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 15.2, Problem 2E
Program Plan Intro
To describe a recursive
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Determine a recurrence relation for the divide-and-conquer sum-computation algorithm. The
problem is computing the sum of n numbers. This algorithm divides the problem into two
instances of the same problem: to compute the sum of the first Ln/2J numbers and compute
the sum of the remaining Ln/2J numbers. Once each of these two sums is computed by
applying the same method recursively, we can add their values to get the sum in question.
T(n) = T(n/2) + 1
T(n) = 2T(n/2) + 2
T(n) = 2T(n/2) + 1
T(n) = T(n/2) + 2
Another recursive algorithm is applied to some data A = (a₁, ..., am) where
m = 2* (i.e. 2, 4, 8,16 ...) where x is an integer ≥ 1. The running time T is
characterised using the following recurrence equations:
T(1) = c when the size of A is 1
T(m) = 2T (2) + c otherwise
Determine the running time complexity of this algorithm.
Write 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*)
Chapter 15 Solutions
Introduction to Algorithms
Ch. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5ECh. 15.2 - Prob. 1ECh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5E
Ch. 15.2 - Prob. 6ECh. 15.3 - Prob. 1ECh. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.4 - Prob. 1ECh. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15 - Prob. 1PCh. 15 - Prob. 2PCh. 15 - Prob. 3PCh. 15 - Prob. 4PCh. 15 - Prob. 5PCh. 15 - Prob. 6PCh. 15 - Prob. 7PCh. 15 - Prob. 8PCh. 15 - Prob. 9PCh. 15 - Prob. 10PCh. 15 - Prob. 11PCh. 15 - Prob. 12P
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- 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…arrow_forwardShow Let f(.) be a computable, strictly monotonic function, that is, f(n+ 1) > f(n) for all n. Show B = {f(n) | n ∈ N} is recursive.arrow_forwardes remaining 8. Consider the function f:NxN-N defined recursively by: 1) Base case: Let meN and define (0,m) = 0 2) Recursive case: For any x,meN, x>0, define f(x,m) = (x-1,m) + (m+m) Prove the following theorem holds using proof by induction: Thereom: For any n,meN, m>0 we have (n.m) I m = n+n Fill in your answer here 9 Help BIU X, x L - ɔE =N E X Formatarrow_forward
- 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…arrow_forward2. Expand the following recurrence to help you find a closed-form solution, and then use induction to prove your answer is correct. T(n) = T(n-1) + 5 for n> 0; T(0) = 8. 3. Give a recursive algorithm for the sequential search and explain its running time.arrow_forwardCreate a 3-by-3 matrix. A = [1 7 3; 2 9 12; 5 22 7]; Calculate the right eigenvectors, V, the eigenvalues, D, and the left eigenvectors, W. [V,D,W] eig(A) %3!arrow_forward
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