Karmarkar, N., Karp, R. M., Lueker, G. S., and Odlyzko, A. M. #1986#. Probabilistic analysis of optimum partitioning. Journal of AppliedProbability, 23:626#645.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and response time goals are achieved through the exploitation of metadata associating clusters with semantic categories. The inter cluster load balancing problem is NPcomplete thus we develop a polynomial time incomplete greedy algorithm, which utilizes ideas from number partitioning algorithms [11, 13, 8] to solve our problem. In a companion paper [18] we take a different approach and utilize the fairness index of [10] as a metric for load balancing. In that paper we also show that our solution is robust with respect to different distributions of the popularity of semantic categories, varying ....

....popularities. This can be done, for example, by minimizing the following quantity: k j j i k i N p N p 1 1 ) min We have experimented with a greedy algorithm, called MinDiff, which solves this minimization problem by using similar ideas as in known number partitioning algorithms [11, 13, 8]. The NP hardness result for BALANCED PARTITION is due to Apostolos Dimitromanolakis. In MinDiff initially all clusters are empty. Then MinDiff considers each semantic category s in turn, and assigns it to the cluster of nodes which minimizes the total difference of balanced popularities among ....

N. Karmarkar, R. Karp, J. Lueker, A. Odlyzko, "Probabilistic Analysis of optimum partitioning" , Journal of Applied Probability, Vol.23, pp. 626-645, 1986.

.... by Johnson et al. to evaluate simulated annealing (1991) Korf to evaluate his improvement to limited discrepancy search (1996) and Walsh to evaluate depth bounded discrepancy search (1997) To encourage difficult search trees by reducing the chance of encountering a perfectly even partitioning (Karmarkar et al. 1986), we used instances with 64 25 digit numbers or 128 44 digit numbers. Common Lisp, which provides arbitrary precision integer arithmetic, was used to implement the algorithms. All results are normalized as if the original numbers were between 0 and 1. To better approximate a normal distribution, ....

Karmarkar, N.; Karp, R. M.; Lueker, G. S.; and Odlyzko, A. M. 1986. Probabilistic analysis of optimum partitioning. Journal of Applied Probability 23:626--645.

.... by Johnson et al. to evaluate simulated annealing (1991) Korf to evaluate his improvement to limited discrepancy search (1996) and Walsh to evaluate depth bounded discrepancy search (1997) To encourage difficult search trees by reducing the chance of encountering a perfectly even partitioning (Karmarkar et al. 1986), we used instances with 64 25 digit numbers or 128 44 digit numbers. Common Lisp, which provides arbitrary precision integer arithmetic, was used to implement the algorithms. All results are normalized as if the original numbers were between 0 and 1. To better approximate a normal distribution, ....

Karmarkar, N.; Karp, R. M.; Lueker, G. S.; and Odlyzko, A. M. 1986. Probabilistic analysis of optimum partitioning. Journal of Applied Probability 23:626--645.

....digits and Log10(Difference) 2 3 4 5 6 Nodes Generated 1,200,000 900,000 600,000 300,000 Standard Variable Choice 2 Variable Choice Figure 1: Searching the greedy spaces using random sampling. many numbers, the probability of a partitioning with a difference of 0 and 1 increases [Karmarkar et al. 1986] . This makes the tree search easier, as the search can terminate once such a partitioning is found. To encourage difficult search trees by reducing the chance of encountering a perfectly even partitioning [Karmarkar et al. 1986] I used instances with 64 25 digit numbers or 128 44 digits ....

....probability of a partitioning with a difference of 0 and 1 increases [Karmarkar et al. 1986] This makes the tree search easier, as the search can terminate once such a partitioning is found. To encourage difficult search trees by reducing the chance of encountering a perfectly even partitioning [Karmarkar et al. 1986] , I used instances with 64 25 digit numbers or 128 44 digits numbers. 1 (Common Lisp, which provides arbitrary precision integer arithmetic, was used to implement the algorithms. Results were normalized as if the original numbers had been between 0 and 1. The Greedy Representation We present ....

