C. D. Bateman and C. S. Helvig and G. Robins and A. Zelikovsky. Provably good routing tree construction with multi-port terminals. In Proceedings of ACM/SIGDA International Symposium on Physical Design, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[22] considered the group Steiner problem on trees and gave an algorithm with a performance ratio of B H(N) B O(lnN) where B is the maximum number of vertices of a group in a subtree of the root, and H(N) is the N th harmonic number. In a recent paper, Bateman, Helvig, Robins and Zelikovsky [6] have given the first algorithm with a sublinear performance guarantee. Their algorithm (with a Java implementation available on the Internet [5] gives an approximation ratio of (1 1 )v . This ratio comes from approximating the group Steiner tree by a 2 star (tree of depth 2) and then ....

C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky. Provably good routing tree construction with multi-port terminals. In Proceedings of A CM/SIGDA International Symposium on Physical Design, Apr. 1997.

....discussed in [54] Very recently, the Group Steiner Tree problem acquired large attention among theory researchers. In fact, after the original appearance of our results, there have been two papers which improve on our bounds in the most general cases. Bateman, Helvig, Robins, and Zelikovsky in [14] approximated the Group Steiner within (1 ln m. They use the fact that the optimal group Steiner tree can be approximated reasonably well by an optimal group Steiner 2 star (a tree of depth at most 2) Their algorithm then approximates this optimal group Steiner 2 star. Garg, Konjevod, and ....

....solution. Even though these results are stronger than ours for general graphs with no assumptions about the number or size of clusters, in cases where there is some a priori knowledge about the number or size of groups, our bounds provide stronger approximation guarantees. The above results of [14] and [39] for Group Steiner can be easily modified to provide similar approximation bounds for GTSP. Even though the Generalized Traveling Salesman problem is a relatively old and quite natural problem, there had not been any worst case performance bounds established for this problem (with the ....

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C. D. Bateman, C. Helvig, G. Robins, and A. Zelikovsky. Provably good routing tree construction with multi-port terminals. In Proceedings of ACM/SIGDA International Symposium on Physical Design, Napa Valley, California, Apr. 1997.

.... for Steiner tree estimation from the University of Virginia [13] and (ii) to the fact that Steiner lengths are measured from the center of the bounding box for all gate pins that are connected to the same net, whereas the actual net only connects to the closest one ( group Steiner problem [2]) 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 5 10 15 20 25 30 Via fill rate ( M3 M4 Utilization factor ratio Fig. 6. The utilization factor at M3 relative to that at M4 for the experiments with virtual vias. A higher number of virtual vias (higher via fill rate) corresponds to a higher layer ....

C. D. Bateman, C. H. Helvig, G. Robins and A. Zelikovsky, "Provably-Good Routing Tree Construction with Multi-Port Terminals," Proc. ACM/SIGDA Intl. Symp. on Physical Design, 1997, pp. 96--102.

....are small, this bound is perhaps of particular interest. Practical heuristics were first considered by Reich and Widmayer [19] These algorithms were experimentally evaluated in [15] and appeared to compute good solutions. Other practical heuristics showing good performance have been proposed [2, 11]. In this paper we give a first (tailored) exact algorithm for solving RGSTP. We show that the problem reduces to the ordinary Steiner tree problem in 3 graphs; the graph is essentially the Hanan grid over all given points. In order to solve this problem efficiently we first show how the ....

C. D. Bateman, C. H. Helvig, G. Robins, and A. Zelikovsky. Provably-Good Routing Tree Construction with Multi-Port Terminals. In ACM/SIGDA International Symposium on Physical Design, 1997.

....[29] considered the group Steiner problem on rooted trees and gave an algorithm with an approximation ratio of B H(N) B O(ln N ) where B is the maximum number of vertices of a group in a subtree of the root, and H(N) is the N th harmonic number. Bateman, Helvig, Robins and Zelikovsky [6] gave the rst algorithm with a sub linear performance guarantee. Their algorithm gives an approximation ratio of (1 ln k 2 ) p k. This ratio comes from approximating the group Steiner tree by a 2 star (tree of depth 2) and then approximating the covering problem on the 2 star within a ....

C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky. Provably good routing tree construction with multi-port terminals. In Proceedings of ACM/SIGDA International Symposium on Physical Design, Apr. 1997.

