Jan-Ming Ho, D. T. Lee, Chia-Hsiang Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM Journal on Computing, 20(5):987--997, October 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....insertion sequence is actually o line) we immediately obtain an improved time bound of e O(n 3 1=6 ) In any constant dimension d, the bound is e O(n ) The minimum diameter spanning tree. An interesting application of the generalized discrete 2 center problem was considered by Ho et al. [20]: given an n point set P IR , nd a spanning tree that minimizes its diameter (i.e. the maximum distance over all pairs of points, where the distance between p and q refers to the sum of the edge lengths, measured in the Euclidean metric, along the path connecting p and q in the tree) As ....

J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong, Minimum diameter spanning trees and related problems, SIAM J. Comput., 20:987-997, 1991.

....graph G = V, E) a degree bound d max (v) N for each vertex v V and a cost c(e) for each edge e E; find a spanning tree T of G of minimum diameter, subject to the constraint that d T (v) dmax (v) for all v T . Much previous research has been done on related problems. In [13], Ho et al. proved that in geometric space, there exists a minimum diameter spanning tree in which there are at most two interior points (non leaf nodes) and the optimal tree can be found in O(n ) time. Hassin and Tamir established in [12] that for a general graph, a minimum diameter spanning ....

....Hassin and Tamir established in [12] that for a general graph, a minimum diameter spanning tree problem is identical to the absolute 1 center problem introduced by Hakimi [11] and as such, a solution can be found in O(mn n logn) where n is the number of nodes and m the number of edges. In [13] and [21] respectively, they prove that minimum diameter, minimum spanning tree and minimum maximum degree, minimum spanning tree are both NP complete. Theorem 1: The decision version of MDDL finding a spanning tree with diameter bound B and a degree constraint dmax (v) for each node, is ....

J. Ho, D. Lee, C. Chang, and C. Wong. Minimum Diameter Spanning Trees and Related Problems. SIAM J.Comput., 20(5):987--997, 1991.

....a network without cycles that minimizes the maximum travel time between any two sites connected by the network. This is of importance e.g. in communication systems where the maximum delay in delivering a message is to be minimized. Ho et al. showed that there always is a mono or a dipolar MDST [HLCW91] For a di#erent proof, see [HT95] Ho et al. also gave an O(n log n) time algorithm for the monopolar and an O(n ) time algorithm for the dipolar case [HLCW91] In addition, they showed that the problem becomes considerably easier when allowing Steiner points, i.e. to find a spanning tree ....

....where the maximum delay in delivering a message is to be minimized. Ho et al. showed that there always is a mono or a dipolar MDST [HLCW91] For a di#erent proof, see [HT95] Ho et al. also gave an O(n log n) time algorithm for the monopolar and an O(n ) time algorithm for the dipolar case [HLCW91] In addition, they showed that the problem becomes considerably easier when allowing Steiner points, i.e. to find a spanning tree with minimum diameter over all point sets P # that contain the input point set P . The reason is that there always is a minimum diameter Steiner tree that is ....

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Jan-Ming Ho, D. T. Lee, Chia-Hsiang Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM Journal on Computing, 20(5):987-- 997, October 1991.

....minimum spanning tree for the points pi. We prove that choosing the centers of the regions as network bases results in a cost at most 3x the optimal. The minimum diameter spanning tree problem (MDSTP) for k disjoint convex regions is a generalization of the minimum diameter spanning tree problem [4] for points. The task is to construct a spanning tree of regions: a node of the tree corresponds to a region, and each edge in the tree connects two regions. The addition of these bridges turns the union of regions into a simply connected set. We wish to choose the bridges such that the diameter ....

....We have ]IN(X1,X2,X3) 3x e, IN (C1 , C2, C3) 2x 3 e, and so the approximation factor is at least 1.52. 6 The minimum diameter spanning tree problem The minimum diameter spanning tree problem (MDSTP) is a generalization of the problem for points considered by Hoet al... [4]. It is defined as follows. Problem MDSTP: Given disjoint compact convex regions Rt, t = 1, k, find a set k S of k I bridges connecting pairs of regions such that U(S) S U Ut= Rt is simply connected and such that 1 Is(S) max Is(P,q)l, p,qCU(S) is minimized, where s (P, q) is the ....

