V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. Journal of Algorithms, vol. 17, 3, pages 447--474, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0) introduce an arc (s; v i ) v i ; t) with capacity jd i j. The original problem is feasible iff in the extended network the maximum s Gamma t flow value is achieved by a flow function which saturates all the arcs emanating from s. The maximum flow can be computed, e.g. by the algorithm of [26] in O(jV jjE j jV j 2 ffl ) for any fixed ffl 0. In conclusion: Theorem 7.1 The directed and undirected Eulerian sandwich problems are polynomial. If in the Eulerian SP the graph G is not connected, then the sandwich graph may also be disconnected. Interestingly, by adding the ....

V. King, S. Rao, and R. E. Tarjan. A faster deterministic maximum flow algorithm. In Proc. 3rd annual ACM-SIAM Symp. on Discrete Algorithms, pages 157--164. ACM Press, 1992.

....an s t maximum flow. In 1961, Gomory and Hu showed how to solve the minimum cut problem with n 1 s t minimum cut computations. Subsequently there was much progress in computing maximum flows, but no one has yet been able to prove a time bound better than O(nm) for any of the best algorithms [1, 9, 10, 25, 41]. Hence we cannot give a bound better than O(n m) for the Gomory Hu algorithm. Gomory Hu stood as the best algorithm for the problem until 1989, when Nagamochi and Ibaraki [47] showed how to find a minimum cut without using maximum flows. Their algorithm (which we will call NI) runs in O(n(m n ....

V. King, S. Rao, and R. Tarjan. A Faster Deterministic Maximum Flow Algorithm. J. Algorithms, 17:447--474, 1994.

....an s t maximum flow. In 1961, Gomory and Hu showed how to solve the minimum cut problem with n 1s tmini mum cut computations. Subsequently there was much progress in computing maximum flows, but no one has yet been able to prove a time bound better than O#nm# for any of the best algorithms [1, 9, 10, 25, 41]. Hence we cannot give a bound better than O#n m# for the Gomory Hu algorithm. Gomory Hu stood as the best algorithm for the problem until 1989, when Nagamochi and Ibaraki [47] showed how to find a minimum cut without using maximum flows. Their algorithm (which we will call NI) runs in O#n#m n ....

V. King, S. Rao, and R. Tarjan. A Faster Deterministic Maximum Flow Algorithm. J. Algorithms, 17:447--474, 1994.

....is easier than solving the MCB problem itself for the component, whichmust be done in a later phase. The Gomory Hu tree for a graph can be constructed by solving n ; 1 network flow problems, n is the number of vertices in the graph. The network flow problem can be solved in O(mn log m=n log n n) [KRT94], so the MCB problem for cographic matroids can be solved in O(mn 2 log n) time. This is considerably faster than any known algorithm for graphic matroids. Thus the slowest step in the algorithm is solving the MCB in the graphic case, for which the best published algorithm is Theta(m 3 n) in ....

King, Rao, and Tarjan. A faster deterministic maximum flowalgorithm. Journal of Algorithms,1994.

....bound Omega Gamma jV j Theta jEj) 20] The preflow push method is more time efficient. The first complete method was designed by Goldberg and Tarjan [21] and uses Karzanov s preflows [23] Goldberg and Tarjan [21] achieve O(jV j jEj log( jV j 2 jEj ) running time and King, Rao, and Tarjan [24] obtain a running time of O(jV j jEj (log jEj jV j log jV j jV j) Goldberg and Rao s algorithm [20] achieves a running time smaller than the Omega Gamma jV j Theta jEj) lower bound at the expense of limiting the capacity of the edges to a maximum value U . The execution time is O(min(jV j ....

V. King, S. Rao, and R. E. Tarjan. A faster deterministic maximum flow algorithm. Journal of Algorithms, 17(3):447--474, 1994.

....and proofs we refer to the original articles that will be cited throughout. A minimum capacity cut ffi(W ) in G separating a prespecified pair of nodes s; t 2 V , i.e. s 2 W; t = 2 W , can be found by various (s; t) max flow min cut techniques. e.g. the algorithm of King, Rao, and Tarjan [KRT94] runs in O(jV jjEj log jEj=jV j log jV j jV j) time. 2 In practical experiments it turned out that a variant of the Goldberg Tarjan preflow push algorithm [GT88] using highest level selection, global relabeling, and gap relabeling is very efficient [DM89, CG95] When we have to solve an (s; ....

