Karp, R. (1978). A Characterization of the Minimum Cycle Mean in a Digraph. Discrete Mathematics, 23, 309--311.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....small modifications to value iteration, optimal cycles (and policies) can be found in (n) iterations. As each numbers in the input, but exponential in the binary representation. iteration takes (m) time, this gives an algorithm that ties the only other algorithm with the same run time of (mn) [Kar78, AMO93]. An algorithm described here has an additional interesting property that it is distributed: each vertex performs a simple local computation and need only communicate to its immediate neighbors via its edges, and by (n) iterations, all vertices will know the highest mean. With (n) more ....

....c R(c) where c ranges over all cycles in G. We call the maximum mean and in solving a DMDP problem we are interested in either the problem of computing or finding an optimal cycle with mean (the problems are equivalent) The DMDP problem was shown solvable in O(mn) time by Karp [Kar78]. Later, algorithms were given with run times of O(mn log n) and these algorithms are believed to be faster than (unmodified) Karp s algorithm in practice [YTO91] as Karp s algorithm takes (mn) irrespective of underlying graph. To the best of our knowledge, all the algorithms with known O(mn) ....

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R. M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309--311, 1978.

....has a path to the maximum average reward cycle reachable. Once optimal cycles for each component are located, constructing an optimal policy is not hard. The DMDP problem, also known as the maximum (equivalently the minimum) mean cycle problem, was rst shown to be solvable in O(mn) time by Karp [66, 25, 1]. There are several other algorithms for solving 118 the problem and a recent computational comparison of these algorithms on a variety of graphs families appears in [30] Further background and related work on the problem is given in Section 6.4.1 and Chapter 7. The basic value iteration ....

....weights, such as the minimum cost to time ratio cycle 140 problem. 7.1.5 The Deterministic Average Reward MDP Problem The deterministic average reward MDP problem is a special fractional linear programming problem. It can be solved in O(minfnm;n 1=2 mlog(nC)g) The rst bound is due to Karp [66], and the next is due to Orlin and Ahuja [67] Applications of the problem are numerous and include network ow problems and systems performance analysis. See for example [1, 30, 52] Several other algorithms have been designed for the DMDP. To the best of our knowledge, all the algorithms with ....

R. M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309-311, 1978.

....field of Network Flows, the optimum cost to time ratio cycle problem. We have a directed graph in which edges have a pair of weights # 1 (b) and # 2 (b) and are seeking a cycle p optimizing a ratio, as follows: min ; 8) see, for example, 7] It can be elegantly solved by linear programming [18,33]. To apply this machinery in the present case, we define the beamlet digraph, making a distinction between a beamlet going from v 0 to v 1 and a beamlet going in the opposite direction. We then of course define # 1 and # 2 consistent with (7) Finally, we employ a fast linear programming code ....

R. M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309--311, 1978.

....small modifications to value iteration, optimal cycles (and policies) can be found in (n) iterations. As each numbers in the input, but exponential in the binary representation. iteration takes (m) time, this gives an algorithm that ties the only other algorithm with the same run time of (mn) [Kar78, AMO93]. An algorithm described here has an additional interesting property that it is distributed: each vertex performs a simple local computation and need only communicate to its immediate neighbors via its edges, and by (n) iterations, all vertices will know the highest mean. With (n) more ....

....c R(c) where c ranges over all cycles in G. We call the maximum mean and in solving a DMDP problem we are interested in either the problem of computing or finding an optimal cycle with mean (the problems are equivalent) The DMDP problem was shown solvable in O(mn) time by Karp [Kar78]. Later, algorithms were given with run times of O(mn log n) and these algorithms are believed to be faster than (unmodified) Karp s algorithm in practice [YTO91] as Karp s algorithm takes (mn) irrespective of underlying graph. To the best of our knowledge, all the algorithms with known O(mn) ....

[Article contains additional citation context not shown here]

R. M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309--311, 1978.

....by the addition of an area term, but this becomes a modelling decision, rather than an uncontrolled feature of the energy itself as it is for linear energies. In [12] a different algorithm was proposed for the optimization of this form of energy. This minimum mean weight cycle algorithm [15] was not general, optimizing only a small subset of the possible instances of the energy, and in addition, had no sensible continuum interpretation. This introduced an undesirable dependence on the discretization, which manifested itself in unfortunate behaviour with respect to boundary length. In ....

....Up to an irrelevant factor, # 1, so the denominator is simply counting the edges in the set C. This discrete problem does not have a continuous counterpart, dependent as it is on the discretization. The algorithm is due to Richard Karp, and we refer the reader to the original paper for proofs [15]. First, fix an arbitrary start vertex s V and define the function F k taking each vertex v V to the minimum of the total weight (defined using #) over the set of paths to v from s that consist of exactly k 0 edges. If no path of k edges exists, then define the weight to be #. Then it ....