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J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM J. Cornput., 20:987-997, 1991.

....2 V ; a cost c(e) 2 Z for each edge e 2 E. Find a spanning tree T of G such that for each v 2 T , degree of v satis es dT (v) dmax(v) and the diameter of T dia(T ) which is the cost of the longest simple path in T , is minimized. Much previous research has been done on related problems. In [11], Ho et al. proved that in geometric space, there exists a minimum diameter spanning tree in which there are at most two interior points (non leaf nodes) and the optimal tree can be found in O(n ) time. Hassin and Tamir established in [10] that for a general graph, a minimum diameter spanning ....

....Hassin and Tamir established in [10] that for a general graph, a minimum diameter spanning tree problem is identical to the absolute 1 center problem introduced by Hakimi [9] and as such, a solution can be found in O(mn n logn) where n is the number of nodes and m the number of edges. In [11] and [15] respectively, they prove that minimum diameter, minimum spanning tree and minimum maximum degree, minimum spanning tree are both NP complete. Theorem 1. The decision version of mddbst nding a spanning tree with diameter bound B and a degree constraint dmax(v) for each node, is ....

J. Ho, D. Lee, C. Chang, and C. Wong. Minimum Diameter Spanning Trees and Related Problems. SIAM J.Comput., 20(5):987-997, 1991.

....MDDL and LDRB problems are NP complete, since they contain the special case in which every node has a degree of two, corresponding to the optimization and the decision version of the Traveling Salesman Problem (TSP) 26] respectively. Previous research has studied several related problems. In [32], Ho et al. proved that in geometric space, there exists a minimum diameter spanning tree in which there are at most two interior points (non leaf nodes) and the optimal tree can be found in O(n ) time. Hassin and Tamir established in [31] that for a general graph, a minimum diameter spanning ....

....Hassin and Tamir established in [31] that for a general graph, a minimum diameter spanning tree problem is identical to the absolute 1 center problem introduced by Hakimi [29] and as such, a solution can be found in O(mn n log n) where n is the number of nodes and m the number of edges. In [32] and [65] respectively, they prove that minimum diameter, minimum spanning tree and minimum maximum degree, minimum spanning tree are both NP complete. Although the special case of MDDL when all nodes have degree of two is hard to solve, the problem might conceivably be less difficult when nodes ....

J. Ho, D. Lee, C. Chang, and C. Wong. Minimum Diameter Spanning Trees and Related Problems. SIAM J.Comput., 20(5):987--997, 1991.

....as a network without cycles that minimizes the maximum travel time between any two sites connected by the network. This is of importance e.g. in communication systems where the maximum delay in delivering a message is to be minimized. Ho et al. showed that there always is a mono or a dipolar MDST [10]. For a di#erent proof, see [9] Ho et al. gave an O(n log n) time algorithm for the monopolar and an O(n ) time algorithm for the dipolar case [10] In the dipolar case the objective is to find the two roots x, y P of the tree such that the function r x r y is minimized, where is ....

....systems where the maximum delay in delivering a message is to be minimized. Ho et al. showed that there always is a mono or a dipolar MDST [10] For a di#erent proof, see [9] Ho et al. gave an O(n log n) time algorithm for the monopolar and an O(n ) time algorithm for the dipolar case [10]. In the dipolar case the objective is to find the two roots x, y P of the tree such that the function r x r y is minimized, where is the Euclidean distance of x and y, and r x and r y are the radii of two disks centered at x and y that cover P . The discrete k center problem is to ....

[Article contains additional citation context not shown here]

Jan-Ming Ho, D. T. Lee, Chia-Hsiang Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM Journal on Computing, 20(5):987--997, October 1991.

