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 5.2, Problem 5E
Program Plan Intro
To find the expected number of inversions by using indicator random variables.
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- 7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).arrow_forwardTrue or False 1. Matrices are often represented by single small letters a, b, c... etc.2. Two m x n matrices A and B are equal if aij=bij for each i & j. (i.e., the two matrices havesame size, and all the corresponding elements are equal).3. Matrices A & B are said to be conformable in the order AB if, and only if, the number ofrows in A is equal to the number of columns in B.4. Suppose Matrix A is having 4 rows and 3 columns, and Matrix B is having 3 rows and 2columns. The product size of AB is a 4 x 2 matrix.5. Suppose B is the matrix obtained from an n x n matrix A by multiplying the entries in arow/column by a non-zero constant and adding the result to the corresponding entries inanother row/column. Then, det(B) = det(A).arrow_forwardGenerate 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 Significant 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. Deliverable(s): Two tabular columns indicating the number of additions, multiplications and divisions for each value of n, for with andwithout pivoting in Pythonarrow_forward
- As an investor, I always check the stock market in order to find good companies to invest in. Recently, I found that the best companies to invest in, are the ones that have largest sum formed by a strictly increasing set of numbers (a set where the next element is always greater than the current element). But before I invest, I need to know the position of the first element of the consecutive increasing numbers. Help me so we can start investing already! Note: If it is already the last element of the row in the array, the next element is the first element of the next row, if there exists a next row. Input 1. Number of rows Description This is the number of rows of the multidimensional array. 2. Number of columns Description This is the number of columns of the multidimensional array. 3. Elements of the multidimensional array Output The first line will contain a message prompt to input the number of rows. The second line will contain a message prompt to input the…arrow_forward20. Given a 4-element array with distinct elements, say {x1, x2, x3, x4}, it is known that x1 is not the minimum. How to use 2 comparisons to find an element that is neither the maximum nor the minimum? Solution:arrow_forwardConsider 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_forward
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