R. Karp, A characterization of the minimum cycle mean in a digraph, Dis. Math. 23 (1978), 309--311.

....case [16, 5, 18] are more complex than the one we describe in section 4. For the sake of clarity, we restrict ourselves to this case. 4. Algorithmic solution To find the global maximum of the energy in equation (2) or equivalently equation (1) we use a graph algorithm due to Richard Karp [13]. This algorithm finds the minimum mean weight cycle in a directed graph. The algorithm requires no initialization by the user. It is also highly parallelizable, with each pixel able to perform its computations independently, reading from, but never writing to, its neighbours. In the experiment, ....

....as P i2I w#e i # jIj . First, define the function F k #v# taking each vertex v 2 V (V is the vertex set) to the weight of the minimum weight path of length k # 0 to v from an arbitrary start vertex s, and define it to be 1 if no path exists of length k. Thenit can be shown (proof is given in [13]) that the weight # of the minimum mean weight cycle is given by = min v2V max k2#0: #n,1## # Fn #v# , F k #v# n , k # (3) where n = jV j. F k #v# can be computed using the recurrence F k #v#= min #u;v#2E F k,1 #u# w##u; v## F 0 #s#=0 F 0 #v#=1 ; 8v 6= s where E is the edge set ....

R. Karp, "A characterization of the minimum cycle mean in a digraph," Dis. Math., Vol. 23, pp. 309--311, 1978.

....) U i (v T ) 1b) In other words, they just di#er by the constant term U i (v 0 ) U i (v T ) for each player i I. Obviously, such a transformation does not change the set of Nash equilibria in S, and consequently, games g and g U are equivalent. It readily follows e.g. by the results of [15] that for any local cost function f satisfying (#) there exists a potential transformation U , such that fU satisfies (#) Obviously, on its turn (#) implies immediately (#) and therefore, these two assumptions are indeed equivalent. Let us also note that all terminal local cost functions are ....

.... a directed path in G from v 0 to v T of cost at most 1 V is obviously equivalent with the well known NP hard problem (P ) Is there a Hamiltonian directed path in G from v 0 to v T Yet, the shortest path problem is polynomial whenever G contains no directed cycle of negative cost (see [15]) Let us remark that in this case the only player can choose any situation, and hence the only Nash equilibria are the shortest paths in G. Such optimality of Nash equilibria does not hold, in general, for games with multiple players. Let us next consider games of two players, I = 1 , such ....

R.M. Karp (1978), A characterization of the minimum cycle mean in a digraph, Discrete Math., 23, 3, 309-311.

....to reduce critical path of six cycles, the application of rephasing reduces the critical path to two cycles. for efficient calculation of iteration bound was done in theoretical computer science, where the problem was most often studied under the name of minimum (cost to time) ratio cycle problem [22], 27] Currently, the most efficient algorithm is one proposed by Hartmann [15] Reiter [41] was the first to point out the relevance of iteration bound to maximal speed of computation in recursive computations. In the early 1980 s, the iteration bound was rediscovered, and thus named, in the ....

R. Karp, "A characterization of the minimum cycle mean in a digraph," Discrete Math., vol. 23, pp. 309--311, 1978.

.... retiming of a unit delay circuit in O(min VW2Elg(VW) VE )steps, where V is the number of processing elements in the circuit and W is the maximum number of registers on a wire in the circuit, by direct application of graph theoretic algorithms for finding the minimum cycle mean in a graph [11, 20]. We demonstrate how to obtain a minimum clock period retiming of a circuit with arbitrary delays in O(VElg D) steps. The best previously known strongly polynomial algorithm for minimum clock period retiming of synchronous circuitry, unit delay or arbitrary delay, required O(VE lg V) steps [17] ....

....consider the problem: Given a sequential circuit G = V, E, d, w) determine a retiming r such that (Gr) is minimum. We consider unit delay circuitry first. In order to compute the minimum feasible period of the circuit we can use Karp s O(VE) algorithm for finding minimum mean cycles in a graph [11]. Then, using Bellman Ford s shortest paths algorithm on G 1 c we can find a retiming r such that (G, is minimum, according to Theorem 3.1. The overall running time is O(VE) which is an improvement over the best previously known strongly polynomial algorithm by a lg V factor, since it ....

R. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309-311, 1978.

....E Gamma E is acyclic. For each e 2 E there is an associated weight f e . We give an O(jV jjEj) algorithm for the problem of finding the minimumvalue of the ratio of e2C f e to the number of arcs of E in a directed cycle C of D. The algorithm is an extension of the algorithm of Karp [2] for finding the minimum cycle mean of a digraph, which reduces to the case where E = E. Karp and Orlin [3] give O(jV j ) and O(jV jjEj log jV j) algorithms to solve a more general parametric shortest path problem (without the assumption that E Gamma E is acyclic) which give the ....

....algorithm for a class of cyclic staffing problems. We show that our algorithm for computing yields an O(n ) algorithm for the same class of problems. Partially supported by NSF grant DMS 8905645. E be a designated subset of arcs. For each e 2 E there is an associated weight f e . Karp [2] gives an O(jV jjEj) algorithm for calculating the minimum cycle mean in a strongly connected digraph based on the following characterization [3, Theorem 1] min jCj F jV j (v) Gamma F k (v) 1) where Gamma is the collection of directed cycles in D, F k (v) is the minimum ....

