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 9.1, Problem 2E
Program Plan Intro
To prove the lower bounds in the worst case for both the maximum and minimum of n numbers is
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Use direct proof to show, "If m+n and n+p are even integers (s and t), where m, n and p are integers then m+p is even"
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Suppose we are to sort a set of 'n' integers using an algorithm with time complexity O (log2 n) that takes 0.5 ms (milliseconds) in the worst case when n = 20,000. How large an element can the algorithm then handle in 1 ms? (Note that only numeric answers can be provided.)
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- What is the difference between Transform and Conquer & Divide and Conquer? Justify your answer by providing at one example for each algorithm in solving a problem.arrow_forwardAssume that n is a power of 2. How many timeswill the body of the for-mloop be executed?Do not use the asymptotic notations to simplify your answer.arrow_forwardWhat is the worst case size of SL and SR (in terms of n) when you divide a list of unsorted items of size n into groups of (2*9)+1 while selecting a pivot? Input your answer as a decimal to three places. E.g. 3n/8 would be 0.375arrow_forward
- Show, in the style of the trace given with the code, how the entropy-optimal sort first partitions the array B A B A B A B A C A D A B R A Use the form to answer the question as same on the picture. lt i gt 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 B A B A B A B A C A D A B R A.arrow_forwardIn terms of the quantity of similarities, what is the bubble sort's worst-case complexity? describe theoreticallyarrow_forwardSuppose we are to sort a set of 'n' integers using an algorithm with time complexity O (log2n) that takes 1 ms (milliseconds) in the worst case when n = 100. How many elements can the algorithm then handle in 4 ms?arrow_forward
- Prove that for any integer n, n6k-1 is divisible by 7 if gcd(n,7)=1 and k is a positive integer.arrow_forwardConsider the problem of finding the smallest and largest elements in an array of n numbers. For solving this problem, design, respectively, a brute-force algorithm, a divide-and-conquer algorithm, and a presorting-based (transform-and-conquer) algorithm. determine and compare the efficiency in Big-O notation of the above three algorithms.arrow_forwardDemonstrate the output of the 1st and 2nd iteration of the given data [8, 0, 4, 2, 1, 9, 7, 5, 6, 3] in A. Quick sort B. Analyze the computational complexity of all the above algorithms in terms of big-O notation?arrow_forward
- Consider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.arrow_forwardUse the mathmetical induction to proove that 3^2n-1 is divisible by 4 whenever n is positive intgerarrow_forwardA sorting algorithm is in O(n2) in the worst case. It takes 3 seconds to sort 10,000 records, i.e. n = 10,000. What is the predicted time for the algorithm to sort 30,000 records? Suppose that you are presented with a machine that is 100 times as fast. How many records will you be able to process on the new machine in 3 seconds?arrow_forward
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