A. V. Goldberg, M. D. Grigoriadis, and R. E. Tarjan, "Use of Dynamic Trees in a Network Simplex Algorithm for the Maximum Flow Problem," Mathematical Programming, vol. 50, pp. 277--290, June 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Irrespective of the delay model employed, this phase can be formulated as the dual of a min cost network flow problem. Using to denote the number of transistors and the number of wires in the circuit, this step in our application has worst case complexity of O( V E log(log V ) [9]. The W phase where transistor gate delays are assumed fixed and their sizes are regarded as variable parameters. As long as Permission to make digital hardcopy of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed ....

....whose dual is a min cost network flow problem [14] X i maximize C i (r(Dmy(i) r(i) For all edges Dmy(i) j, j) FSDU(Dmy(i) j) r(j) 0. 10) Note that the D phase optimization is in the form of the dual of a minimum cost network flow problem, [9]. Also the constant terms in the RHS of the contraints in the D phase can be integerized by ap propriate scaling, i.e. by multiplying every constant term by some power of 10 and then rounding off the product. By choosing appropriate powers of 10 arbitrary accuracy canbe maintained with almost no ....

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A. V. Goldberg, M. D. Grigoriadis, and R. E. Tarjan, "Use of Dynamic Trees in a Network Simplex Algorithm for the Maximum Flow Problem," Mathematical Programming, vol. 50, pp. 277--290, June 1991.

....of the rst edge on the path to the root, from which we immediately get a parent pointer. Unfortunately, the above axiomatic interface has been found too limited for many application of dynamic trees, and instead authors have worked directly with the Sleator and Tarjan s underlying representation [30, 5, 21, 24, 23, 14, 4, 1, 16, 9, 8, 7, 22]. In particular, this is the case for the previous solutions to the dynamic center [6] and median problems [3] and we believe part of the reason for their worse bounds and more complex solutions is diculties in working directly with Sleator and Tarjan s underlying representation. Of course, one ....

A.V. Goldberg, M.D. Grigoriadis, and R.E. Tarjan. Use of dynamic trees in a network simplex algorithm for the maximum ow problem. Math. Programming, 50:277-290, 1991.

....algorithms are known for many types of flow problem, a commonly used alternate method for solving these problems is provided by the network simplex algorithm [2] a specialization of the simplex method from linear programming. Polynomial bounds are known for variants of this algorithm (e.g. see [8, 10]) but these bounds are generally asymptotically larger than those of the combinatorial algorithms (or of interior point linear programming algorithms [13] Nevertheless network simplex remains an important practical alternative [1, 10] apparently because its running time in practice is smaller ....

....from the data structure we described. We now describe how to perform the pivot. The first thing that must be done is to determine the saturated edge of T that is replaced by the pivot edge. This can be done in time O(log n) per pivot using 5 the dynamic tree data structure of Sleator and Tarjan [8, 12], or alternatively in time O(m 1 2 ) using our tree partition data structure. Once we have determined the swap made in T by the pivot, we must update the data structures used to find the next pivot. As seen in the previous section, the restricted partition and topology tree take O( # m) time to ....

A. V. Goldberg, M. D. Grigoriadis, and R. E. Tarjan. Use of dynamic trees in a network simplex algorithm for the maximum flow problem. Mathematical Programming 50 (1991) 277--290. 12

....King [16] independently to represent trees whose vertices have associated lists. Additional related work deals with a class of dynamic trees in which vertex or edge values are combined along paths, rather than over an entire tree. Dynamic trees of this kind arise in various network flow algorithms [10, 11, 12, 13, 20, 22, 24] and in other settings[4, 6] Two different representations of such trees have been proposed. The first, by Sleator and Tarjan [20, 21, 24] decomposes each tree into vertex disjoint paths and represents these paths by search trees, either biased search trees [3] or splay trees [21] With the ....

....used in parallel tree processing [17] This representation, too, has an O(log n) worst case time bound per tree operation. Both the Sleator Tarjan representation and the Frederickson representation can be applied to the problem we consider here, achieving the same O(log n) time bound (see e.g. [10]) But we regard these solutions are inferior, for two reasons. First, both data structures must be extended to handle tree vertices of ordinary degree. Second, both structures, even without the unbounded degree extension, are noticably more complicated than the Euler tour structure, which ....

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A. V. Goldberg, M. D. Grigoriadis, and R. E. Tarjan. "Use of dynamic trees in a network simplex algorithm for the maximum flow problem," Math. Prog. 50 (1991), 277-290.

....endpoints fs u ; s v g as appropriate. Thus, to split node v at edge e, we execute Split(v; s v ) If T has n nodes and hence n Gamma 1 edges, then T 0 has 2n Gamma 2 nodes. Note that every node in T 0 has degree at most three. A similar idea has been used by Goldberg, Grigoriadis and Tarjan [12] in a different extension of dynamic trees that supports computing minima and maxima over subtrees. Our extension requires some additional ideas. Figure 4 gives an example of an edge ordered tree. The transformed tree T 0 is maintained with a standard Sleator Tarjan dynamic tree. The node path ....

A. V. Goldberg, M. D. Grigoriadis, and R. E. Tarjan. Use of dynamic trees in a network simplex algorithm for the maximum flow problem. Math. Prog., to appear.

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A. V. Goldberg, M. D. Grigoriadis, and R. E. Tarjan, "Use of Dynamic Trees in a Network Simplex Algorithm for the Maximum Flow Problem," Mathematical Programming, vol. 50, pp. 277--290, June 1991.

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