N. Alon, J.H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley & Sons, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....e . ntro uction The success of the probabilistic method illustrates the effectiveness of probabilistic techniques in handling combinatoric dependencies, even in non probabilistic settings [1, 17]. Although the probabilistic method yields non constructive proofs, these proofs can often be derandomized by the method of conditional probabilities [14, 17, 1] Thus, the probabilistic method has played a role in the design and analysis of algorithms. One of the main areas of application is the ....

.... method illustrates the effectiveness of probabilistic techniques in handling combinatoric dependencies, even in non probabilistic settings [1, 17] Although the probabilistic method yields non constructive proofs, these proofs can often be derandomized by the method of conditional probabilities [14, 17, 1]. Thus, the probabilistic method has played a role in the design and analysis of algorithms. One of the main areas of application is the design and analysis of combinatorial approximation algorithms [15, 14, 3] Typically an approximation algorithm derived by the method begins by solving a linear ....

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, New York, 1992.

....erc cs.uga.edu Herbert S. Wilf Department of Mathematics University of Pennsylvania Philadelphia, PA 19104 6395 wilf math.upenn.edu Dedicated to the memory of Gian Carlo Rota Abstract We answer a question posed by Lampert and Slater [7] Consider a sequence of real numbers q n in the interval [0, 1] defined by q 0 =0,q 1 = 1, and, for n # 1, q n 1 equals an average of preceding terms in the sequence. The weights used in the average are provided by a triangular array p n,k of probabilities whose row sums are 1. What is the limiting behavior of a sequence q n so defined For the ....

....and the sequence q n exhibits oscillatory behavior up to a certain computable point, then it will exhibit oscillatory behavior from then on. We carry out the computations necessary to verify that the Lampert Slater sequence satisfies the hypotheses of the latter theorem. A result on martingales [1] is used to prove the close concentration of the weights p n,k . AMS MOS Subject Classification (1990) 05A16, 60C05, 60G46 1 1 Introduction In [7] the following question is raised. Begin with n players, and repeat the following knockout procedure while there remain two or more players. ....

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Noga Alon, Joel H. Spencer, and Paul Erdos, The Probabilistic Method, John Wiley & Sons, Inc., New York, 1992. 17

....2 Definitions This section contains the definitions we need from first order logic and finite model theory. A more thorough treatment of first order logic can be found in Enderton [4] of finite model theory in Ebbinhaus and Flum [3] and of the probabilistic method in Alon, Spencer, and Erdos [1] . We concentrate on first order logic over the basic operations #,U, That is, we are interested in sentences made up of = equality) # (linear order) U (an unary predicate) the binary connectives # (disjunction) and # (conjunction) negation) and the first order quantifiers ....

....it could be dense. By the latter, we mean that between any 2 Z chains, there s another. When n 1 # p(n) # n 1 2 , almost surely isolated 1 s occur. Using the notation above, we have E(X) ##as n ##. Since all the events are independent, Var[X] # E[X] By the Second Moment Method (see [1] , chapter 4 for details) Pr[X =0] # Var [ X ] E [ X ] 2 # E X ] E [ X ] 2 = 1 E X # 0 the electronic journal of combinatorics 4 (1997) #R23 7 [000 ) 000] Figure 1: A model of T 0 [00 ) 00100 ) 00100 ) z a Z chain (00100 ) 00100 ) 00] Figure 2: A model of T 1 Thus, ....

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, Inc, New York, 1992.

....number of 1 s that occur (i.e. the total number of elements for which the unary predicate holds) Then, E(X i ) p(n) and by linearity of expectation, E(X) X i E(X i ) np(n) We have E(X) 1 as n 1. Since all the events are independent, Var[X ] E[X ] By the Second Moment Method (see [1], chapter 4 for details) Pr[X = 0] Var[X] E[X] 2 E[X] E[X] 2 = 1 E[X] 0 Thus, Pr[X 0] 1. So, almost surely, arbitrarily many 1 s occur. We can write this in first order logic as a schema of sentences ff r , each of which states there is at least r 1 s : ff r : 9x 1 : x r ....

....of the unary predicate of U 2 . By Theorem 5, Duplicator has a winning strategy for any q move EF game played on M 1 and M 2 . So, for any move of Spoiler in such a interval, Duplicator has a winning move. This gives U 1 j t U 2 . a Using the Janson Inequalities The Janson Inequalities (see [1], chapter 8 for more details) says that events that are mostly independent sometimes have probability nearly equal to the truly independent case. We will use these inequalities to give the limiting probability that a sequence of pairs a are the only occuring of length up to t. Lemma 1 Let c be ....

