Karp, R. M.: Probabilistic recurrence relations. J. ACM 41 (1994) 1136-1150.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....position changes in discrete time intervals. If the particle is currently at position s it moves to position s X where X is a random variable ranging over the integers 1; n s such that E[X] g(n s) The following result due to Karp, Upfal and Widgerson was rst stated in [10] see also [9] for additional information on probabilistic recurrences) Lemma 7 (Karp, Upfal, Widgerson [10] Let T be the random variable denoting the number of steps in which the particle reaches the position n. Then E(T ) 1 dx=g(x) We can use Lemma 7 to analyze greedy routing when r 1. More ....

R. Karp, \Probabilistic Recurrence Relations", in Proceedings of 23rd Annual Symposium on Theory of Computing, pages 190-197, 1991.

....in discrete time intervals. If the particle is currently at position s it moves to position s X where X is a random variable ranging over the integers 1; n Gamma s such that E[X] g(n Gamma s) The following result due to Karp, Upfal and Widgerson was first stated in [10] see also [9] for additional information on probabilistic recurrences) Lemma 7. Karp, Upfal, Widgerson [10] Let T be the random variable denoting the number of steps in which the particle reaches the position n. Then E(T ) dx=g(x) We can use Lemma 7 to analyze greedy routing when r 1. More ....

R. Karp, "Probabilistic Recurrence Relations", in Proceedings of 23rd Annual Symposium on Theory of Computing, pages 190-197, 1991.

....side is the I O cost for sampling and partitioning, the second term is the I O cost for sorting the samples, and the last term is for the recursive calls. In the recurrence the terms are actually random variables. It suffices to use Karp s method for solving probabilistic recurrence relations [21] to get the optimal solution = O( log . with high probability. The distribution approach used here is different from those of the distribution sort algorithms for the various I O and memory hierarchy modds [3,27,37,38] and the distribution sweeping algorithms discussed in Section 2, but it has ....

R. M. Karp, "Probabilistic Recurrence Relations," Proc. 2Srd ACM STOC (1991), 190 197.

....side is the I O cost for sampling and partitioning, the second term is the I O cost for sorting the samples, and the last term is for the recursive calls. In the recurrence the [q[ terms are actually random variables. It suffices to use Karp s method for solving probabilistic recurrence relations [21] to get the optimal solution T(z, O(ulog ) with high probability. The distribution approach used here is different from those of the distribution sort algorithms for the various I O and memory hierarchy models [3,27,37] but has the same asymptotic I O complexity. In the distribution ....

R. M. Karp, "Probabilistic recurrence relations," Proc. 23rd ACM STOC, New Orleans, LA (1991).

.... Elementary calculations show that the probability that an edge colours itself at each round is never less than a constant of value e (1 o(1) It follows by well known results on probabilistic recurrence relations that with high probability every edge is coloured within O(log n) rounds [6, 15]. Notation When we write a b, we mean a = b(1 o(1) The set f1; 2; ng will be denoted by [n] 3 A Large Deviation Inequality A key ingredient of our proof is a large deviation inequality for functions of independent random variables, which was recently developed by the second author. ....

R.M. Karp, Probabilistic Recurrence Relations, 23rd STOC, 1991, 190--197.

....we shall refer to this as the trivial algorithm. The trivial algorithm always computes a valid colouring regardless of the composition of the initial lists, and does so in O(log n) rounds with high probability that is, with probability approaching 1 as the number of vertices increases [10, 13, 4]. It is apparent that the trivial algorithm is distributed, since each vertex only relies on information from the neighbouring vertices. The well known distributed algorithm for the same problem given by Luby [15] amends the trivial algorithm in the following way: at the beginning of each round ....

....the algorithm sufficiently to show that with high probability there exist a round i = O(k) such that, for every vertex u, a i (u) deg i (u) 1: 1) This will occur after the algorithm has switched to its trivial phase. Since the behaviour of the trivial algorithm in this situation is known [10, 13, 4], 5 we can then immediately conclude that the trivial algorithm will with certainty complete the colouring and it will do so within O(log n) rounds with high probability. As stated in the introduction, with a little bit more work it is possible to show that the running time is actually O(k log ....

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R.M. Karp, Probabilistic recurrence relations, in Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC 91), pages 190--197, New Orleans.

....average size of these indices is a fraction of n. Our reference source here is the Master Theorem MT, for short as it can be found in [16] Other references in this subject include the (classic) Master Theorem [1, 2, 3] several improvements [10, 17, 18, 19] as well as other related results [11]. Assume that we have the recurrence Fn = t n W Delta FSn , with t n 0 and Sn = Z Delta n O(1) for some 0 Z 1. If Fn describes the cost to solve with a certain algorithm a problem of size n, then t n customarily called toll function is the cost of the divide and combine steps, W ....

R.M. Karp. Probabilistic recurrence relations. Journal of the ACM, 41(6):1136--1150, November 1994.

