Efficient algorithms for a constrained k-tree core problem in a tree network

Wang, Biing-Feng; Peng, Shietung; Yu, Hong-Yi; Ku, Shan-Chyun

doi:10.1016/j.jalgor.2004.12.002

Cited by 8 publications

(12 citation statements)

References 26 publications

(34 reference statements)

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“…Can one design more efficient parallel algorithms for them? An O(pn 3 )-time algorithm was described in [19] for the p-median problem with distance constraint on trees for general p.…”

Section: Future Workmentioning

confidence: 99%

“…They gave an O(n 2 )-time algorithm for the case of p = 2. Recently, Wang et al [19] presented an O(pn 3 )-time algorithm for general p. Tamir et al [17] introduced another generalization of the p-median problem which limits the eccentricity of p-medians and gave an O(pn 2 )-time algorithm for it. The eccentricity of a subset S ⊆ V is defined as Ecc G (S) = max v∈V d (v, S).…”

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confidence: 99%

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Efficient algorithms for two generalized 2-median problems and the group median problem on trees

Chan

et al. 2009

Theoretical Computer Science

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a b s t r a c tThe p-median problem on a tree T is to find a set S of p vertices on T that minimizes the sum of distances from T 's vertices to S. In this paper, we study two generalizations of the 2-median problem, which are obtained by imposing constraints on the two vertices selected as a 2-median: one is to limit their distance while the other is to limit their eccentricity. Previously, both the best upper bounds of these two generalizations were O(n 2 ) [A. Tamir, D. Perez-Brito, J.A. Moreno-Perez, A polynomial algorithm for the p-centdian problem on a tree, Networks 32 (1998) 255-262; B.-F. Wang, S.-C. Ku, K.-H. Shi, Cost-optimal parallel algorithms for the tree bisector problem and applications, IEEE Transactions on Parallel and Distributed Systems 12 (9) (2001) 888-898].In this paper, we solve both in O(n log n) time. We also study cases when linear time algorithms exist for the two generalizations. For example, we solve both in linear time when edge lengths and vertex weights are all polynomially bounded integers. Furthermore, we consider the relaxation of the two generalized problems by allowing 2-medians on any position of edges, instead of just on vertices, and we give O(n log n)-time algorithms for them. A problem, named the tree marker problem, arises several times in our approaches to the two generalized 2-median problems, and we give an O(n log n)-time algorithm for this problem. We also use this algorithm to speedup an algorithm of Gupta and Punnen [S.K. Gupta, A.P. Punnen, Group center and group median of a tree, European Journal of Operational Research 65 (1993) 400-406] for the group median problem, improving the running time from O(kn)to O(n + k log n), where k is the number of groups in the input.

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“…Can one design more efficient parallel algorithms for them? An O(pn 3 )-time algorithm was described in [19] for the p-median problem with distance constraint on trees for general p.…”

Section: Future Workmentioning

confidence: 99%

mentioning

confidence: 99%

Efficient algorithms for two generalized 2-median problems and the group median problem on trees

Chan

et al. 2009

Theoretical Computer Science

Self Cite

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“…Finally, we compute a (k, l)-coredian tree of T from a set of the k-coredian subtrees obtained in the last step. The authors in [10] proved that the upper bound of S is n, where n is the number of nodes in the tree network. They also have shown that a set S can be constructed in O(n 2 ).…”

Section: Constructing the (K L) Coredian Treementioning

confidence: 99%

“…Jennings [6] presents distributed algorithms for finding center path and core path in asynchronous networks in O( ) time with O(n) messages, where is the diameter of the network. Additional results on center/core paths can be found in [6,[8][9][10][11][23][24][25].…”

Section: Introductionmentioning

confidence: 96%

“…Becker et al [11] dealt with the (k, l)-core tree problem and presented an O(n 2 log n) time algorithm for arbitrary edge lengths and an O(n 2 ) time algorithm for equal edge lengths. Recently, Wang et al [10] presented two algorithms for constructing a (k, l)-core tree. The first algorithm has O(n 2 ) time complexity for the case in which each edge has an arbitrary length.…”

Section: Introductionmentioning

confidence: 99%

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The (k,l) Coredian Tree for Ad Hoc Networks

Dvir

Segal

2008

2008 the 28th International Conference on Distributed Computing Systems Workshops

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In this paper, we present new efficient strategy for constructing a wireless tree network containing n nodes of diameter ∆ while satisfying the QoS requirements such as bandwidth and delay. Given a tree network T , a coredian path is a path in T that minimizes the centdian function, a k-coredian tree is a subtree of T with k leaves that minimizes the centdian function, and a (k, l)-coredian tree is a subtree of T with k leaves and diameter l at most that minimizes the centdian function. The (k, l)-coredian tree can serve as a backbone for a network, where intermediate nodes belong to the backbone and the leaves serve as the heads of the clusters covering the rest of the network. We show that a coredian path can be constructed at O(∆) time with O(n) messages and a k-coredian tree can be constructed at O(k∆) time with O(kn) messages. While a (k, l)-coredian tree can be constructed at O(n 2 ) time with O(n 2 ) messages. A simulation is presented for various values of n and k. 1

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