....spanning tree for the points p i . We prove that choosing the centers of the regions as network bases results in a cost at most 3 p 2 the optimal. The minimum diameter spanning tree problem (MDSTP) for k disjoint convex regions is a generalization of the minimum diameter spanning tree problem [4] for points. The task is to construct a spanning tree of regions : a node of the tree corresponds to a region, and each edge in the tree connects two regions. The addition of these bridges turns the union of regions into a simply connected set. We wish to choose the bridges such that the ....

.... N (x 1 ; x 2 ; x 3 ) 3 p 2 ; N (c 1 ; c 2 ; c 3 ) 2 p 3 3 ; and so the approximation factor is at least 1:52. 6 The minimum diameter spanning tree problem The minimum diameter spanning tree problem (MDSTP) is a generalization of the problem for points considered by Ho et al. [4]. It is de ned as follows. Problem MDSTP: Given disjoint compact convex regions R i , i = 1; k, nd a set S of k 1 bridges connecting pairs of regions such that U(S) S [ S k i=1 R i is simply connected and such that S (S) max p;q2U(S) j S (p; q)j; is minimized, where S (p; ....

[Article contains additional citation context not shown here]

J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM J. Comput., 20:987-997, 1991.

....cost c(e) # Z for each edge e # E. Find a spanning tree T of G such that for each v # T , degree of v satisfies dT (v) # dmax(v) and the diameter of T dia(T ) which is the cost of the longest simple path in T , is minimized. Much previous research has been done on related problems. In [11], Ho et al. proved that in geometric space, there exists a minimum diameter spanning tree in which there are at most two interior points (non leaf nodes) and the optimal tree can be found in O(n 3 ) time. Hassin and Tamir established in [10] that for a general graph, a minimum diameter spanning ....

....Hassin and Tamir established in [10] that for a general graph, a minimum diameter spanning tree problem is identical to the absolute 1 center problem introduced by Hakimi [9] and as such, a solution can be found in O(mn n 2 logn) where n is the number of nodes and m the number of edges. In [11] and [15] respectively, they prove that minimum diameter, minimum spanning tree and minimum maximum degree, minimum spanning tree are both NP complete. Theorem 1. The decision version of mddbst finding a spanning tree with diameter bound B and a degree constraint dmax(v) for each node, is ....

J. Ho, D. Lee, C. Chang, and C. Wong. Minimum Diameter Spanning Trees and Related Problems. SIAM J.Comput., 20(5):987--997, 1991.

....regions in the Euclidean space, and are interested in finding a spanning tree connecting these regions so that the maximum length of shortest path between two points in S k i=1 R i is minimized. The minimum diameter spanning tree problem(MDSTP) is formally defined as follows. Refer to the paper [2] on minimum diameter spanning trees for the points. Problem (MDSTP) Given a set S of regions R i for i = 1; k, find a spanning tree T D for S such that max shortest path p2F jpj is minimized, where F = S k i=1 R i ) T D . In this problem, a node in the tree corresponds to a ....

J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM J. Comput., 20:987--997, 1991.

....of the tree. When all path weights are zero, this problem reduces to the ordinary Steiner tree problem and for large path weights the problem reduces to the rectilinear Steiner arborescence problem. It should be noted that similar problems in which the diameter (i.e. longest path) of the tree [11] or the longest segment of the tree is minimized [3, 8] are not solvable on the Hanan grid. Weighted Sum of Elmore Delays This is a variant of the previous problem in which the path lengths are computed using the so called Elmore delay function [21] It was shown by Boese et al. 1] that an ....

Jan-Ming Ho, D. T. Lee, Chia-Hsiang Chang, and C. K. Wong. Minimum Diameter Spanning Trees and Related Problems. SIAM Journal on Computing, 20(5):987--997, 1991.

....diameter of a tree is just the length of its longest path. Since geometric spanning trees are often used in applications such as VLSI, in which the time to propagate a signal through the tree is proportional to its diameter, it makes sense to look for a spanning tree of minimum diameter. Ho et al. [68] give an algorithm for this problem, based on the following fact: Lemma 14 (Ho et al. Any point set has some minimum diameter spanning tree in which there are at most two interior points. Proof: We start with any minimum diameter spanning tree T , and perform a sequence of diameterpreserving ....

