Algorithms And Data Structures

Read e-book online Statistical methods in algorithm design and analysis PDF

By Weide B.W.

Using statistical equipment within the layout and research of discrete algorithms is explored. one of the layout instruments are randomization, score, sampling and subsampling, density estimation, and "cell" or "bucket" suggestions. The research concepts comprise these in response to the layout equipment in addition to using stochastic convergence suggestions and order records.

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2. 3. 080 is written in C++. 0c for integer and polynomial arithmetic, parts of which are described in Shoup (1995). It uses Karatsuba’s (Karatsuba & Ofman 1962) method for multiplying large integers. Next we discuss three methods employing fast polynomial arithmetic. D. Paterson & Stockmeyer’s (1973) method: We assume that (n + 1) = m2 is a square (padding f with leading zeroes if necessary), and write f = (i) mi x , with polynomials f (i) ∈ R[x] of degree less than m for 0≤i

Then f (x + b) ∞ ≤ f (x + b) 1 ≤ (|b| + 1)n f 1 ≤ (n + 1)(|b| + 1)n f ∞ . For b = ±1, the following sharper bound is valid: f (x ± 1) Proof. Let f = f (x + b) 1 0≤i≤n ∞ ≤ f (x ± 1) ≤ 2n+1 f ∞ . fi xi . Then fi · (x + b)i = 1 ≤ 0≤i≤n 1 |fi |(1 + |b|)i ≤ (|b| + 1)n f 0≤i≤n Moreover, we have f (x + b) 1 ≤ f (1 + |b|)i = f ∞ 0≤i≤n and the claim for |b| = 1 follows. ✷ ∞ (1 + |b|)n+1 − 1 , |b| 1 . 1 Computing Taylor Shifts For all b, B, n ∈ N>0 , the polynomial f = B bound within a factor of at most n + 1: f (x + b) 1 (b + 1)i ≥ f =B 0≤i≤n ∞ (b 43 xi achieves the first + 1)n .

24. Consider the following random experiment. An urn contains b black and w white balls, and we draw k ≤ w + b balls without replacement. (i) If w ≥ b, then the probability that at most k/2 balls are white is at most 1/2. (ii) If w ≥ 4b and k ≥ 8, then the probability that at most k/2 balls are white is at most 1/4. Proof. 31 in von zur Gathen & Gerhard (1999) for a proof of (i). Let X denote the random variable counting the number of white balls after k trials. Then X has a hypergeometric distribution prob(X = i) = w i b k−i w+b k for 0 ≤ i ≤ k, with mean µ = EX = k · prob(X = k) = 0≤i≤k 4 kw ≥ k.

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Statistical methods in algorithm design and analysis (thesis) by Weide B.W.


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