C. Thomborson, B. Alpern, and L. Carter, "Rectilinear Steiner tree minimization on a workstation, preprint (1992).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....sets) We describe and study an algorithm that is effective on 30 input terminals. The fastest algorithms in the literature are only applicable to problems containing up to roughly 17 points. See Yang and Wing[11] Sidorenko[8] Lewis, Pong, and Van Cleave[7] and Thomborson, Alpern, and Carter[9]. From the standpoint of VLSI design, there are two important contributions of this work. The first is the design and construction of a tool that can efficiently compute rectilinear Steiner minimal trees for smallsized terminals sets. The second is a comparison of the exact solution to Steiner ....

C. Thomborson, B. Alpern, and L. Carter, "Rectilinear Steiner tree minimization on a workstation, preprint (1992).

....to the rectilinear Steiner tree (RST) problem, which is NP complete [10] This suggests that no polynomial time algorithm can solve the RST problem exactly. Nonetheless, exponentialtime algorithms have been devised that can solve the RST problem exactly for small instances in reasonable time [9, 23, 25], as have many efficient polynomial time heuristics (see Hwang, Richards, and Winter [16] that produce good suboptimal solutions. The time complexity of the vast majority of these algorithms is a function of size of the instance, not of the routing area as in maze routing algorithms. No such ....

C. D. Thomborson, B. Alpern, and L. Carter, Rectilinear Steiner tree minimization on a workstation, in Proceedings of the DIMACS Workshop on Computational Support for Discrete Mathematics, 1992.

....a set of 27 terminals. lem in G solves the original geometric RSMT instance. The most efficient algorithm for solving the Steiner problem on G is the dynamic programming algorithm of Dreyfus and Wagner [3] which has time complexity O(n 2 3 n ) when applied to G. Thomborson, Alpern, and Carter [12] present some improvements to the Dreyfus Wagner algorithm that do not change the algorithm s time complexity, but do improve its efficiency in practice. Ganley and Cohoon [4] present a more direct, geometric algorithm that has time complexity O(n3 n ) and is faster in practice than the ....

....Hwang s theorem [10] so this term is dominated by the decomposition summation. 7 Empirical results We have implemented the SFDP algorithm in order to compare it empirically with the FDP algorithm [4] the Dreyfus Wagner algorithm [3] and the algorithm of Thomborson, Alpern, and Carter [12]. Figure 5 plots the running time of each algorithm as a function of the number n of input terminals. As can be seen, the FDP algorithm is faster than both the Dreyfus Wagner algorithm and the Thomborson, Alpern, and Carter algorithm, but still has the same slope on the logarithmic scale of the ....

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C. D. Thomborson, B. Alpern, and L. Carter. Rectilinear Steiner tree minimization on a workstation. In Proceedings of the DIMACS Workshop on Computational Support for Discrete Mathematics, 1992.

....whose time complexity is O(2 p k log k ) but their algorithm is designed such that either the time complexity or the optimality of the computed tree is probabilistic. Furthermore, while their algorithm has not been implemented, it is not expected to be competitive for small terminal sets [10]. Salowe and Warme [9] present an algorithm that works very well in practice it can efficiently solve problems with up to 30 terminals but the only known bound on its time complexity is O(2 2 k ) The other major approach to solving the RSMT problem exactly is to reduce it to the Steiner ....

....O(k 2 3 k (k 2 log k)2 k ) time and O(k 2 2 k ) space, the FDP algorithm requires O(k3 k k2 k ) time and O(2 k ) space. Furthermore, the FDP algorithm is far faster in practice than the Dreyfus Wagner algorithm, and is very easy to implement. Thomborson, Alpern, and Carter [10] present several optimizations to the Dreyfus Wagner algorithm that reduce the constant factors in its time complexity. These optimizations rely on explicit memory management and loop reordering to maximize locality of reference, and on some geometric properties of the Hanan grid graph. Many of ....

C. D. Thomborson, B. Alpern, and L. Carter, Rectilinear Steiner tree minimization on a workstation, in Proceedings of the DIMACS Workshop on Computational Support for Discrete Mathematics, 1992.

....main memory and disk. When this happens, the most expedient solution often is to find a computer with more memory. Although not illustrated in our examples, the techniques described in this paper (both ours and the previous tiling references) can be applied to some problems with great success [TAC94]. The standard techniques cited in section 2 can greatly reduce the amount of data that needs to be paged out to disk and brought back into main memory during the course of a computation. However, if the transformed program still relies on the demand paging mechanisms provided by the computer ....

Thomborson, C., B. Alpern and L. Carter, "Rectilinear Steiner Tree Minimization on a Workstation," Computational Support for Discrete Mathematics, N. Dean and G. Shannon editors, Volume 15 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, American Mathematics Society (1994).

....H PRAM model [HR92] is another model that assumes a tree organization for the processors. Our model differs from these other models by allowing data to be stored in the intermediate nodes of the tree. This allows us to use the same structure to model the memory hierarchy, which in our experience [AABCH91, TAC92, ACFS93] is crucially important for developing high performance programs. 3 Choosing a Specific Model A three step approach can be used to derive a specific PMH model for a given architecture. First, choose an overall tree structure to reflect the data transfer bandwidths among the processors and between ....

Thomborson, C., B. Alpern and L. Carter, "Rectilinear Steiner Tree Minimization on a Workstation, " DIMACS Workshop on Computational Support For Discrete Mathematics, March 1992, Also IBM RC 17680.

....be modified so that the interpreter would run efficiently on a PC with 32 MB of RAM. As another example, with Larry Carter and Bowen Alpern of the IBM TJ Watson Research Laboratory, I developed a code for computing rectilinear Steiner minimal trees on k 24 pins using the Dreyfus Wagner recurrence[6]. Before storage optimization and data compression, our data structures required 2k 2 2 k Gamma1 16 bit integer words. For k = 23, this is 4.5 GB, or slightly more than the 32 bit addressing limit of the workstations (and most supercomputers) available until very recently. In reflecting on ....

Clark Thomborson, Bowen Alpern, and Larry Carter. Rectilinear Steiner tree minimization on a workstation. In N. Dean and G.E. Shannon, editors, Computational Support for Discrete Mathematics, volume 15 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 119--136. American Mathematical Society, 1994.

....are often possible. Appendix I gives examples. 3. Accommodate the memory hierarchy. Proper cache usage often affects performance by a factor of two, and in some cases, such as LAPACK s vanilla BLAS, by as much as a factor of eight. Proper consideration of paging can have arbitrarily large payoff [Tho92]. A computer s translation look aside buffer (TLB) can also play a significant role, e.g. a factor of two or three. We found that the general advice, Strive for temporal and spatial locality is insufficient. A more accurate view is that at each level of the memory hierarchy, hardware moves data ....

....and number of blocks it holds, and the time required to transfer a block to or from the next higher module. We have used the MH model for developing or consulting on a variety of programs, including ones for in core and out of core matrix operations, integer sorting, a dynamic programming problem [Tho92], a variety of implicit and explicit differential equations, and algorithms for 2 D molecular dynamics, signal processing and a VLSI application. 10 Appendix I argues that LAPACK s single parameter is insufficient. 11 More accurately, the MH model is intended to guide the programmer in ....

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C. Thomborson, B. Alpern, and L. Carter. Rectilinear Steiner tree minimization on a workstation. Research Report RC17680, IBM Watson Research Center, Yorktown Heights, New York, March 1992. Presented at DIMACS Workshop on Computational Support For Discrete Mathematics.

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