Narenda Karmarkar, Richard M. Karp, George S. Lueker, and Andrew M. Odlyzko. Probabilistic analysis of optimum partitioning. Journal of Applied Probability, 23:626--645, 1986.

.... Johnson et al. to evaluate simulated annealing [1991] Korf to evaluate his improvement to limited discrepancy search [1996] and Walsh to evaluate depth bounded discrepancy search [1997] To encourage difficult search trees by reducing the chance of encountering a perfectly even partitioning [Karmarkar et al. 1986] , we used instances with 64 25 digit numbers or 128 44 digit numbers. 1 (Common Lisp, which provides arbitrary precision integer arithmetic, was used to implement the algorithms. Results were normalized as if the original numbers had been between 0 and 1. To better approximate a normal ....

Narenda Karmarkar, Richard M. Karp, George S. Lueker, and Andrew M. Odlyzko. Probabilistic analysis of optimum partitioning. Journal of Applied Probability, 23:626--645, 1986.

....E[ff 2;n ] O(1) O(n Gamma1 ) O(n Gammac log n ) O(ae n ) for some c 0, 0 ae 1. The O(ae n ) result for optimal scheduling is not discussed here (see Coffman and Lueker (1991) Section 4.3) and in fact remains a conjecture. The strongest results of this type are those of Karmarkar et al. 1986) who showed that the median of the final difference, 2ff 2;n , is bounded by a constant times n=2 n . 4. Policy Free Error Asymptotics From (2.7) it is clear that the relative size of the error ff m;n compared to the makespan L m;n itself is asymptotically negligible as n 1 for the greedy ....

Karmarkar, N., Karp, R. M., Lueker, G. S., and Odlyzko, A. M (1986), "Probabilistic Analysis of Optimum Partitioning," J. Appl. Prob., 23(3):626--645.

....each with 128 44 digit numbers. ulated annealing [1991] Korf to evaluate his improvement to limited discrepancy search [1996] and Walsh to evaluate depth bounded discrepancy search [1997] To encourage difficult search trees by reducing the chance of encountering a perfectly even partitioning [Karmarkar et al. 1986] , we used instances with 64 25 digit numbers or 128 44 digits numbers. Common Lisp, which provides arbitrary precision integer arithmetic, was used to implement the algorithms. To better approximate a normal distribution, the logarithm of the partition difference was used as the leaf cost. The ....

Narenda Karmarkar, Richard M. Karp, George S. Lueker, and Andrew M. Odlyzko. Probabilistic analysis of optimum partitioning. Journal of Applied Probability, 23:626--645, 1986.

....4 3 1 0 3 1 0 0 0 0 0 0 0 2 Repeated differencing Two coloring based on differencing steps u = 2 Figure 1: Illustration of the KK heuristic. Since the publication of the KK algorithm (Karmarkar and Karp, 1982) no superior deterministic heuristic for Number Partitioning has been developed. Indeed, Karmarkar et al. 1986) later declared: The expected residue u produced by the KK algorithm] was a great improvement over other results, but] is still much greater than the optimum difference which is shown in this paper to be likely. It would be very interesting, though possibly very difficult, to improve upon that ....

....For a given problem instance size n and probabilities p lo 1 and p hi 1, we aim to find some u lo and u hi such that for large n, Prfu opt u lo g p lo and Prfu opt u hi g p hi , where u opt is the optimum residue. Both are easily derived from asymptotic probability bounds derived by Karmarkar et al. 1986). The first of these bounds, p lo , is an upper bound on the probability that u opt u lo . It is assumed that the elements of A are uniformly distributed between 0 and 1. p lo j 2 n u lo s 24 n . The other bound, p hi , is an upper bound on the probability that u 0 opt u hi , where ....

Karmarkar, N., Karp, R. M., Lueker, G. S., and Odlyzko, A. M. (1986). Probabilistic analysis of optimum partitioning. Journal of Applied Probability, 23:626--645.

