M. Hofri (1987). Probabilistic analysis of algorithms. Springer-Verlag, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are independent. Define M 1 = 0, then Mn d = n X i=2 X i (3.2) X i ) independent, and IE Mn = Hn Gamma 1; VarMn = Hn Gamma H (2) n (3. 3) where H (k) n = P n j=1 1 j k ; Hn = H (1) n = ln n fl O Gamma n Gamma1 Delta , H (2) n Gamma n 1 i(2) 2 6 (cf. Hofri [Ho87]) Define the normalized version d Mn : Mn Gamma IE Mn p VarMn ; 3.4) then as in section 2 we obtain the normal approximation but with logarithmic rate only. 1. Analysis of recursive algorithms by the contraction method 6 Theorem 3.1 For all n 2 IN and some constant C 0 holds ae i d ....

M. Hofri. Probabilistic analysis of algorithms. Springer, 1987

....logarithmic moment generating function in our admission rule. The LD admission rule thus takes the mean as well as the higher moments of the measured aggregate arrivals into account. Hoeffding Bound Approach Floyd [15] studies measurement based admission control based on the Hoeffding bound [41, 42]. The Hoeffding bound is a Chernoff style bound for sums of bounded, independent random variables. Recall that U j ; j = 1; J , are steady state random variables denoting the rate at which connection j traffic enters the multiplexer. Furthermore, recall that c j and r j denote the peak ....

M. Hofri. Probabilistic Analysis of Algorithms. Springer Verlag, 1987.

....literature on theoretical computer science, especially on average case analysis of algorithms, can therefore 5 be used fruitfully in asymptotic enumeration. One notable survey paper in that area is that of Vitter and Flajolet [371] There are also presentations of relevant methods in the books [177, 209, 235, 236, 237, 223]. Section 18 is a guide to the literature on these topics. The aim of this chapter is to survey the most important tools of asymptotic enumeration, point out references for the results and methods that are discussed, and to mention additional relevant papers that have other techniques that might ....

.... equations, special functions, and so on) are available in the books [54, 100, 114, 115, 315, 344, 354, 372, 382, 385] Integral transforms are treated extensively in [89, 95, 116, 299, 365] Books that deal with asymptotics arising in the analysis of algorithms or probabilistic methods include [11, 55, 108, 209, 223, 240, 241, 270, 338]. Nice general introductions to combinatorial identities, generating functions, and related topics are presented in [81, 351, 377] Further material can be found in [13, 88, 99, 173, 188, 335, 336] A very useful book is the compilation [168] While it does not discuss methods in too much detail, ....

M. Hofri, Probabilistic Analysis of Algorithms, Springer, 1987.

....have been used for real applications. This suggests that worst case examples are not realistic. In [11] the problem of generating random benchmarks for DL systems was addressed. The idea of generating random benchmarks is to perform experimentally probabilistic analyses for DL algorithms (see [18]) In DL a Tbox defines the logical rules of a given context of application by propositions, called concepts, and binary relations, called roles, together with possible logical interactions between them (implications, exclusions. It can be seen as the rule of the game. With one Tbox can be ....

M. Hofri. Probabilistic analysis of algorithms. Springer-Verlag, New York, 1987.

....length of each bad interval and the first interval h 0; 2 l i (4n) s Gamma 2 Gamma 2 (4n) s ji is good, the lemma follows also in this case. 2 Take n 2s random independent ff i and let V (x) P n 2s i=1 N ff i (x) We will need the following elementary inequality (see e.g. [9]) Lemma 21. Hoeffding s inequality) Let X 1 ; X k be independent random variables with values in the interval [0; 1] and S = P k i=1 X i . Let = E [S=k] Then P [S Gamma k kt] exp Gamma Gamma Omega Gamma kt 2 ) Delta : Now we have Majority vs. Threshold 19 Lemma 22. If m ....

M Hofri, Probabilistic Analysis of Algorithms, Springer-Verlag, 1987.

....Thus, for n 2 ) P rob(9(x; y) jA(x; y)j (1 ffi)ffll) 2 n exp( Gamma2n) 0:25 (3) Observe that this corresponds to the probability that L does not have property (1) We need a similar statement for property (2) for which we use Hoeffding inequality Lemma 1. 1 Hoeffding(1963) 4] [5] Let Y 1 ; Y l be independent random variables with values in the interval [0; z] Let = E( 1 l Sigma j=l j=1 Y i ) Then P rob( 1 l Sigma j=l j=1 Y i d) d Gammad ( z Gamma z Gamma d ) z Gammad ) l=z For d 2 [1; z= this simplifies to P rob( 1 l Sigma j=l ....

