Concept explainers
(a)
To determine the asymptotic bounds for the recurrence relation using master method.
(a)
Explanation of Solution
Given Information: The recurrence relation is
Explanation:
For a divide and conquer recurrence of the form
Case 1: If
Case 2: If
Case 3: If
The values of
Therefore,
So, case 3 of the master method applies.
Hence,
(b)
To determine the asymptotic bounds for the recurrence relation using master method.
(b)
Explanation of Solution
Given Information: The recurrence relation is
Explanation:
The values of
Therefore,
So, case 3 of the master method applies.
Hence,
(c)
To determine the asymptotic bounds for the recurrence relation using master method.
(c)
Explanation of Solution
Given Information: The recurrence relation is
Explanation:
The values of
Therefore,
So, case 2 of the master method applies.
Hence,
(d)
To determine the asymptotic bounds for the recurrence relation using master method.
(d)
Explanation of Solution
Given Information: The recurrence relation is
Explanation:
The values of
Therefore,
So, case 3 of the master method applies.
Hence,
(e)
To determine the asymptotic bounds for the recurrence relation using master method.
(e)
Explanation of Solution
Given Information: The recurrence relation is
Explanation:
The values of
Therefore,
So, case 1 of the master method applies.
Hence,
(f)
To determine the asymptotic bounds for the recurrence relation using master method.
(f)
Explanation of Solution
Given Information: The recurrence relation is
Explanation:
The values of
Therefore,
So, case 2 of the master method applies.
Hence,
(g)
To determine the asymptotic bounds for the recurrence relation using master method.
(g)
Explanation of Solution
Given Information: The recurrence relation is
Explanation:
The recurrence relation is not in the form of master theorem. Therefore, it cannot be solve by master theorem.
Solve the recurrence relation
as follows:
Therefore, the asymptotic notation of the recurrence
Want to see more full solutions like this?
Chapter 4 Solutions
Introduction to Algorithms
- Let T(n) be defined by the recurrence relation T(1) = 1, and T(n) = 32T(n/2) + n^k for all n > 1. Determine the integer k for which T (n) = Θ(n^(5)*log n).arrow_forwardGive asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T (n) is constant for n ≤ 3. Make your bounds as tight as possible, and justify your answers (you can use any of the methods we discussed in class). 1). T(n)=T(n/2)+lgn.arrow_forwardSolve the following recurrences assuming that T(n) = Θ(1) for n ≤ 1. a) T (n) = 3T (n/π) + n/π b) T(n) = T(log n) + log narrow_forward
- Give tight asymptotic upper bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for sufficiently small n. Please give me a step by step answers, thanks.arrow_forwardWe have already had a recurrence relation of an algorithm, which is T (n) = 4T (n/2) + n log n. We know T (1) ≤c.(a) express it as T (n) = O(f (n)), by using the iteration method.(b) Prove, by using mathematical induction, that the iteration rule you have observed in (a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T (n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction. Then by finding out the eventual number of t and substituting it into your general form of T (n), you get the O(·) notation of T (n).]arrow_forwardConsider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.arrow_forward
- Let T(n) be the maximum number of guesses required to correctly identify a secret word that is randomly chosen from a dictionary with exactly n words. Determine a recurrence relation for T(n), explain why the recurrence relation is true, and then apply the Master Theorem to show that T(n) = Θ(log n).arrow_forwardAssume that T(n) = n for n≤2. For the questions below, find the tightest asymptotic upper bound and show the method you used to obtain it. 1) T(n)=2T(√(n)) (square root of n) 2) T(n)=T(n−1)+10 3) T(n)=(T(n/2))2arrow_forwardSolve the following recurrences using recursion tree method and write the asymptotic time-complexity. 1. T(n) = 3T (n/4) + n^22. T(n) = T (n/5) + T(4n/5) + n3. T(n) = 3T(n − 1) + n^4 4. T(n) = T (n/2) + n^2arrow_forward
- Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T (n) is constant for n ≤ 3. Make your bounds as tight as possible, and justify your answers T (n) = 3T (n/5) + log^2 narrow_forwardUse the master method to give tight asymptotic bounds for the following recurrence T(n) = 2T(n/4) + 1 Group of answer choices 1. ϴ(n0.5lgn) 2. ϴ(n0.5) 3. ϴ(n2) 4. ϴ(n)arrow_forwardConsider the following recurrence algorithm Algorithm 1 REVERSE 1. function REVERSE(A)2. if |A| = 1 then3. return A4. else5. Let B and C be the first and second half of A6. return concatenate REVERSE(C) and REVERSE(B)7. end if8. end function Let T(n) be the running time of the algorithm on an instance of size n. Write down the recurrence relation for T(n) and solve itarrow_forward
- C++ Programming: From Problem Analysis to Program...Computer ScienceISBN:9781337102087Author:D. S. MalikPublisher:Cengage Learning