V. King, S. Rao, and R. Tarjan (1994), "A faster deterministic maximum flow algorithm", Journal of Algorithms 17, 447--474.

....costs, MF is the time to solve a max flow problem, and AP is the time to solve an assignment problem. The best current bounds on SP are O(m n log n) 9] O(m log log C) 22] and O(m n p log C) 1] The best current bounds on MF are O(mn log(n 2 =m) 14] O(nm log m=n log n n) [25], O(mn log m=n n n 2 log 2 n) for any constant 0 [35] O(nm log(2 n p log U=m) 2] and O(minfn 2=3 ; p mgm log(n 2 =m) log U) 13] The best current bounds on AP are O(nSP) and O( p nm log(nC) 10, 12, 33] Cycle Canceling Algorithms Cut Canceling Algorithms Class of ....

V. King, S. Rao and R. Tarjan, "A faster deterministic maximum flow algorithm," Journal of Algorithms 17 (1994) 447--474.

....denote the worst case complexity of finding a maximum flow in a network with m arcs, n nodes, and arbitrary positive capacities. Currently, the best known bounds on MF(m, n)areO(mn log(n 2 m) O(mn log m n log n n) and O(mn log m n n n 2 log 2 # n) for any constant # 0, due to [22] [39], and [47] respectively. We let MF(m, n, U) denote the complexity assuming the capacities are integers between 0 and U . Currently, the best known bounds for MF(m, n, U)are O(mn log(n # log U (m 2) and O(min n 2 3 , # m m log(n 2 m)logU) due to [3] and [21] respectively. 2.2.4 ....

V. King, S. Rao, and R. E. Tarjan. A faster deterministic maximum flow algorithm. Journal of Algorithms, 17:447--474, 1994.

....flows. Combined with a structure called Gomory Hu trees, these methods involved n different calls to an algorithm to find a maximum flow between a pair of vertices on an n node graph 1 . The fastest flow algorithm currently available is based on the Push Relabel technique of Goldberg and Tarjan [10, 19] and run in time O(nm) on an n node m edge graph 2 . Hao and Orlin [15] extended this method to piggyback all the n flow computations in asymptotically the same time for one, giving an algorithm for finding the minimum cut in time O(nm) An alternate set of algorithms for finding the ....

V. King, S. Rao, and R. E. Tarjan, A Faster Deterministic Maximum Flow Algorithm, J. Algorithms 17 (1994), pp. 447-474.

....= n, we get optimal cuts. Theorem 2 Algorithm EFFICIENT finds a set of k cuts, one for each value of k, 2 k n; each cut is within a factor of (2 Gamma 2=k) of the optimal k cut. The algorithm requires a total of n Gamma 1 max flow computations. Using the current best known max flow algorithm [GT, KTR], our algorithm has a running time of O(mn 2 n 3 ffl ) for any fixed ffl 0. 4 Algorithm SPLIT Perhaps the first heuristic that comes to mind for finding a k cut is the following: Algorithm SPLIT: Start with the given graph. In each iteration, pick the lightest cut, in the current ....

V. King, S. Rao, and R. Tarjan, "A faster deterministic maximum flow algorithm, Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, 1992, pp. 157-164.

....time is no more than a single maximum flow computation. The running time of the current best algorithm for maximum flow is slightly more than O(mn) This was first achieved by Goldberg and Tarjan [GT] and the complexity of their algorithm is O(mn log(n 2 =m) Their result was later improved by [Al, CH, CHM, KRT, PW] For many years, since the paper of Gomory and Hu appeared, the only approach to the minimum cut problem was via maximum flows. This has changed in the last years, and several papers have shown that a minimum cut can be solved more efficiently than maximum flow. Nagamochi and Ibaraki [NI] gave an ....

V. King, S. Rao and R. Tarjan, A faster deterministic maximum flow algorithm, Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, 1992, pp. 157-164.