....a # j = 1 , then Erdos [103] showed that c(n) # cn # as n ## (9.80) for a positive constant c. Although the recurrence (9.79) is similar to that of Eq. 9.76) the results are di#erent (no oscillations can occur for a recurrence given by Eq. 9.79) and the methods are dissimilar. Karp [221] considers recurrences of the type T (x) a(x) T (h(x) where x is a nonnegative real variable, a(x) # 0, and h(x) is a random variable, 0 # h(x) # x, with m(x) being the expectation of h(x) Such recurrences arise frequently in the analysis of algorithms, and Karp proves several theorems ....

R. M. Karp, Probabilistic recurrence relations, Proc. 23rd ACM Symp. Theory of Computing, 1991, pp. 190--197.

.... O(1) # 30.3 n, and the total expected time is O(n) Proof: The time can be expressed as a random variable which satisfies a probabilistic recurrence T (S) # 371 S 64 T (R) 8) where R is a random variable with expected size (1 49 256) S . By the theory of probabilistic recurrences [15], the expected value of T (S) can be found using the deterministic recurrence T (n) 371n 64 T ( 1 49 256)n) 9) which solves to the formula given in the theorem. # Although the constant factor in the analysis of this algorithm is disappointingly large, we believe this algorithm should be ....

R. M. Karp. Probabilistic recurrence relations. J. ACM 41 (1994) 1136--1150.

....# 30.3 n, and the total expected time is O(n) Proof: The time can be expressed as a random variable which satisfies a probabilistic recurrence T (S) # 371 S 64 T (R) 10) where R is a random variable with expected size (1 49 256) S . By the theory of probabilistic recurrences [16], the expected value of T (S) can be found using the deterministic recurrence T (n) 371n 64 T ( 1 49 256)n) 11) which solves to the formula given in the theorem. # 15 Although the constant factor in the analysis of this algorithm is somewhat disappointingly large, we believe this ....

R. M. Karp. Probabilistic recurrence relations. J. ACM 41 (1994) 1136-- 1150.

....s X s2S ff 2jSj Delta s min ae ffl 8 P ae ; ff 2k oe k X s=1 Delta s min ae ffl 8 P ae ; ff 2k oe ffl Phi: Since ff 2 Gamma1 ffl Gamma1 ln(2mffl Gamma1 ) the claimed bound on the expected decrease of Phi follows. We use a result due to Karp [16] to analyze the number of iterations used by the randomized version of Improve Packing. Let ffi Phi denote the ratio of upper and lower bounds on the potential function Phi during a single execution of Improve Packing. Each iteration of the algorithm when P2 is not satisfied is expected to ....

.... Delta s min ae ffl 8ae 0 ; ff 2k oe k X s=1 Delta s min ae ffl 8ae 0 ; ff 2k oe ffl Phi: Since ff = Omega Gamma ffl Gamma1 Gamma1 log(mffl Gamma1 ) we get the claimed decrease in Phi. To analyze the number of iterations, we once again apply the result of Karp [16]. This implies that the randomized version of Improve Cover is expected to terminate in O( P ae )ffl Gamma3 log(mffl Gamma1 ) kffl Gamma1 0 ) iterations, and is a factor of ffl Gamma1 faster if the initial solution is 6ffl optimal for ffl 1=12. We use this randomized ....

R. M. Karp. Probabilistic recurrence relations. In Proceedings of the 23rd Annual ACM Symposium on the Theory of Computing, pages 190--197, 1991.

....all flows simultaneously. Roughly speaking, we loose a factor of k due to the fact that we update only a single flow, but gain a factor of k back due to the fact that we can use a larger oe. But now our oracle runs k times faster for the same improvement. Using an analysis technique due to Karp [11], this leads to a conclusion that the algorithm will terminate in expected O(ffl Gamma3 k) iterations, where the running time of each iteration is dominated by the computation of a single commodity minimum cost flow. Thus, we have the following: Theorem 3.1 Min cost multicommodity flow can ....

R.M. Karp. Probabilistic recurrence relations. In Proc. 23rd Annual ACM Symposium on Theory of Computing, pages 190--197, 1991.

.... Elementary calculations show that the probability that an edge colours itself at each round is never less than a constant of value e 2 (1 o(1) It follows by well known results on probabilistic recurrence relations that with high probability every edge is coloured within O(log n) rounds [6, 15]. Notation When we write a # b, we mean a = b(1 o(1) The set 1, 2, n will be denoted by [n] 3 A Large Deviation Inequality A key ingredient of our proof is a large deviation inequality for functions of independent random variables, which was recently developed by the second ....

R.M. Karp, Probabilistic Recurrence Relations, 23rd STOC, 1991, 190--197.

