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 7.1, Problem 1E
Program Plan Intro
To illustrate the operation of PARTITION on the array
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Chapter 7 Solutions
Introduction to Algorithms
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- 7. (a) Show that E3x(Px x Px). (b) Show that {Qx,Vy(Qy → Pz)} EVx Px.arrow_forwardTitle: Comparison of Exhaustive, Greedy, and Dynamic Programming approaches for the Partition problem.Description: In this assignment, we are given a set S containing n integers and we are expected to output a partition with two subsets such that they have the minimumdifference in their sums. For this purpose, we can use the simple exhaustive approach, which takes every subset A of S and compare the difference d of the sums: d =sum(A)-sum(S-A). The greedy approach is to use the numbers in S in pairs in descending order as described in class (you can also see the Wikipedia entry for thepartition problem). The dynamic programming algorithm that we covered in class is also to be implemented.We are also asked to compare the three algorithms in their runtime (in msecs) and difference found (d) for various values of k=log2(n). Your program will produce theseperformance measures, and then using any utility such as Excel, draw the plots of their runtimes versus k and d versus k (use…arrow_forwardQ1. Design a generic matrix multiplier. Here ‘generic’ means the design should work for anymatrix multiplication X∙Y = Z, where the matrices X, Y, and Z are of sizes a×b, b×c, and a×crespectively. Now, pick a, b, c from the last three digits of your roll number. ( If any of the digitsis 0, take 3 instead of it; if two digits are 0s, then take 3 for the first 0, and 5 for the second 0)For Example, if the last three digits are 123 then a = 1, b = 2, and c = 3; if the last three digitsare 204 then a = 2, b = 3, c = 4; f the last three digits are 100 then a = 1, b = 3, c = 5.arrow_forward
- Consider an n by n matrix, where each of the n2 entries is a positive integer. If the entries in this matrix are unsorted, then determining whether a target number t appears in the matrix can only be done by searching through each of the n2 entries. Thus, any search algorithm has a running time of O(n²). However, suppose you know that this n by n matrix satisfies the following properties: • Integers in each row increase from left to right. • Integers in each column increase from top to bottom. An example of such a matrix is presented below, for n=5. 4 7 11 15 2 5 8 12 19 3 6 9 16 22 10 13 14 17 24 1 18 21 23 | 26 | 30 Here is a bold claim: if the n by n matrix satisfies these two properties, then there exists an O(n) algorithm to determine whether a target number t appears in this matrix. Determine whether this statement is TRUE or FALSE. If the statement is TRUE, describe your algorithm and explain why your algorithm runs in O(n) time. If the statement is FALSE, clearly explain why no…arrow_forward5. Consider removing the "strict" restriction in our definition of strict majority element. That is, we define r as a majority element of an array A if greater than or equal to half of the elements of A are equal to r. (In our more formal definition, we replace the > by > .) The claim that A has at most one strict majority element is no longer true if we remove the “strict". That is, it might be the case that A has more than one majority element. (Can you think of an example? What restrictions on n must there be to have more than 1?) Since the claim is no longer true, our proof for the strict majority element can no longer work. Which line of the proof is the earliest line that is no longer true (i.e. it no longer follows from the previous line or from given assumptions). You should assume that we replace our contradiction assumption with “suppose that array A has at least two distinct majority elements r and y". a. The definitions of A, and Ay b. That n 2 |Az|+ |Ay| c. That the…arrow_forwardA[i][j] = 1 where i = j how do I translate this to c++ programing?It is for a two dimentional matrix, trying to print a diagonal with ones and rest zeros. ex. 1 0 0 0 1 0 0 0 1 ************************************* could you show me this as a pointer function?arrow_forward
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