameter of at most l which minimizes the sum of the distances of the vertices from the selected subtree. We call such a subtree the (k, l)-core of T. We provide two algorithms; the first one for unweighted trees has time complexity of O(n 2 ), whereas the second one for weighted trees has time complexity of O(n 2 log n). The idea for both the algorithms is that, by starting from the tree T, we construct new rooted trees where the maximum length of a path is at most l. Then, for each new tree, we can apply a greedy-type procedure to find a subtree containing the root with at most k leaves and which minimizes the sum of the distances.

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“…Notice that if k ϭ 2 the problem reduces to finding an l-core of T, that is, a path P in T that minimizes DISTSUM, with length at most l [2].…”

Section: Notation and Definitionsmentioning

confidence: 99%

Section: Definition 3 Given a Weighted Tree T A Central Vertex V Ofmentioning

confidence: 99%

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Efficient algorithms for finding the (k, l)‐core of tree networks

et al. 2002

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“…While there have been other many other generalizations of the center and centroid (some of which we discuss in Section 5), none directly generalize central paths or spines. There have been some very interesting generalizations of a core [19,20,23,24,1]. In particular, the k-tree core is similar to our T D -median, except that instead of bounding the maximum degree, this concept considers subtrees with at most k leaves.…”

Section: Introductionmentioning

confidence: 99%

Generalized Centrality in Trees

Mulder¹,

Pelsmajer²,

Reid³

2010

Advances in Interdisciplinary Applied Discrete Mathematics

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In 1982, Slater defined path subgraph analogues to the center, median, and (branch or branchweight) centroid of a tree. We define three families of central substructures of trees, including three types of central subtrees of degree at most D that yield the center, median, and centroid for D = 0 and Slater's path analogues for D = 2. We generalize these results concerning paths and include proofs that each type of generalized center and generalized centroid is unique. We also present algorithms for finding one or all generalized central substructures of each type.

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“…Also, when we specify db(v) = 2 for all v ∈ V (T ), the output will be a path. Problems of the type OptSubtree are of interest in facilities planning and location (see for example, [1,4]). …”

Section: Introductionmentioning

confidence: 99%