R.M. Karp, A characterization of the minimum cycle mean in a digraph, Discrete Mathematics 23 (1978) 309--311.

....The timing formula for a loop statement with a loop bound N models a loop unrolled N times. This approach is exact but is computationally intractable for a large N. In [11] Lim et al. give an efficient approximate loop timing analysis method using a maximum cycle mean algorithm due to Karp [9]. In summary, in order to perform timing analysis within the ETS framework, the following three components should be decided: 1) determination of the PA structure, 2) definition of the operation on PAs, and (3) definition of the pruning condition. Once these three components are determined, the ....

....node represents several nodes before the merging operations are performed, we have to modify the weights between the merged node and other nodes in the merged IDG. Note that our definition of a weight between two nodes means the minimum distance 1 2 3 4 5 6 7 1 [1,19] 2,20] 3,211 [5,231 [7,211 [9,25] 2 [1,11 [2,181 [4,201 [6,201 [8,241 3 [1,17] 3,191 [5,19] 7,231 4 [2,161 [2,16] 6,20] 5 [o, oI [4,41 6 [4,4] M 1 M2 M3 M4 MI [1,19] 3,21] 5,23] M2 [2,18] 4,20] M3 [2,16] Figure 8. A tabular form representation of dis tance bounds. between the issues of first instructions of ....

R.M. Karp. A Characterization of the Minimum Cycle Mean in a Digraph. Discrete Mathematics, 23:309 311, 1978.

....7.1 Let G (V, E, w) be a weighted directed graph. The maximum cycle mean of G, denoted recta(G) is the maximum average weight of an edge in a directed cycle of G. hat is, w(O) Iol : o is a directed cycle of G) We remark that the maximum cycle mean can be computed in polynomial time [14]. To analyze internal synchronization systemsFthe definition of patterns and views is extended so that the fire steps are points with the usual attributes (i.e. Fprocessor of oc currenceFlocal time of occurrenceFand for pattersFreal time of occurrence) We extend the standard bounds mapping ....

R. M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics1'23:309 3111'1978.

....i (v 0 ) U i (v T ) 1b) In other words, they just di er by the constant term U i (v 0 ) U i (v T ) for each player i 2 I. Obviously, such a transformation does not change the set of Nash equilibria in S, and consequently, games g and g U are equivalent. It readily follows e.g. by the results of [15] that for any local cost function f satisfying ( there exists a potential transformation U , such that fU satis es ( Obviously, on its turn ( implies immediately ( and therefore, these two assumptions are indeed equivalent. Let us also note that all terminal local cost functions are ....

.... a directed path in G from v 0 to v T of cost at most 1 jV j is obviously equivalent with the well known NP hard problem (P ) Is there a Hamiltonian directed path in G from v 0 to v T Yet, the shortest path problem is polynomial whenever G contains no directed cycle of negative cost (see [15]) Let us remark that in this case the only player can choose any situation, and hence the only Nash equilibria are the shortest paths in G. Such optimality of Nash equilibria does not hold, in general, for games with multiple players. Let us next consider games of two players, I = f0; 1g, such ....

R.M. Karp (1978), A characterization of the minimum cycle mean in a digraph, Discrete Math., 23, 3, 309-311.

....Our discussion of the min cost flow problem has already started in Section 2.1. Recall that solving the directed augmentation problem simply amounts to finding a negative cost dicycle in a given arc weighted digraph. Since one can detect a negative dicycle in O(minfnm; p nmlog(nW )g) time [9, 15], Algorithm II solves the min cost flow problem with integral data 9 in time O(mlog(nUW ) minfnm; p nmlog(nW )g) Here, n and m denote the number of nodes and arcs of the given network, respectively. Interestingly, among cycle canceling algorithms the running time of the generic Algorithm II is ....

R. M. Karp, A characterization of the minimum cycle mean in a digraph, Discrete Mathematics 23 (1978), 309 -- 311.

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Karp, R. (1978). A Characterization of the Minimum Cycle Mean in a Digraph. Discrete Mathematics, 23, 309--311.

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Richard M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309--311, 1978. (Ref: p. 83.)

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R. Karp. A Characterization of the Minimum Cycle Mean in a Digraph. Discrete Mathematics, 23:309--311, 1978.

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R. M. Karp, "A characterization of the minimum cycle mean in a digraph," Discrete Mathematics, vol. 23, pp. 309--311, 1978.

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R. Karp, "A characterization of the minimum cycle mean in a digraph," Discrete Mathematics, vol. 23, pp. 309--311, 1978.

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Karp R., "A characterization of the minimum cycle mean in a digraph", Discrete Mathematics, North Holland Pub. Co., no. 23, pp 309-311, 1978.

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R. M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Math., 23(3):309--311, 1978.

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R.M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309-311, 1978.

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R.M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309-311, 1978.

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R. M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23:309--311, 1978.

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Karp, R.M.: "A characterization of the minimum cycle mean in a digraph" Discrete Mathematics 23, 1978, 309-311.

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