....needed are upper bounds on the expected values of the h i (n) Joint work with Marek Karpinski. The problem here is to find an upper bound for T (n) with little information about the distribution of h(n) h 1 (n) h k (n) Karp studied Equation (1) with the following assumptions [1]: i) a(n) 0, ii) h(n) is a random variable over [0; n] iii) E[h(n) m(n) where 0 m(n) n, and m(n) and m(n) n are non decreasing. Under these conditions, if one defines u(n) to be the least non negative solution of (n) a(n) m(n) we have two results according to the function ....

Karp (Richard M.). -- Probabilistic recurrence relations. In Proceedings of the Twenty Third Annual ACM Symposium on Theory of Computing. pp. 190--197. -- ACM Press, 1991.

....T in which each bucket has no more than b distinct keys therefore produces the desired grouped output. Let j 0 be the number of distinct keys hashed into some new bucket from a bucket with j distinct keys. The properties of universal hashing [6] show that E[j 0 ] j=b. Theorem 1. 1 of Karp [13] thus shows that Pr [T blog b Gc c 1] G=b blog b Gc c for any positive integer c. Therefore, with high probability, O(scan(N) log b G) I Os suffice to group I. We use global compaction and percolation to optimize space usage. 6 Conclusion Our functional approach produces external graph ....

R. M. Karp. Probabilistic recurrence relations. J. ACM, 41(6):1136--50, 1994.

....the total expected number of terms added to h is O(mn d 1 ) Moreover, the probability that more than 2m(s 1) calls to Reduce1 will be required before h contains all the terms of h is bounded by e Gammas . To see this, we may apply Karp s method of probabilistic recurrence relations. In (Karp, 1991), recurrence relations of the form T (x) a(x) T (r(x) are used to analyze the running 16 D. ANGLUIN AND D. SLONIM time of recursive randomized algorithms. In this relation, a(x) is a nonnegative real function of x corresponding to the amount of effort expended on the original problem, and ....

....x] corresponding to the size of the next subproblem to solve recursively. Let (x) be an upper bound on the expectation of random variable r(x) and let u(x) be the least nonnegative solution to the deterministic equivalence relation (x) a(x) x) Then the following theorem (Theorem 1 in (Karp, 1991)) is useful in showing the bound stated above: Theorem 6 (Karp) Suppose there is a constant d such that a(x) 0; x d and a(x) 1; x d. Let c t = minfx j u(x) tg. Then, for every positive real x and every positive integer w, Pr[T (x) u(x) w] x) x w Gamma1 (x) c u(x) To ....

Karp, R.M. (1991). Probabilistic recurrence relations. In Proceedings of the Twenty Third Annual ACM Symposium on Theory of Computing, (pp. 190-197). New Orleans, LA: ACM Press.

....and by the SERC Grant GR E 68297 2 Dept. of Computer Science, University of Karlsruhe, and International Computer Science Institute, Berkeley, California. 1 Introduction Two classes of probabilistic recurrence relations occur frequently in the analysis of divide and conquer algorithms cf. Ka 91] A problem instance z of size x is divided into subproblems h 1 (z) h k (z) On a sequential computer the subproblems has to be solved one after another. Therefore the running time of a divide and conquer algorithm is the solution of a recurrence of the form T (z) a(z) T (h 1 (z) ....

.... max(T (h 1 (z) T (h k (z) 2) We consider the case where the h i (z) are random variables. In this case the running time T (z) is also a random variable, and we estimate bounds on its probability distribution for both cases, 1) and (2) This work is an extension of Karp s results [Ka 91] and solves the open questions of [Ka 91] Throughout the paper we use the following notations and assumptions: ffl a(z) is a function on the size of z and does not depend on the distribution of z. We will denote this fact by writing a(x) instead of a(z) 3 ffl The size(h i (z) are random ....

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Karp, R. M., Probabilistic Recurrence Relations, Proc. 23 rd ACM STOC (1991), pp. 191-197.

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Karp, R. M.: Probabilistic recurrence relations. J. ACM 41 (1994) 1136-1150.

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Richard M. Karp. Probabilistic recurrence relations. In Proceedings of the twenty-third annual ACM symposium on Theory of computing, pages 190--197. ACM Press, 1991.

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R. M. Karp, Probabilistic recurrence relations, in proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC 91), pages 190--197, New Orleans, Louisiana.

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Richard M. Karp, "Probabilistic Recurrence Relations", Proc. 23 ACM Symp. on The Theory of Computing , pp. 190--197, 1991.

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R.M. Karp, Probabilistic recurrence relations, in Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC 91), pages 190--197, New Orleans.

No context found.

R. M. Karp, Probabilistic recurrence relations, in proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC 91), pages 190--197, New Orleans, Louisiana.

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KARP, R. (1991), Probabilistic Recurrence Relations, in "Proceedings of the 23rd Symposium on Theory of Computing", pp 190--197.

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R. Karp, "Probabilistic Recurrence Relations", 23rd STOC, pp. 190--197, 1991.

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