.... for Steiner tree estimation from the University of Virginia [12] and (ii) to the fact that Steiner lengths are measured from the center of the bounding box for all gate pins that are connected to the same net, whereas the actual net only connects to the closest one ( group Steiner problem [2]) Layer Sai Halasz U i U i 1 Only M3 U 3 U 4 M1 M2 0.85 0.07 min 1.10 M2 M3 0.85 0.56 avg 1.74 M3 M4 0.85 1.10 max 2.30 Table 2: The utilization factor U i on layer i, relative to U i 1 : Sai Halasz model of constant relative factors versus experimental values that are not constant. ....

C. D. Bateman, C. H. Helvig, G. Robins and A. Zelikovsky, "Provably-Good Routing Tree Construction with Multi-Port Terminals", Proc. ACM/SIGDA Intl. Symp. on Physical Design, 1997, pp. 96--102.

....the group Steiner problem on trees and gave an algorithm with a performance ratio of B Delta H(N) B Delta O(ln N ) where B is the maximum number of vertices of a group in a subtree of the root, and H(N) is the N th harmonic number. In a recent paper, Bateman, Helvig, Robins and Zelikovsky [6] have given the first algorithm with a sublinear performance guarantee. Their algorithm (with a Java implementation available on the Internet [5] gives an approximation ratio of (1 ln k 2 ) p k. This ratio comes from approximating the group Steiner tree by a 2 star (tree of depth 2) and ....

C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky. Provably good routing tree construction with multi-port terminals. In Proceedings of ACM/SIGDA 7 International Symposium on Physical Design, Apr. 1997.

....of o(log k) see [219, 240, 363, 270] A (k Gamma 1) approximation algorithm is given by Reich and Widmayer [335] and Ihler [218] See also Ihler, Reich, and Widmayer [221] Slav ik [363] gives an O(log k) approximation algorithm for the special case of an edge weighted tree. Bateman et al. [58] give the first sublinear approximation factor for general graphs, with an approximation factor of (1 ln k 2 ) Delta p k. Charikar et al. 87] give a k ffl approximation algorithm that runs in polynomial time, as well as an O(log 2 n) approximation algorithm that runs in ....

C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky. Provably good routing tree construction with multi-port terminals. In Proc. ACM/SIGDA International Symposium on Physical Design, page ??, Apr. 1997.

....vertex from each of the groups. This problem is hard to approximate within logarithmic factors even on stars (via reduction from set cover [11] The problem was introduced by Reich and Widmayer [20] and finds applications in VLSI design. Some related problems are discussed in [14] Bateman et al. [6] gave the first non trivial approximation algorithm with a ratio of (1 ln k=2) p k. Charikar et al. 7] using a reduction to directed Steiner trees gave an algorithm with a ratio of k ffl for any fixed ffl 0. They also showed that it is possible to obtain an O(log 2 k) approximation in ....

C.D. Bateman, C.S. Helvig, G. Robins, and A. Zelikovsky. "Provably good routing tree construction with multi-port terminals", Proceedings of the 28th ACM/SIGDA International Symposium on Physical Design (Apr 1997).

....robins cs.virginia.edu, 804) 982 2207, http: www.cs. virginia.edu robins 1 Optional Steiner nodes may be included in order to reduce the cost of the spanning tree interconnecting the groups of N (see Figure 1) The Group Steiner Problem captures practical scenarios in VLSI circuit design [4, 9, 19], where circuit modules may be rotated and flipped when positioned on a VLSI chip. This induces multiple potential connection points for a given circuit module, namely, one for each of the eight possible orientations [4, 19] see Figure 1(b) These locations correspond to a group of up to eight ....

....Steiner Problem captures practical scenarios in VLSI circuit design [4, 9, 19] where circuit modules may be rotated and flipped when positioned on a VLSI chip. This induces multiple potential connection points for a given circuit module, namely, one for each of the eight possible orientations [4, 19] (see Figure 1(b) These locations correspond to a group of up to eight nodes in the Group Steiner Problem formulation when there is only one connection point for the given module (see Figure 1(c) This formulation also captures the pin assignment step in VLSI physical design [18] where the ....

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C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky. Provably-good routing tree construction with multi-port terminals. In Proc. International Symposium on Physical Design, pages 96--102, Napa Valley, CA, April 1997.

