Alon, N., and Spencer, J. H. The Probabilistic Method, second ed. WileyInterscience Series in Discrete Mathematics and Optimization. New York, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A 1 , A n be a set of bad events with Pr[A i ] p for all i. If each A i is mutually independent of all but at most d of the other events A j and ep(d 1) 1, then with positive probability no bad event occurs. Many applications of the LLL can be found in the literature (see, e.g. [2, 4, 5, 9, 11, 12, 14, 18, 20, 21, 22, 26, 27, 29, 30]) To turn proofs using the Lovasz Local Lemma into efficient algorithms, even random ones, proved to be difficult for many of these applications. In a breakthrough paper [8] Beck presented a method of converting some applications of the Lovasz Local Lemma into polynomial time algorithms (with ....

N. Alon and J. H. Spencer. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1992.

....QLP by de nition, thus value(f ) E[value(f ) Here we have used a probabilistic argument to assert the nonexistence of a combinatorical object with some property, the labeling f , with absolut certainty rather than a probability. This is a simple application of the Probabilistic Method, [1]. As a consequence, if we provide a solution to QLP we could have solved IQLP . But IQLP is clearly NP hard, so is QLP . 4 Semide nite Relaxation and Analysis Our aim is to perturb the matrix W , so that it becomes positive semide nite (a square matrix A is called positive semide nite, A 0, ....

N. Alon and J.H. Spencer. The Probabilistic Method. Wiley Interscience Series in Discrete Mathematics and Optimization, New York Chichester Weinheim Brisbane Singapore Toronto, 2000.

....) We also use the special symbol e to represent a value function whose value is 1 for every state. Thus, e : X R such that e(x) 1 for all x 2 X. The following technical background propositions, simply extending results for MDPs presented in Bertsekas [3] and (for the last proposition) Alon[1], are proven in Appendix B, for completeness. Here, let V ( and V ( be value functions, and i be a policy pair. x) for all x 2 X. Then, for all x 2 X, V ) x) T ) x) and (T A ; B V ) x) T ) x) Proposition 2.3. For any r 2 R, and e the unit value ....

N. Alon, J. H. Spencer, and P. Erd} os, The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wieley and Sons, New York, 1992.

....value function for any i satisfies, at any x 2 X, jV B(x)j V max = R max = 1 ) We also write e for the value function whose value is 1 for every state. We prove the following technical background propositions in the full paper, following analogous proofs for MDPs by others [1] [17]. Here, V ( and V ( are arbitrary value functions, and h i is an arbitrary policy pair. Proposition 1. Suppose V (x) V (x) for all x 2 X. We then have (T V ) x) T for all x 2 X. Also, we have (T B V ) x) T B V ) x) for all x 2 X. Proposition 2. For ....

Noga Alon, Joel H. Spencer, and Paul Erdos, The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wieley and Sons, New York, 1992.

....We show that with weak designs, one can have d much smaller than is possible with the corresponding designs. The weak designs used in our first extractor are constructed using an application of the Probabilistic Method, which we then derandomize using the Method of Conditional Expectations (see [ASE92] and [MR95, Ch. 5] We then apply a simple iteration to these first weak designs to obtain the weak designs used in our second extractor. We also prove a lower bound showing that our weak designs are near optimal. Entropy loss. Since a (k; extractor Ext: f0; 1g Thetaf0; 1g is given k ....

....so that 1. Each set contains exactly one element from each block, and 2. ae Delta (i Gamma 1) Supppose we have S 1 ; S i Gamma1 ae [d] satisfying the above conditions. We prove that there exists a set S i satisfying the required conditions using the Probabilistic Method [ASE92] (see also [MR95, Ch. 5] Let a 1 ; a be uniformly and independently selected elements of B 1 ; B , respectively, and let S i = fa 1 ; a g. We will argue that with nonzero probability, Condition 2 holds. Let Y j;k be the indicator random variable for whether a k 2 S ....

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Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, Inc., 1992.

....This yields two improved extractors: The first extractor extracts a constant fraction of the min entropy. The weak designs underlying this construction are constructed using an application of the 8 Probabilistic Method, which is derandomized using the Method of Conditional Expectations (see [ASE92] and [MR95] ch. 5) A simple iteration is applied to these weak designs to construct new weak designs, which are used to construct an extractor which improves upon previous construction in case we want to extract all the min entropy of the source. The shortcome of these probabilistic ....

....S 1 ; S i Gamma1 , for every new subset S i , the expected value of P j i 2 jS i S j j is smaller (or equal) to aem. Hence, with non zero probability, there exists a set S i satisfying the requirements. The construction is derandomized by the Method of Conditional Expectations (see [ASE92, MR95]) In order to find the i th subset we must compute and store all previous i Gamma 1 subsets (so we can calculate the appropriate conditional expectation) Thus, the algorithm requires time and space polynomial in m even if it is required to output only one specific subset. Hence, the ....

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, Inc., 1992.

