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 7.4, Problem 2E
Program Plan Intro
To show that the best case running time of the quick sort is
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Perform experimental analysis to test the hypothesis that Python’s sorted method runs in O(n log n) time on average.
Show that:
logn = O(n)
For 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 operations
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Introduction to Algorithms
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- What 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_forwardWhat is the worst-case running time complexity of matrix substraction? select one: a.O(n^2.5) b.O(2n) c.O(n^2) d.O(3^n)arrow_forwardQuestion: What is the worst-case running time of Find-Index-2(A[1 : n])? What about its worst-caseexpected running time? Remember to prove your answer formally.arrow_forward
- Suppose that, in a divide-and-conquer algorithm, we always divide aninstance of size n of a problem into 10 subinstances of size n/3, and thedividing and combining steps take a time in Θ(n2) in n>1, and it performs one basic operation if n = 1. Write a recurrenceequation for the running time of T(n), and solve the equation.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_forwardShow how to reduce HAMCIRCUIT to HAMPATH in polynomial time.arrow_forward
- Implement The Python code to implement multiplication of these two sparse matrices is given below. raw1 [0, 1, 2, 2, 3, 4]colm1 [1, 3, 0, 4, 0, 1]raw2 [0, 1, 3, 4]colm2[1, 1, 1, 1]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_forwardDevelop and implement an ADT for preprocessing a string to support efficiently answering queries of the form How many times does a given k-gram appear ? Each query should take time proportional to k log N in the worst case, where N is the length of the string.arrow_forward
- For the pseudo-code below derive the simplified asymptotic running time in Q(?) notation. for i ->1 .. n do j -> n while i < j*j do j -> j – 2arrow_forwardFor the problem, give pseudocode for your solution, and remember to include a proof of correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute force O(n 2 ) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative. In IR2 , we define a slab to be a pair of parallel lines. Given a set of points P in IR2 , find the narrowest slab containing P, where the width of the slab is the vertical distance between its bounding lines.arrow_forwardShow that in the worst situation, Quicksort uses N 2/2 compares, but that random shuffling prevents this from happening.arrow_forward
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