J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM J. Comput., vol. 20, 1991, pp. 987--997.

....by mapping the clients as nodes V in the graph G, and the network links connecting the clients as edges E. The multicast tree can be obtained by building a minimum diameter spanning tree on G and then the multicast delay bound is minimized. An algorithm has been proposed and proved by Ho et al.[9], which runs in O(n 3 ) where n = jN j, for building a minimum diameter spanning tree in a graph. 3.3 Minimization of Multicast Network Resource Usage and Delay Bound A natural extension is then to minimize in both measures. For examples, we can try to find the minimum spanning tree with least ....

....Minimization of Multicast Network Resource Usage and Delay Bound A natural extension is then to minimize in both measures. For examples, we can try to find the minimum spanning tree with least diameter or we can try to find the minimum diameter spanning tree with least total edge weight. Ho et al.[9] have shown that it is NP complete to find a tree satisfying given bounds on these two measures. So, it is not an computational efficient attempt to find such a multicast tree. ....

J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong, Minimum diameter spanning trees and related problems, SIAM Journal of Computing, vol. 20, 1991, pp. 987-997.

....is a connected subgraph T = V; E T ) without cycles. The diameter of T , D(T ) is defined as the longest of the shortest paths in T among all the pairs of nodes in V . The minimum diameter spanning tree (MDST) problem is to find a spanning tree of G of minimum diameter. Ho, Lee, Chang and Wong [HLCW] consider the case where the graph G is a complete Euclidean graph, induced by a set S of n points in the Euclidean plane. They call this special case the geometric MDST problem. They prove that in this geometric problem there is an optimal tree which is either monopolar or dipolar. A spanning ....

....a monopole such that each of the remaining points is connected to it; and it is said to be dipolar if there exist a pair of points in S called a dipole such that all the remaining points are directly connected to exactly one of the two points in the dipole. Based on the latter property, Ho et al. [HLCW] develop an O(n 3 ) algorithm to find a spanning tree of minimum diameter of a Euclidean graph. They also mention that the above results extend to any graph whose edge lengths satisfy the triangle inequality. In this note we consider the general case where the edge lengths do not necessarily ....

[Article contains additional citation context not shown here]

J.-M. Ho, D.T. Lee, C.-H. Chang and C.K. Wong, "Minimum diameter spanning trees and related problems," SIAM J. Computing 20 (1991) 987-997.

....s to t i in T such that P e2T r(e) C. 3.2.1 Complexity of the Routing Tree Problem In this subsection, we shall discuss the complexity of the above problems. Work on spanning tree construction with simultaneous multiple objective optimization (e.g. delay and wire length) has been reported in [3, 10, 16, 18]. While the literature deals with the case where the objectives are mutual dependent (e.g. delay and wire length are both measured by geometric distance) our formulation considers the case where the objectives are independent , which, as mentioned earlier, is essential for the timing driven ....

....linear assignment in Section 3.4. With the cost functions (1) and (2) we use the following heuristic to construct bounded delay routing trees. The heuristic, P rimBDRT , follows the approach of the Prim s minimum spanning tree construction [29] and is a variation of the tree construction used in [3, 10, 16, 18] which measure both routing and delay costs based on geometric distance. We grow a tree T = V T ; ET ) incrementally, starting from the source s. At each step, we choose an edge e = u; v) where u 2 V T and v 2 V nVT , such that the routing cost of the edge is minimum and the delay constraint ....

J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong, "Minimum diameter spanning trees and related problems," SIAM J. Comput., vol. 20, no. 5, pp. 987--997, 1991.

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Jan-Ming Ho, D. T. Lee, Chia-Hsiang Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM Journal on Computing, 20(5):987--997, October 1991.

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Jan-Ming Ho, D. T. Lee, Chia-Hsiang Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM Journal on Computing, 20(5):987--997, October 1991.

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J.-M. Ho, D. T. Lee, C.-H. Chang, and C. K. Wong. Minimum diameter spanning trees and related problems. SIAM Journal on Computing, 20(5):987--997, 1991.

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