J.Spencer, Ten lectures on the probabilistic method, SIAM, Philadelphia, 1987.  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 ....

J. Spencer. Ten Lectures on the Probabilistic Method. 2nd Edition. SIAM, Philadelphia, 1994.

.... i p i The task for linear discrepancy is to round the real numbers p 1 ; p n up or down in such a way as to minimize the total error on an edge of H. An investigation of the relationship between linear and hereditary discrepancy was undertaken by Lov asz, Spencer and Vesztergombi [9] [15]. They proved that lindisc(H) herdisc(H) We note in passing that a key lemma in Baranyai s proof of the existence of a factorization of the complete uniform hypergraph [1] can be viewed as a special case of this statement. Lov asz, Spencer and Vesztergombi went on to ask: What is the maximal ....

....on ground set [n] then lindisc(H) herdisc(H) Conjecture 1 was proved to hold for two special types of hypergraphs, both of hereditary discrepancy 1. For interval hypergraphs (the vertex set is [n] and the edge set consists of the integer intervals [i; j] for 1 i j n) Spencer [15] gave a short argument (a gem shown to him by Lov asz) For hypergraphs with edge set consisting of initial segments in either of two given orderings of [n] Knuth gave a complicated proof [8] another proof was given independently by J. Ossowski. Knuth dubbed this the two way rounding ....

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J. Spencer, Ten Lectures on the Probabilistic Method, 2nd edition, SIAM, 1994. 7

....c) O q c ln c Proof. By Lemma 6 we may apply Theorem 2 with = 2 , D = 2K ln( p0n and p 0 . This yields a fair p coloring having discrepancy at most Dc p i n in color i 2 [c] The claim follows from c 31:4. ut This is quite close to the optimum. An extension of Spencer s [Spe87] proof shows that hypergraphs arising from Hadamard matrices have c color discrepancy ( c ) It is a famous open problem already for 2 colors whether colorings having discrepancy O( n log( can be computed eciently. Therefore a constructive version of Theorem 4 is not to be expected at ....

J. Spencer. Ten Lectures on the Probabilistic Method. SIAM, 1987.

....edge S h, where we write #(S) for x#S #(x) The hereditary discrepancy of is the maximum discrepancy of any restriction to a subset Y X. Discrepancy and hereditary discrepancy are important notions in combinatorics and discrete geometry; for more information we refer, e.g. to [1] or [2]. Throughout the note, the asymptotic notation is used under the assumption that n # #. All logarithms have natural base. We denote by disc(F) and herdisc(F) the discrepancy and the hereditary discrepancy , respectively. The following question is a folklore in discrepancy theory (as far as we ....

J. Spencer. Ten lectures on the probabilistic method. CBMS-NSF. SIAM, Philadelphia, PA, 1987.

....for Pi and by Sn we denote the random set of solutions of Pi . We are interested in establishing a condition subject to which the probability that Pi has a solution goes to 0 as n 1. Such a condition is readily provided by the First Order Moment method (for excellent expositions see [2] and [27]) That is, by first noting that E[jSn j] Pr[ Pi ] Delta jS n ( Pi)j) 1) and then noting that Pr[ Pi has a solution] Pr[ Pi ] Delta I Pi ) 2) where for an instantiation of the random variable Pi the indicator variable I Pi is defined as I Pi = ae 1 if Pi has a ....

J. H. Spencer, Ten Lectures on the Probabilistic Method, 2nd edition, SIAM, Philadelphia (1994).

....give asymptotic results that apply to graphs of order n and maximum degree d. They rely heavily on Lov asz s local Lemma, and are shown thanks to proofs that are similar to the ones given by Alon et al. AMR91] about acyclic coloring. We rst recall Lov asz s local Lemma below ( EL75] see also [Spe87]) Lemma 1 (Lov asz s local Lemma [EL75] Let A 1 ; A 2 ; An be events in an arbitrary probability space. Let the graph H = V; E) on the nodes f1; 2 : ng be a dependency graph for the events A i (that is, two events A i and A j will share an edge in H i they are dependent) If there ....

J. Spencer. Ten Lectures on the Probabilistic Method. SIAM, Philadelphia, 1987.

....= 0, and 11 ffl We let g(n) O(f(n) mean lim sup n 1 [g(n) f(n) 1, and ffl We let g(n) f(n) mean lim n 1 [g(n) f(n) 1. We also have E mean expected value. This section will outline several applications of the probabilistic method (see [2] or, for a short introduction, [25] or, even shorter, 22] We are looking at random trees from a largely graph theoretic point of view: for more on random graphs and random trees, see [3] 23] 16] or [13] or, for readers who prefer shorter introductions, 14] or [19] 3.1 MSO Zero One Laws One bit of nomenclature before we ....