M. Hofri, Probabilistic Analysis of algorithms, Springer Verlag, New York, 1987, pp 104

....ffl For large j, we can find an approximate value of the sum. If we use Stirling s approximation for the factorial, we find that A(j) P 1 t=j (t) t Gamma2 p 2t t 1=2 e Gammat e Gammat P 1 t=j 1 p 2t 5=2 We can approximate the sum with an integral using Euler s summation formula [6] to get: A(j) 2 3 p 2 j Gamma3=2 Gamma 3 4 j Gamma5=2 Gamma 1 O(j Gamma2 ) Delta (5) Table 3 lists the first few values of the distribution A(j) We summed the first 400 values of C(t) and then added the approximation of A i (401) to calculate A i (1) We calculated the ....

Micha Hofri. Probabilistic Analysis of Algorithms. Springer-Verlag, 1987.

....logarithmic moment generating function in our admission rule. The LD admission rule thus takes the mean as well as the higher moments of the measured aggregate arrivals into account. Hoeffding Bound Approach Floyd [18] studies measurement based admission control based on the Hoeffding bound [19, 20]. The Hoeffding bound is a Chernoff style bound for sums of bounded, independent random variables. Recall that U j ; j = 1; J , are steady state random variables denoting the rate at which connection j traffic enters the multiplexer. Furthermore, recall that c j and r j denote the peak ....

M. Hofri. Probabilistic Analysis of Algorithms. Springer Verlag, 1987.

.... the sum of the values of the elements in S exceeds M=g 2 t is smaller than the probability that the sum of f=g independently selected elements X i , with 0 X i M=f and with expected value M=g 2 , exceeds M=g 2 t (see [22] and the references therein) Applying Hoeffding s Inequality [6] this probability can be estimated as follows. Pr( X i X i M=g 2 t) exp( Gamma2 Delta f 3 Delta t 2 = g Delta M 2 ) So, we should take t(f; g; M ) c Delta M Delta ln 1=2 M Delta g 1=2 =f 3=2 ; for some constant c, to get the desired bound on the probability. In ....

Hofri, M., Probabilistic Analysis of Algorithms, Springer, 1987.

....are well known bounds on the tail of the distribution of a sum of random variables. Both bounds can be found in many standard works on probability theory. Our basic formulation of the Chernoff bounds was taken from a paper by Hagerup and Rub [20] and the Hoeffding inequality from a book by Hofri [21]. In addition, we often use a pair of formulas of the form P (A f(1 ffl) Delta D) C, P (A f(1 Gamma ffl) Delta D) C, with A, D and C expressions and f a function. Such formulas are presented concisely with the following notation: P (A f(1 ffl) Delta D) C. A random variable B is a ....

....sum Z of n mutually independent random variables X i with mean i = E[X i ] Strong bounds on the tail probabilities are given by the Hoeffding inequalities. Let = P i i =n. First we give some results for the case that 0 X i 1 for all i. The strongest form of the Hoeffding inequality is [21] P (Z n Delta ( t) h ( t) t Delta ( 1 Gamma ) 1 Gamma Gamma t) 1 Gamma Gammat i n : 4) Stronger results can be derived with the following generalized Hoeffding inequality , which incorporates the effect of the variance on the tail estimates [22] for X i a ....

M. Hofri. Probabilistic Analysis of Algorithms. Springer Verlag, 1987.

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M. Hofri (1987). Probabilistic analysis of algorithms. Springer-Verlag, New York.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer--Verlag, 1987.

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M Hofri. Probabilistic Analysis of Algorithms. Springer-Verlag, 1987.

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Hofri, M., Probabilistic Analysis of Algorithms, Springer, 1987.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer--Verlag, 1987.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer, 1987.

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Hofri, M., Probabilistic Analysis of Algorithms, Springer-Verlag, 1987. 20

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Micha Hofri. Probabilistic Analysis of Algorithms. Springer-Verlag, 1st. edition, 1987.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer-Verlag, 1987.

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Hofri, M., Probabilistic Analysis of Algorithms, Springer-Verlag, 1987.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer--Verlag, 1987.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer-Verlag, 1987.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer--Verlag, 1987.

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M. Hofri. Probabilistic Analysis of Algorithms. Springer--Verlag, 1987.

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Hofri, M., Probabilistic Analysis of Algorithms, Springer-Verlag, 1987.

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