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 27.2, Problem 4E
Program Plan Intro
The following is pseudocode and analysis of multithreaded
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Chapter 27 Solutions
Introduction to Algorithms
Ch. 27.1 - Prob. 1ECh. 27.1 - Prob. 2ECh. 27.1 - Prob. 3ECh. 27.1 - Prob. 4ECh. 27.1 - Prob. 5ECh. 27.1 - Prob. 6ECh. 27.1 - Prob. 7ECh. 27.1 - Prob. 8ECh. 27.1 - Prob. 9ECh. 27.2 - Prob. 1E
Ch. 27.2 - Prob. 2ECh. 27.2 - Prob. 3ECh. 27.2 - Prob. 4ECh. 27.2 - Prob. 5ECh. 27.2 - Prob. 6ECh. 27.3 - Prob. 1ECh. 27.3 - Prob. 2ECh. 27.3 - Prob. 3ECh. 27.3 - Prob. 4ECh. 27.3 - Prob. 5ECh. 27.3 - Prob. 6ECh. 27 - Prob. 1PCh. 27 - Prob. 2PCh. 27 - Prob. 3PCh. 27 - Prob. 4PCh. 27 - Prob. 5PCh. 27 - Prob. 6P
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- Given an n×n matrix M in which every entry is either a 0 or 1. Present an algorithm that determines if ∃i, 1 ≤ i ≤ n, such that M[i, j] = 0 and M[j, i] = 1, ∀j, 1 ≤ j ≤ n ∧ j 6= i, using examining an entry of M as the key operation. Your algorithm must examine at most 3n − ⌊lg n⌋ − 3 entries of M.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## Count the number of unique paths from a[0][0] to a[m-1][n-1]# We are allowed to move either right or down from a cell in the matrix.# Approaches-# (i) Recursion- Recurse starting from a[m-1][n-1], upwards and leftwards,# add the path count of both recursions and return count.# (ii) Dynamic Programming- Start from a[0][0].Store the count in a count# matrix. Return count[m-1][n-1]# T(n)- O(mn), S(n)- O(mn)# def count_paths(m, n): if m < 1 or n < 1: return -1 count = [[None for j in range(n)] for i in range(m)] # Taking care of the edge cases- matrix of size 1xn or mx1 for i in range(n): count[0][i] = 1 for j in range(m): count[j][0] = 1 for i in range(1, m): for j in range(1, n): # Number of ways to reach a[i][j] = number of ways to reach # a[i-1][j] + a[i][j-1] count[i][j] = count[i - 1][j] + count[i][j - 1]…arrow_forward
- The transposition routine is given in [FXT: aux2/transpose2.h]. We only use even powers of two so thetransposition is that of a square matrix.As for dyadic convolutions we do not need the data in a particular order so we also include a version ofthe matrix algorithm that omits the final transposition:arrow_forwardCan you find a more efficient sparse matrix representation (in terms of space and computational overhead)?arrow_forwardGiven an array A of n distinct positive integers and an integer B, find the number of unordered pairs of indices(i and j) such that i != j and that A[i] + A[j] > B using an algorithim in time nlogn. Prove that the algorithm you have is in time nlognarrow_forward
- Write a JAVA program to find the transpose of the matrixarrow_forwardComputer Science Given an N x N matrix M with binary entries i.e every entry is either 1 or 0. You are told that every row and every column is sorted in increasing order. You are required to output a pair (i,j) with 1 <= i and j <= n corresponding to the entry of the matrix satisfying Mij = 1 and Mrs = 0 for all 1 <= r <= i and 1 <= s <= j except for Mij Informally this includes the entry of M = 1 and is closest to the top left corner. for example: M = [ 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1] output is (2,3) or (1,4) M = [ 0 1 1 1 1 1 1 1 1] output could be (1,2) or (2,1) Design a divide and conquer algorithm, explain correctness and runtime of the algorithm.arrow_forwardConstruct a square matrix with NN rows and NN columns consisting of nonnegative integers from 00 to 10^{18}1018, such that its determinant is equal to 11, and there are exactly A_iAi odd numbers in the ii-th row for each ii from 11 to NN, or report there isn't such a matrix. Standard input The first line contains a single integer NN. Each of the next NN lines contains a single integer A_iAi. Standard output If there is no solution, output \text{-}1-1. Otherwise, print NN lines, each consisting of NN integers, representing the values of the constructed matrix. If there are multiple solutions, print any. Constraints and notes 2 \le N \le 502≤N≤50 1 \leq A_i \leq N1≤Ai≤N For 40\%40% of the test files, N \le 17N≤17.arrow_forward
- Write a Python script for each that will apply the algorithm to any given zero-one matrix. Attached in what is have so far for the code. The pseudocode is also attached. Please make sure that the code I have is what the psuedocode says. If not, please make edits to it so that it is correct.arrow_forwardGive a clear description of an efficient algorithm for finding the k smallest elements of a very large n-element vector. Compare its running time with that of other plausible ways of achieving the same result, including that of applying k times your solution for part (a). [Note that in part (a) the result of the function consists of one element, whereas here it consists of k elements. As above, you may assume for simplicity that all the elements of the vector are different.]arrow_forwardWrite the Python code to find the transitive closure when given zero-one matrix. DO NOT use the Warshall Algorithm in this code. Pseudo code: A := MR B := A for i := 2 to n A:= A ⊙ MR B:= B ∨ A return B (the zero-one matrix for R*)arrow_forward
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