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 8.1, Problem 3E
Program Plan Intro
To show that there is no comparison sort
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Given the runtime, f(n) = n° + n +n+1 show that
a. f(n) = 0(n*)
b. f(n) = e(n³)
c. f(n) = (n²)
Please solve the recurrence:
J(n) = J(n/2) + J(n/3) + J(n/6) + n
When the order of increase of an algorithm's running time is N log N, the doubling test leads to the hypothesis that the running time is a N for a constant a. Isn't that an issue?
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Introduction to Algorithms
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- Generate ten random numbers (s1-s10) using the linear congruently generator using the seed (s0) as 5, and with the three parameters a, b, and m as 14, 15, and 21, respectively. (Note: si+1 = (a*si + b) mod m). SHOW WORK.arrow_forwardWhich of these statements is false regarding this: T(n) = O(f(n)) means that oooo T(n) is an upper bound on the running time that holds for all inputs of size n T(n) doesn't grow faster than f(n) T(n) has order of f(n) T(n) grows faster than f(n)arrow_forwardGiven an n-element sequence of integers, an algorithm executes an O(n)-time computation for each even number in the sequence, and an O(logn)-time computation for each odd number in the sequence. What are the best-case and worst-case running times of this algorithm? Why? Show with proper notations.arrow_forward
- In our shuffling algorithm, let's say that you select a random integer between 0 and N-1 as opposed to one between i and N-1. Demonstrate that none of the N! possibilities are equally probable to be the resulting order. Run the test for this version of the prior exercise.arrow_forwardGiven the function T(n) = n3 + 20n + 5, show that T(n) is O(n3)arrow_forwardFor the following pseudo-code, what is the time complexity function T(n) and the order (O)? You can ignore the over head operations and just count the basic operationsarrow_forward
- Analyze the following code fragment and provide an asymptotic (Θ) bound on the running time as a function of n. You do not need to give a formal proof, but you should justify your answer.arrow_forwardLet Tk (n) denote the value returned by A(k, n). This gives T0(n) = 2 + n, T1(0) = 0, Tk (0) = 1 for k ≥ 2, and Tk (n) = Tk−1(Tk (n − 1)) for k > 0 and n > 0.arrow_forwardGiven an n-element array X of integers, Algorithm A executes an O(n) time computation for each even number in X and an O(log-n) time computation for each odd number in X. What are the best case and worst case for running time of algorithm C?arrow_forward
- Prove by induction that fib(0)+fib(1)+…+fib(n) = fib(n+2)-1, fib(0)=0 fib(1)=1arrow_forwardModify the decimal addition algorithm of Figure 1.2 so that it does not print out nonsignificant leading zeroes; that is, the answer to Exercise 5 would appear as 178rather than 0178.arrow_forwardCan someone give me an example of a simple collinear for loop algorithm that can be run in O(n^3) sequential time?arrow_forward
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