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 28.2, Problem 1E
Program Plan Intro
To show that multiplying and squaring matrices have the same difficulty: an
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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…
What is the time complexity for computing the dot product of two n-dimensional vectors? I.e. the operation
is x¹y, where x, y are two n-dimensional column vectors.
3. Explain why the next matrix in Floyd’s algorithm can be written over its predecessor (i.e., when calculating D(k) from D(k-1), we do not need another matrix but can change D(k-1) into D(k).
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- Find the generalized time functions of the following code snippet Matrics_Multiplication (A, B) { // nx m matrix A // m x p matrix B // n×p matrix C // C= AB 1. Let C be a new matrix of the appropriate size 2. For i from 1 to n: For j from 1 to p: 3. 4. Let sum = 0 For k from 1 to m: 5. Set sum + sum + Ajk × Bj 6. 7. Set Cj - sum 8. Return C }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_forwardFollowing code is a part of an algorithm where A = [a; ;] is an N x N matrix with a;i = a(i, j), x, y, b are vectors of length N. Count the total number of operations (addition/subtraction and multi- plication/division) for the code below in terms of N (Give a simplified answer in terms of N): Begin Algorithm fct = 2 sum for i = 1 to N x(i) y(i) = Fa end i loop for j = 1 to N suml = 0 for k = j+1 to N suml = suml+a(j,k) * y(k) end k loop r(j) end j loop sum = 10 + sum1 * fct b(j)-sum a(j, j) End Algorithmarrow_forward
- Array P = [40, 30, 25, 10, 35, 5, 20] Suppose the dimension of 6 matrices (A1, A2 … A6) are given by array P A1 is a P[0] x P[1] matrix A2 is a P[1] x P[2] matrix . . . A6 is a P[5] x P[6] a) Find the minimum number of scalar multiplications necessary to calculate the product of all the 6 matrices (A1.A2.A3.A4.A5.A6) and show the parenthesization for this multiplication. • Solve the problem manually (you need not to write any code) using bottom-up tabulation approach. Compute and show the ‘m’ matrix and ‘s’ matrix to solve your problem.arrow_forwardSparse matrix-vector multiplication in Python is similar to numpy.dot operation, however the provided @ operator is used. Let us consider multiplying a 5 by 5 sparse matrix with a 5 by 3 sparse matrix shown below. 01000 00010 10001 10000 01000 100 001 x 000 010 100 The Python code to implement multiplication of these two sparse matricesarrow_forwardWrite pseudo-code algorithms for each of the matrix operations (matrix addition, scalar multiplication and product of two matrices),and matrix should be 3*3arrow_forward
- Write pseudo-code algorithms for matrix addition, scalar multiplication, and product of two matrices in 3*3 matrix.arrow_forwardThis question is about the Matrix Chain Multiplication problem. You are given four matrices with the following dimensions.arrow_forwardTrue or False The inverse of a matrix M can be calculated in Python with the command Inverse=np.linalg.inv(M)arrow_forward
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