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, Inc, New York, 1992.

....e . 3 . ntro uction The success of the probabilistic method illustrates the effectiveness of probabilistic techniques in handling combinatoric dependencies, even in non probabilistic settings [1, 17]. Although the probabilistic method yields non constructive proofs, these proofs can often be derandomized by the method of conditional probabilities [14, 17, 1] Thus, the probabilistic method has played a role in the design and analysis of algorithms. One of the main areas of application is the ....

.... method illustrates the effectiveness of probabilistic techniques in handling combinatoric dependencies, even in non probabilistic settings [1, 17] Although the probabilistic method yields non constructive proofs, these proofs can often be derandomized by the method of conditional probabilities [14, 17, 1]. Thus, the probabilistic method has played a role in the design and analysis of algorithms. One of the main areas of application is the design and analysis of combinatorial approximation algorithms [15, 14, 3] Typically an approximation algorithm derived by the method begins by solving a linear ....

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, New York, 1992.

.... finding a deterministic version of the skeletal graph G s used in step 3 of the construction would imply deterministically constructing a graph with no clique and no independent set larger than O(logn) Finding a polynomial time procedure for this problem has been a long standing open problem [1]. It appears that while this problem is not solved, major changes to our construction would be needed to make it fully deterministic (and polynomial time) What is the true performance guarantee of the randomized and polynomially amplified version of the algorithm Our result gives only a lower ....

Noga Alon, Joel Spencer, and Paul Erdos. The Probabilistic Method. Wiley, 1992.

....2 Definitions This section contains the definitions we need from first order logic and finite model theory. A more thorough treatment of first order logic can be found in Enderton [4] of finite model theory in Ebbinhaus and Flum [3] and of the probabilistic method in Alon, Spencer, and Erdos [1]. We concentrate on first order logic over the basic operations f; U; g. That is, we are interested in sentences made up of = equality) linear order) U (an unary predicate) the binary connectives (disjunction) and (conjunction) negation) and the first order quantifiers 9 ....

....be dense. By the latter, we mean that between any 2 Z chains, there s another. When n Gamma1 p(n) n Gamma1=2 , almost surely isolated 1 s occur. Using the notation above, we have E(X) 1 as n 1. Since all the events are independent, Var[X] E[X] By the Second Moment Method (see [1], chapter 4 for details) Pr[X = 0] Var[X] E[X] 2 E[X ] E[X] 2 = 1 E[X ] the electronic journal of combinatorics 4 (1997) #R23 7 [000 Delta Delta Delta ) Delta Delta Delta 000] Figure 1: A model of T 0 [00 Delta Delta Delta ) Delta Delta Delta 00100 Delta Delta Delta ....

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, Inc, New York, 1992.

....theory (Section 2.1) Since we rely heavily on the definitions from [8] we have included them in Section 2.3. A more thorough treatment of first order logic can be found in Enderton [5] of finite model theory in Ebbinghaus and Flum [4] and of the probabilistic method in Alon, Spencer, and Erdos [1]. 2.1 Definitions from First Order Logic We concentrate on first order logic over the basic relational symbols f; U; g. That is, we are interested in sentences made up of = equality) linear order) U (an unary predicate) the binary connectives (disjunction) and (conjunction) ....

....M j= T 0 , then the order type of M is ( Delta for some order type . When n Gamma1 p(n) n Gamma1=2 , almost surely isolated 1 s occur. Using the notation above, we have E(X) 1 as n 1. Since all the events are independent, Var[X] E[X] By the Second Moment Method (see [1], chapter 4 for details) Pr[X = 0] Var[X] E[X] 2 E[X] E[X] 2 = 1 E[X] 0 Thus, Pr[X 0] 1. Let B i be the event that i and i 1 are 1 s, let Y i be its random indicator variable, and Y = P i Y i . Then, E(Y i ) Pr[B i ] p 2 and E(Y ) np 2 0. So, almost surely, 1 s ....

Noga Alon, Joel H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, Inc, New York, 1992.

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N. Alon, J.H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley & Sons, 1992.

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N. Alon, J. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, 1992.

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Nago Alon, Joel H. Spencer, and Paul Erdos, editors. The Probabilistic Method. Wiley, 1992.

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N. Alon, J. H. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, Inc, 1991. 5

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N. Alon, J. Spencer, and Paul Erdos, The Probabilistic Method, 1992, John Wiley and Sons. 7

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Noga Alon, Joel H. Spencer, and Paul Erdos, editors. The Probabilistic Method. Wiley, 1992.

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N. Alon, J. Spencer, and Paul Erdos. The Probabilistic Method. John Wiley and Sons, 1992.

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