....given an undirected weighted graph G = V; E) and M V , find a minimum cost tree which spans all of M . This work was supported by a Packard Foundation Fellowship and by National Science Foundation Young Investigator Award MIP 9457412. A preliminary version of this work has appeared in [3] [8] The corresponding author is Professor Gabriel Robins, Department of Computer Science, University of Virginia, Charlottesville, VA 22903 2442, robins cs.virginia.edu, 804) 982 2207, http: www.cs.virginia.edu robins 1 Nodes in V Gamma M (referred to as Steiner nodes) may be optionally ....

....6 extends our basic group Steiner approach into a bounded radius formulation that minimizes tree cost as well as source to sink pathlengths in a provably good manner. Finally, we discuss our implementation and experimental results in Section 7. A preliminary version of this work appeared in [3] [8] 2 Depth Bounded Steiner Trees In this section, we introduce the concept of Steiner depth bounded 5 trees. Our motivation for using depth bounded trees is two fold: 1) optimal depth 2 bounded trees can be used to approximate optimal group Steiner trees to within a factor of 2 Delta p k, ....

C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky. Provably-good routing tree construction with multi-port terminals. In Proc. International Symposium on Physical Design, pages 96--102, Napa Valley, CA, April 1997.

....OPT have the same length 1. Indeed, if there is an edge of length less than the maximum length, then we may increase its length up to the maximum without increasing the bottleneck cost. We will use the following two lemmas which are well known graph theoretical facts (e.g. Lemma 2 can be found in [1]) Fact 1 Given a tree with n leaves and all internal nodes of degree 3, except the root r that may have degree 2 or 3. There exists a leaf at most log 2 n edges away from r. Fact 2 Any tree with n leaves contains a node c (called center) such that removing c splits the tree into connecting ....

C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky, Provably-Good Routing Tree Construction with Multi-Port Terminals, in Proc. International Symposium on Physical Design, Napa Valley, CA, April 1997, pp. 96--102. 12

....OPT have the same length 1. Indeed, if there is an edge of length less than the maximum length, then we may increase its length up to the maximum without increasing the bottleneck cost. We will use the following two lemmas which are well known graph theoretical facts (e.g. Lemma 5 can be found in [1]) Lemma 4 Given a tree with n leaves and all internal nodes of degree 3, except the root r that may have degree 2 or 3. There exists a leaf at most log 2 n edges away from r. Lemma 5 Any tree with n leaves contains a node c (called center) such that removing c splits the tree into connecting ....

D. Bateman, C. Helvig, G. Robins and A.Zelikovsky. Provably good routing tree construction with multi-port terminals. Proc. of IEEE-ACM Int. Symp. on Physical Design, 96--102, 1997.

....tree which contains at least one node from each group N i . As in the classical Steiner problem, we allow optional Steiner nodes in order to reduce the cost of the spanning tree interconnecting the groups (Figure 1(c) The Group Steiner Problem captures practical scenarios in VLSI layout design [2, 15]. For example, a circuit module may be rotated and flipped when positioned on a VLSI chip. This induces multiple potential connection points for the given circuit module, This work was supported by a Packard Foundation Fellowship and by National Science Foundation Young Investigator Award ....

.... Steiner tree (solid dots represent Steiner nodes) The first published approximation algorithm for the Group Steiner Problem produced solutions O(k) times worse than optimal [7, 8, 15] Recently, we improved this result by giving a heuristic with an approximation bound of O(k 1 2 ) times optimal [2]. The main result of this paper is a new series of heuristics with an improved performance ratio 1 of O(k ffl ) for arbitrarily small values of ffl 0, where k is the number of groups. On the negative side, this problem cannot be efficiently approximated with a performance ratio of less than ....

C. D. Bateman, C. S. Helvig, G. Robins, and A. Zelikovsky. Provably-good routing tree construction with multi-port terminals. In Proc. ACM/SIGDA International Symposium on Physical Design, pages 96--102, Napa Valley, CA, April 1997.

No context found.

C. D. Bateman and C. S. Helvig and G. Robins and A. Zelikovsky. Provably good routing tree construction with multi-port terminals. In Proceedings of ACM/SIGDA International Symposium on Physical Design, 1997.

No context found.

M. Bellare, C. Helvig, G. Robins, and A. Zelikovsky, `Provably good routing tree construction with multiport terminals', in Twenty-Fifth Annual ACM Symposium on Theory of Computing, pp. 294--304, (1993).

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