....a partition is perfect in a fraction, 1=2) log 2 (l) that is, 1=l) of the 2 n possible partitions. The expected number of perfect partitions, hSoli is therefore simply, hSoli = 2 n l : The exact analysis of number partitioning problems has been considerably more difficult. Karmarkar et al. [11] have determined bounds on the probability distribution for the optimum partition for a bag of real numbers drawn from the interval [0; 1] Using some complicated analysis based on second moments, they showed that the size of the median optimal partition is Theta( p n=2 n ) but were unable ....

....a critical value of this parameter [7] Korf (personal communication) has predicted that a phase transition occurs when the median optimal difference is 1, and that this coincides with a peak in search cost. Using the asymptotic value for the median optimal difference due to Karmarkar et al. [11], Korf suggested that the phase transition occurs when p n:l 2 n = 1. This agrees asymptotically with = 1. 4 Phase transition To determine the critical value of , we plot in Figure 4 the probability that a bag with an even sum has a perfect partition against for n from 6 to 30, and log 2 ....

N. Karmarkar, R. Karp, J. Lueker, and A. Odlyzko. Probabilistic analysis of optimum partitioning. Journal of Applied Probability, 23:626--645, 1986.

....3 1 0 3 1 0 0 0 0 0 0 0 2 Repeated differencing Two coloring based on differencing steps u = 2 Figure 1: Illustration of the KK heuristic. Since the publication of the KK algorithm (Karmarkar and Karp, 1982) no superior deterministic heuristic for Number Partitioning has been developed. Indeed, Karmarkar et al. 1986) later declared: The expected residue u produced by the KK algorithm] was a great improvement over other results, but] is still much greater than the optimum difference which is shown in this paper to be likely. It would be very interesting, though possibly very difficult, to improve upon that ....

....For a given problem instance size n and probabilities p lo 1 and p hi 1, we aim to find some u lo and u hi such that for large n, Prfu opt u lo g p lo and Prfu opt u hi g p hi , where u opt is the optimum residue. Both are easily derived from asymptotic probability bounds derived by Karmarkar et al. 1986). The first of these bounds, p lo , is an upper bound on the probability that u opt u lo . It is assumed that the elements of A are uniformly distributed between 0 and 1. p lo j 2 n u lo r 24 n . The other bound, p hi , is an upper bound on the probability that u 0 opt u hi , where ....

Karmarkar, N., Karp, R. M., Lueker, G. S., and Odlyzko, A. M. (1986). Probabilistic analysis of optimum partitioning. Journal of Applied Probability, 23:626--645.

....in a finite set A and a magnitude m(a) 2 [0; 1] for each a 2 A. The optimal solution s to a given 2.2 Previous Work 3 problem instance (A; m) is a subset A 0 A such that the difference cost(s) j X a2A 0 m(a) Gamma X a2A GammaA 0 m(a)j is as small as possible. 3 Karmarkar et al. [10] have shown that the median of the distribution of expected costs of optimal solutions shrinks in O( p n=2 n ) Number partitioning can be understood as the optimization problem implied by the classic NP complete partition decision problem (sp12, p. 223 in Garey and Johnson [2] partition ....

Narenda Karmarkar, Richard M. Karp, George S. Lucker, and Andrew M. Odlyzko. Probabilistic analysis of optimum partitioning. Journal of Applied Probability, 23:626--645, 1986. REFERENCES 75

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Karmarkar, N., Karp, R. M., Lueker, G. S., and Odlyzko, A. M. #1986#. Probabilistic analysis of optimum partitioning. Journal of AppliedProbability, 23:626#645.

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N. Karmarkar, R. Karp, J. Lueker, and A. Odlyzko. Probabilistic Analysis of Optimum Partitioning. J. Appl. Prob.,

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KARMARKAR, N., KARP, R. M., LUEKER, G. S., and ODLYZKO, A. M. "Probabilistic Analysis of Optimum Partitioning." Journal of Applied Probability. Vol. 23, pp. 626--645, 1986.

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