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 5.3, Problem 6E
Program Plan Intro
To explain the implementation of PERMUTE-BY-SORTING to handle the case of two or more priorities are identical.
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Write the quicksort algorithm and explain how it performs sorting. Analyze its time complexity in the worst-case. Please show all work.
Consider median-3 quick sort and bubble sort. When using these two algorithms to sort a list of n elements that have already been sorted in increasing order, which of the following is true for the worst-case time complexity?
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.
Chapter 5 Solutions
Introduction to Algorithms
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- An array of n elements is almost sorted if and only if every element is at most k spots away from its actual location. Assuming that you can only perform pairwise comparisons, formally prove that any algorithm which can sort almost sorted arrays must have running time S2(n log k), You may assume that n is a multiple of k.arrow_forwardAm i correct to assume that the possible outcomes for the algorithm of topological Sorting are: 4,3,6,2,8,7,5,1 and 3,6,4,1,2,8,7,5 Are there other possible outcomes?arrow_forwardIn this problem, consider a non-standard sorting algorithm called the Slow Sort. Given anarray A[1 : n] of n integers, the algorithm is as follows: Slow-Sort(A[1 : n]):1. If n < 100, run merge sort (or selection sort or insertion sort) on A.2. Otherwise, run Slow-Sort(A[1 : n −1]), Slow-Sort(A[2 : n]), and Slow-Sort(A[1 : n −1]) again. Question: Write a recurrence for Slow-Sort and use the recursion tree method to solve this recurrenceand find the tightest asymptotic upper bound on the runtime of Slow-Sort.arrow_forward
- For the 8-queens problem, define a heuristic function, design a Best First Search algorithm in which the search process is guided by f(n) = g(n) + h(n), where g(n) is the depth of node n and h(n) is the heuristic function you define, and give the pseudo code description.arrow_forwardIn this problem, consider a non-standard sorting algorithm called the Slow Sort. Given anarray A[1 : n] of n integers, the algorithm is as follows: Slow-Sort(A[1 : n]):1. If n < 100, run merge sort (or selection sort or insertion sort) on A.2. Otherwise, run Slow-Sort(A[1 : n −1]), Slow-Sort(A[2 : n]), and Slow-Sort(A[1 : n −1]) again. Question: Prove the correctness of Slow-Sort.arrow_forward(a) Carry out the Select algorithm on the following set, using k = 19 (return the element of rank 19). Show your steps! Ensure that you run the partition algorithm in such a way that you maintain theelements in their original relative order.56; 78; 34; 19; 67; 32; 13; 12; 90; 92; 50; 51; 30; 1; 99; 58; 43; 42; 24; 65; 21; 25; 68; 69; 101 (b) Carry out the Randomized Select algorithm from class on the set of elements from part (a), usingk = 19. Show your steps! When selecting a random pivot, suppose you always chose the last element inthe array. Ensure that you run the partition algorithm in such a way that you maintain the elements intheir original relative orderarrow_forward
- If a binary search algorithm is applied to a sorted array of 1000 elements, how many comparisons will be needed in the worst case scenario to find an element?arrow_forwardLet C(n) denote the number of comparisons performed by quicksort in sorting an array of size n in the worst case. Write the recurrence relation for C(n) and solve.arrow_forwardA magic square of order n is an arrangement of the integers from 1 to n2 in an n × n matrix, with each number occurring exactly once, so that each row, each column, and each main diagonal has the same sum Design and implement an exhaustive-search algorithm for generating all magic squares of order n.arrow_forward
- QuickSort is run for an array A in a manner that PARTITION consistently produces a 5:1 split for the (sub)arrays to be sorted (recursively) next. Picture the recurrence tree for this scenario ... What will be the length of the longest path from the root to a leaf (= deepest depth)in this tree? Group of answer choices log_5/6 (n) ... "log of n with base 5/6" log_6/5 (n) ... "log of n with base 6/5" log_6 (n) ... "log n with base 6" n^log_5(6) ... "n to the power of log_5(6)"arrow_forwardGiven 2 sorted arrays (in increasing order), find a path through the intersection that produces the maximum sum and return the maximum sum. That is, we can switch from one array to another array only atcommon elements. If no intersection element is present, we need to take the sum of all elements from the array with greater sum. Sample Input:61 5 10 15 20 2552 4 5 9 15Sample Output :81arrow_forwardWe argued in class that if the input is not uniformally distributed, the worst-case running time of Bucket Sort (using Insertion Sort in each bucket) in O(n^2). The reason: all those numbers could fall in the same bucket, and then the time of Insertion sorting then dominates. How many out of the total n numbers could fall in one bucket and the algorithm still run in linear time if the rest are evenly distributed? (Choose one option, no points if you choose more than one.) A. O(1)B. O(√n)C. O(On)D. O(log n)E. None of the abovearrow_forward
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