....only dense instances makes a problem easier because such graphs inherit most of the good properties of dense random graphs. In this paper we try to analyze the di#culty of approximating some of the above problems for random sparse graphs (drawn from the standard G n,p model with p = c n [5, 2]) and expanders. A natural question is to ask whether there is any relation between the approximability of the maximization version of the problems, and whether we can infer some consequence for the minimization version from our understanding of these maximization versions. It thus makes sense to ....

....Theorem 1 also gives an explicit construction of disperser graphs. 4 Theorem 2. A constant # exists such that for any # 0, for any n and any d # # # 2 , an # disperser graph with n vertices and maximum degree at most d can be constructed in poly(n) time. Definition 6 (Random sparse graphs [5, 2]) We consider the standard class of random graphs G n,p which have n nodes and each potential edge exists with probability p. Although the random sparse graphs considered in this paper are expected to be non connected, with high probability they have good mixing properties. Lemma 1 (Cherno# ....

N. Alon and J.H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York, 1992. With an appendix by P. Erdos, A Wiley-Interscience Publication.

....arbitrary ds=Re relation routing problem. Because of the definition of R, there exists a simple path collection for any such problem with congestion at most R ds=Re and dilation at most R. Consider now the case that log R s R. Then we want to show with the help of the Lovasz Local Lemma (see [AES92], p.55) that there exists a simple path collection PG in H with congestion O(s) and dilation at most R. Lemma 4.2 (Lov asz) Let A 1 ; Am be a set of bad events, each A i occuring with probability at most p and depending on at most b other events in fA 1 ; Am g. If ep(b 1) 1, ....

N. Alon, P. Erdos, J. Spencer. The Probabilistic Method. Wiley Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, 1992.

....graph can be constructed deterministically in time poly(N) 4 with the corresponding designs. The weak designs used in the first extractor of Theorem 1 are constructed using an application of the Probabilistic Method, which we then derandomize using the Method of Conditional Expectations (cf. [ASE92] and [MR95, Ch. 5] We then apply a simple iteration to these first weak designs to obtain the weak designs used in the second extractor. We also prove a lower bound showing that our weak designs are near optimal. The second improvement is achieved by using a specific error correcting code rather ....

....set contains exactly one element from each block, and 2. P j i 2 jS i S j j ae Delta (i Gamma 1) 13 Supppose we have S 1 ; S i Gamma1 ae [d] satisfying the above conditions. We prove that there exists a set S i satisfying the required conditions using the Probabilistic Method [ASE92] (see also [MR95, Ch. 5] Let a 1 ; a be uniformly and independently selected elements of B 1 ; B , respectively, and then let S i = fa 1 ; a g. We will argue that with nonzero probability, Condition 2 holds. Let Y j;k be the indicator random variable for the event ....

[Article contains additional citation context not shown here]

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, Inc., 1992.

....A 1 ; A n be a set of bad events with Pr[A i ] p for all i. If each A i is mutually independent of all but at most d of the other events A j and ep(d 1) 1, then with positive probability no bad event occurs. Many applications of the LLL can be found in the literature (see, e.g. [2, 4, 5, 9, 11, 12, 14, 18, 20, 21, 22, 26, 27, 29, 30]) To turn proofs using the Lovasz Local Lemma into efficient algorithms, even random ones, proved to be difficult for many of these applications. In a breakthrough paper [8] Beck presented a method of converting some applications of the Lovasz Local Lemma into polynomial time algorithms (with ....

N. Alon and J. H. Spencer. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1992.

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Alon, N., and Spencer, J. H. The Probabilistic Method, second ed. WileyInterscience Series in Discrete Mathematics and Optimization. New York, 2000.

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Alon, N., and Spencer, J. H. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. New York, 1992. With an appendix by Paul Erd}os.

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N. Alon and J. H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], New York, second edition, 2000. With an appendix on the life and work of Paul Erdos.

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N. Alon and J. H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, second edition, 2000. 4

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Noga Alon and Joel H. Spencer, The probabilistic method, second ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience [John Wiley & Sons], New York,

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N. Alon, J. Spencer: The Probabilistic Method, Wiley Interscience Series in Discrete Mathematics and Optimization (1992).

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Noga Alon and Joel H. Spencer, The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1992, with an appendix by Paul Erdos, a Wiley-Interscience Publication.

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N. Alon and J. H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience

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N. Alon and J. Spencer. The Probabilistic Method. Wiley--Interscience series in discrete mathematics and optimization. John Wiley & Sons, second edition, 2000.

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N. Alon and J. H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], New York, second edition, 2000. With an appendix on the life and work of Paul Erd}os.

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N. Alon and J. H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], New York, second edition, 2000. With an appendix on the life and work of Paul Erdos.

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N. Alon and J. Spencer, The probabilistic method, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, New York, 1992. 3.2.3

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Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. WileyInterscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, Inc., 1992.

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Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, Inc., 1992.

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N. Alon, J. Spencer, P. Erd}os, The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons (1992).

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