.... 4 1 n c Gamma1 : Obtaining a better o(1) factor is straightforward by increasing the number of intervals. A fractional maximum flow can be found by the push relabel method of Goldberg and Tarjan [8] whose currently fastest implementation has running time T 1 (n; m) O(nm log m n log n n) [15]. In that case even when our algorithm is used to obtain a 4 1 2 n approximation, the running time is dominated by a single max flow computation. Alternatively, the new maxflow algorithm of Goldberg and Rao [10] may be used with T 1 (n; m) O(min(n 2=3 ; m 1=2 )m log( n 2 m ) log U ) ....

V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. In Proceedings of the 3rd ACM-SIAM Symposium on Discrete Algorithms, pages 157--164, January 1992.

....proofs we refer to the original articles that will be cited throughout. A minimum capacity cut ffi (W ) in G separating a prespecified pair of nodes s; t 2 V , i.e. s 2 W; t = 2 W , can be found by various (s; t) min cut techniques, e.g. when applying the the algorithm of King, Rao, and Tarjan [18] in O(jV jjEj log jEj=jV j log jV j jV j) In practical experiments it turned out that a variant of the Goldberg Tarjan preflow push algorithm [10] using highest level selection, global relabeling, and gap relabeling is very efficient [6, 5] We are using this algorithm requiring O(jV j 2 p ....

V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. Journal of Algorithms, 17:447--474, 1994.

.... fl = o(n) When fl = Omega Gamma n) CELL runs in time O(m n n fl log(n 2 = fl) which is never worse than the algorithm of Goldberg and Tarjan [7] There have been a number of recent results that improve the running time of Goldberg and Tarjan s preflow push algorithm, such as [1] 2] 3][9][11] The excess scaling algorithm of [1] runs in time O(mn n 2 p logU) where U is the largest arc capacity. The randomized algorithm of [3] runs in time O(mn n 2 log 3 n) with high probability. The deterministic algorithms of [11] and [8] run in time O(nm log m=n log n n) n 2 log ....

V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. In Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, pages 157--164, 1992.

....preflow push algorithm, such as [1] 2] 3] 9] 11] The excess scaling algorithm of [1] runs in time O(mn n 2 p logU) where U is the largest arc capacity. The randomized algorithm of [3] runs in time O(mn n 2 log 3 n) with high probability. The deterministic algorithms of [11] and [8] run in time O(nm log m=n log n n) n 2 log 2 ffl n) for any constant ffl. Some of these algorithms can be used to compute the flow in the reduced graph and some cannot, depending on the rules used to select vertices and edges for push operations. The dependence is made clearer in subsequent ....

....log(n 2 = fl) Ahuja and Orlin give an excess scaling algorithm that runs in time O(mn n 2 log U ) U the maximum capacity of an edge. This algorithm does not have any special edge selection rule, and can be applied to compute the flow G R . On the other hand, the algorithms of [3][8][11] achieve their running times by edge selection rules that are incompatible with the selection rule given in Section 8, and so they cannot be used unless an alternate edge selection rule is devised. ....

V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. Improved version of an algorithm that appeared in Proc. 3rd ACMSIAM Symp. on Discrete Algorithms, 1992, 157--164.

....authentication mechanisms for a wide range of systems, even those based on technologies other than public keys. One natural direction for future research is to find approximation algorithms that supersede ours in accuracy, efficiency, or both. David Johnson suggested computing a maximum flow [GT88, AOT89, KRT92] with capacity constrained nodes for finding the number of disjoint paths from the source to the target. A maximum flow is not guaranteed to include only paths (or for that matter, any paths) of length at most the specified path bound, even if run on a restricted graph consisting of only those ....

V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. In Proceedings of the 3rd ACM Symposium on Discrete Algorithms, pages 157--164, 1992.

....of rescheduling. We then apply these ideas to network flow. Cheriyan and Hagerup [6] introduced an on line game on a bipartite graph as a fundamental step in improving algorithms for computing the maximum flow in networks. They described a randomized strategy to play the game. King, Rao and Tarjan [11] studied a modified version of this game, called node kill , and gave a deterministic strategy. We obtain an improved deterministic algorithm for the node kill game (and hence for maximum flow) in all but the sparsest graphs. The running time achieved is O(mn log m=n n n 2 log 2 ffl n) ....

....log n) total computation. Our algorithms extend to more general versions of the problem where both tasks and servers may arrive and depart, and where tasks have an associated weight and the maximum weighted load must be minimized. 1. 2 Network Flow The node kill game, introduced by King et al. [11], is played on a bipartite graph G = U; V; E) where jU j = jV j = n and jEj = m. We refer to the nodes in U as items and the nodes in V as columns. The rules of the game are as follows. 1. At the start of each turn, player A (the algorithm) must designate a current edge fu; vg for each item u. ....