J. Spencer, Ten Lectures on the Probabilistic Method (SIAM, 2nd ed., 1995).

....### 1. A nonconstructive algorithm for sorting n elements in k rounds is an algorithm that is proven to exist, but its existence proof does not reveal how to produce it. For example, the graph on n vertices that represents the rst round may be proven to exist by the probabilistic method [5, 20]. 2. A constructive algorithm for sorting in k rounds is a sequence of algorithms # with the following properties: 1) The algorithm # sorts n elements in k rounds. 2) There is a polynomial time algorithm that, given n (in unary) produces # . 3. A randomized algorithm to sort n ....

J. Spencer. Ten Lectures on the Probabilistic Method. Conf. Board of the Math. Sciences, Regional Conf. Series, AMS and MAA, 1987.

....large, 4 c # c O(1) n f(n, cn) # ( 3 ln(2) 8)n. and 3 ln(2) 8 are approximately 0.343859 and 0.509833, respectively. The upper bound is proved by a simple first moment argument, and the lower bound by analyzing an algorithm; both techniques are exactly those demonstrated in [Spe94, Lecture 6] to analyze the Gale Berlekamp switching game. Our next results relate to the low density case, when c is above but close to the critical value 1. How does f(n, cn) depend on c = 1 # for small # Theorem 3. For any fixed # 0, 1 # 3)n f(n, 1 #)n) also, there exist absolute ....

Joel H. Spencer, Ten lectures on the probabilistic method, second ed., SIAM, 1994.

....is obtained using some computation on the n given vectors and the random vectors. It is not obvious how to derandomize the above randomized algorithms, i.e. to obtain a good set of random vectors deterministically. A natural way to derandomize is to use the method of Conditional Probabilities[14, 16]. The problem that occurs then is to compute the conditional probabilities in polynomial time. Our Contribution. The main contribution of this paper is a technique which enables derandomization of all approximation algorithms based upon semidefinite programming listed above. This leads to ....

J. Spencer, Ten Lectures on the Probabilistic Method, SIAM, Philadelphia, 1987.

....1 2) provided the code length is Theta(log j Sigmaj) Our linear time algorithm constructs these codes in two steps. Step 1. j Sigmaj codes with the above property are constructed in this step. Each code has length O(log j Sigmaj) This is done using the method of Conditional Probabilities [6, 8], which is easily seen to run in O( j Sigmaj 1=8 ) log j Sigmaj) o(j Sigmaj) time (the number of ordered triples is cubic in j Sigmaj , each triple needs to be checked each time a bit is set, the total number of bits over all codes is O(j Sigmaj log j Sigmaj) checking a triple ....

J. Spencer, Ten Lectures on the Probabilistic Method, SIAM, Philadelphia, 1987.

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J.Spencer, Ten lectures on the probabilistic method, SIAM, Philadelphia, 1987.

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J. Spencer (1994) Ten Lectures on the Probabilistic Method. SIAM: Philadelphia, Pennsylvania.

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J. Spencer, \Ten lectures on the probabilistic method", CBMS-NSF Regional Conference Series in Applied Math., number 52, SIAM.

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. J. Spencer. Ten Lectures on the Probabilistic Method. SIAM, Philadephia, 1987.

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J. Spencer. Ten Lectures on the Probabilistic Method. SIAM, Philadelphia, 1987. 23

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J. H. Spencer, Ten Lectures on the Probabilistic Method, SIAM, Philadelphia, 1987.

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J. Spencer. Ten Lectures on the Probabilistic Method. SIAM, Philadelphia, 1987. 27

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J. Spencer. Ten lectures on the probabilistic method. CBMS-NSF. SIAM, Philadelphia, PA, 1987. 2

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J. Spencer, Ten Lectures on the Probabilistic Method (SIAM, 2nd ed., 1995).

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J. Spencer. Ten Lectures on the Probabilistic Method. SIAM, 1987.

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J. Spencer. Ten Lectures on the Probabilistic Method. SIAM, Philadelphia, 1987.

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J. Spencer. Ten lectures on the probabilistic method. CBMS-NSF. SIAM, Philadelphia, PA, 1987. 3

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J. Spencer, Ten Lectures on the Probabilistic Method, SIAM, 1987.

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J. Spencer (1994) Ten Lectures on the Probabilistic Method. SIAM Philadelphia, Pennsylvania.

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