To show that multiplying and squaring matrices have the same difficulty: an M(n) -time matrix-multiplication algorithm indicates O(M(n)) - time squaring algorithm, and as S ( n ) -time squaring algorithm indicates an O(S(n)) -time matrix-multiplication algorithm.

Question

Want to see the full answer?

Answer 1

Question

Chapter 28.2, Problem 1E

Program Plan Intro

To show that multiplying and squaring matrices have the same difficulty: an M(n) -time matrix-multiplication algorithm indicates O(M(n)) - time squaring algorithm, and as S(n) -time squaring algorithm indicates an O(S(n)) -time matrix-multiplication algorithm.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

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).

We are given a code symmetry, that takes an n x n matrix as its argument. It returns true if the matrix is symmetric and false if it is not. The run time of the code is calculated by counting how many times the /= comparison is executed. procedure symmetry(M) for x:=1 to (n-1) for y:=(i+1) to n if mxy /= myx then return false else return true (a) Calculate the best case run time for symmetry (will be a number). (b) Calculate the worst case run time for symmetry (will be a polynomial in n, it will be helpful to draw matrices of different sizes and compare the number of comparisons to find a pattern). (c) Determine the worst case run time in big O notation (d) Using the following definition of big O notation, prove the answer to (c) Definition: Let f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) if there are constants C and k such that whenever x > k. [This is…

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…