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 25.1, Problem 8E
Program Plan Intro
To modify the fastest all pair shortest path
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Generate random matrices of size n ×n where n = 100, 200, . . . , 1000.Also generate a random b ∈ Rnfor each case. Each number must beof the form m.dddd (Example : 4.5444) which means it has 5 Signif-icant digits in total. Perform Gaussian elimination with and withoutpartial pivoting for each n value (10 cases) above. Report the numberof additions, divisions and multiplications for each case in the form ofa table. No need of the code and the matrices / vectors.
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…
Get the time complexity function from the pseudocode for the addition of the 2 matrices below, and prove whether the big-oh is O(n^2) so that it satisfies the rule f(n) <= c g(n);
/ add two matricesfor(i = 0 ; i < rows; i++){for(j = 0; j < columns; j++)matrix2[i][j] = matrix1[i][j] + matrix2[i][j];}// display the resultfor(i = 0 ; i < rows; i++){for(j = 0; j < columns; j++){printf("%d ", matrix2[i][j]);}printf("\n");}
Chapter 25 Solutions
Introduction to Algorithms
Ch. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - Prob. 5ECh. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8ECh. 25.1 - Prob. 9ECh. 25.1 - Prob. 10E
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - Prob. 5ECh. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - Prob. 9ECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25 - Prob. 1PCh. 25 - Prob. 2P
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- Get the time complexity function from the pseudocode for the addition of the 2 matrices below, and prove whether the big-oh is O(n^2) so that it satisfies the rule f(n) <= c g(n) for(i = 0 ; i < rows; i++){for(j = 0; j < columns; j++)matrix2[i][j] = matrix1[i][j] + matrix2[i][j];} for(i = 0 ; i < rows; i++){for(j = 0; j < columns; j++){printf("%d ", matrix2[i][j]);}printf("\n");}arrow_forwardGiven an array of numbers X₁ = {x₁, x2, ..., n } an exchanged pair in X is a pair xi, xj such that i x¡ . Note that an element x; can be part of up to n - 1 exchanged pairs, and that the maximal possible number of exchanged pairs in X is n(n − 1)/2, which is achieved if the array is sorted in descending order. Give a divide-and-conquer algorithm that counts the number of exchanged pairs in X in O(nlogn) time.arrow_forwardWrite an algorithm called MatrixRowMultiple(A[0…n – 1, 0… n – 1]) which takes an n x n matrix A and outputs if true, if each element in the first row is twice the value of the corresponding element of row 2. Otherwise, it returns false. (ii) Determine the worst case complexity of the algorithm which you have developed.arrow_forward
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