[Article contains additional citation context not shown here]

V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. In Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, pages 157--164, 1992.

....flows. Combined with a structure called Gomory Hu trees, these methods involved n different calls to an algorithm to find a maximum flow between a pair of vertices on an n node graph 1 . The fastest flow algorithm currently available is based on the Push Relabel technique of Goldberg and Tarjan [18, 30] and run in time O(nm) on an n node m edge graph 2 . Hao and Orlin [24] extended this method to piggyback all the n flow computations in asymptotically the same time for one, giving an algorithm for finding the minimum cut in time O(nm) An alternate set of algorithms for finding the ....

V. King, S. Rao, and R. E. Tarjan, A Faster Deterministic Maximum Flow Algorithm, J. Algorithms 17 (1994), pp. 447-474.

.... Ahuja et al. 3] O(nm log(n p log U=m) 1989 Cheriyan Hagerup [7] E(nm n 2 log 2 n) 1990 Cheriyan et al. 8] O(n 3 = log n) 1990 Alon [4] O(nm n 8=3 log n) 1992 King et al. 37] O(nm n 2 ffl ) 1993 Phillips Westbrook [44] O(nm(log m=n n log 2 ffl n) 1994 King et al. [38] O(nm log m= n log n) n) 1997 Goldberg Rao [24] O(min(n 2=3 ; m 1=2 )m log(n 2 =m) log U) Table 2: History of the capacitated bound improvements. The dates correspond to the first publication of the result; citations are to the most recent versions. Bounds containing U apply to the ....

V. King, S. Rao, and R. Tarjan. A Faster Deterministic Maximum Flow Algorithm. J. Algorithms, 17:447--474, 1994.

.... log n) 1986 Goldberg Tarjan [26] O(nm log(n 2 =m) 1987 Ahuja Orlin [2] O(nm n 2 log U) 1987 Ahuja et al. 3] O(nm log(n p log U=m) 1989 Cheriyan Hagerup [7] E(nm n 2 log 2 n) 1990 Cheriyan et al. 8] O(n 3 = log n) 1990 Alon [4] O(nm n 8=3 log n) 1992 King et al. [37] O(nm n 2 ffl ) 1993 Phillips Westbrook [44] O(nm(log m=n n log 2 ffl n) 1994 King et al. 38] O(nm log m= n log n) n) 1997 Goldberg Rao [24] O(min(n 2=3 ; m 1=2 )m log(n 2 =m) log U) Table 2: History of the capacitated bound improvements. The dates correspond to the first ....

V. King, S. Rao, and R. Tarjan. A Faster Deterministic Maximum Flow Algorithm. In Proc. 3rd ACM-SIAM Symposium on Discrete Algorithms, pages 157--164, 1992.

....two specified vertices, s and t, is called the minimum s t cut problem, and is closely related to the minimum cut problem. The classical Gomory Hu algorithm [21] solves the minimum cut problem using n Gamma 1 minimum s t cut computations. The fastest current algorithms for the s t cut problem [1, 6, 7, 19, 30] use flow techniques, in particular the push relabel method [19] and run in (nm) time. For the minimum cut problem, Hao and Orlin [23, 25] have given an algorithm (ho) based on the pushrelabel method, that shows how to perform all n Gamma 1 minimum s t cuts in time asymptotically equal to that ....

V. King, S. Rao, and R. Tarjan. A Faster Deterministic Maximum Flow Algorithm. J. Algorithms, 17:447--474, 1994.

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V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. Journal of Algorithms, vol. 17, 3, pages 447--474, 1994.

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V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. In Proceedings of the 3rd ACM-SIAM Symposium on Discrete Algorithms, pages 157--164, January 1992.

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# V. King, S. Rao, and R. Tarjan, "A Faster Deterministic Maximum Flow Algorithm," Proc. Third ACM Symp. Discrete Algorithms, pp. 157164, 1992.

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V. King, S. Rao, and R. Tarjan. A Faster Deterministic Maximum Flow Algorithm. J. Algorithms, 17:447--474, 1994.

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