Introduction to Algorithms
Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 8.4, Problem 5E
Program Plan Intro

To describe an algorithm that sorts these numbers in linear average case time.

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Given a vector of real numbers r = (r1, V2, . . ., rn). We can convert this vector into a probability vector P = (P1, P2, . . ., Pn) using the formulation: p; = e¹¹/(Σï-₁ e¹¹), for all i. Write a Python function vec_to_prob(r) that takes the vector r as input and returns the vector p. Both r and p will be numpy arrays. You can assume r is non-empty. Sample inputs and outputs: Input: np.array([4, 6]), output: [0.11920292 0.88079708] • Input: np.array([3.4, 6.2, 7.1, 9.8]), output: [0.00151576 0.02492606 0.06130823 0.91224995] Hint: use numpy.sum [ ] # Write your function here Let's test your function. [ ] # Convert input from list to np.array first before calling your function to avoid errors print (vec_to_prob(np.array([4, 6]))) print (vec_to_prob (np. array ( [3.4, 6.2, 7.1, 9.8]))) print (vec_to_prob (np.array([3, 5.5, 0])))
We have considered, in the context of a randomized algorithm for 2SAT, a random walk with a completely reflecting boundary at 0—that is, whenever position 0 is reached, with probability 1 we move to position 1 at the next turn. Consider now a random walk with a partially reflecting boundary at 0– whenever position 0 is reached, with probability 1/4 we move to position 1 at the next turn, and with probability 3/4 we stay at 0. Everywhere else the random walk either moves up or down 1, each with probability 1/2. Find the expected number of moves to reach n starting from position i using a random walk with a partially reflecting boundary. (You may assume there is a unique solution to this problem, as a function of n and i; you should prove that your solution satisfies the appropriate recurrence.)
Write Algorithm to Generating a random number from a distribution described by a finite sequence of weights.in: sequence of n weights W describing the distribution (Wi ∈ N for i = 0, . . . , (n −1) ∧ 1 ≤ n−1 i=0 Wi)out: randomly selected index r according to W (0 ≤ r ≤ m − 1)
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