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 7.4, Problem 1E
Program Plan Intro
To show that the recurrence relation
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Use the substitution method to show that for the recurrence equation:
T( 1 )=1
T( n )=T( n/3 ) + n the solution is T( n )=O ( n )
Solve the recurrence: T(n) = T(n/2) + 4n
T(1) = 1
Solve the following recurrences exactly:(a) T(1) = 8, and for all n ≥ 2, T(n) = 3T(n − 1) + 15.(b) T(1) = 1, and for all n ≥ 2, T(n) = 2T(n/2) + 6n − 1 (n is a power of 2)
Chapter 7 Solutions
Introduction to Algorithms
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- Use the master method to give tight asymptotic bounds for the following recurrence T(n) = 2T(n/4) + nº.5 (nº.5Ign) e(nº.5) e(n) ○ e(n²)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_forwardSolve the first-order linear recurrence T(n) = 3T(n − 1) +8, T(0) = 6 by finding an explicit closed formula for T(n) and enter your answer in the box below. T(n) =arrow_forward
- Find the order of growth for solutions of the following recurrences. a. T (n) = 4T (n/2) + n, T (1) = 1 b. T (n) = 4T (n/2) + n2, T (1) = 1 c. T (n) = 4T (n/2) + n3, T (1) = 1arrow_forwardSolve the recurrence; T(n)=2T(n/2) + cn T(1) = carrow_forwardSolve recurrence equation T(n) = T(n-1) +2arrow_forward
- Solve the recurrence relation: T (n) = T (n/2) + T (n/4) + T (n/8) + n. Use the substitution method, guess that the solution is T (n) = 0 (n log n). Solve the recurrence relation T (n) = T ( √n) + c. n > 4 Derive the runtime of the below codearrow_forwardSolve the recurrence relation: T (n) = T (n/2) + T (n/4) + T (n/8) + n. Use the substitution method, guess that the solution is T (n) = O (n log n)arrow_forwardLet f (f(n) and g(n)) be asymptotically nonnegative functions. Using the basic definition of Θ notation, prove that max(f(n), g(n)) = Θ(f(n) + g(n)),arrow_forward
- Given the following recurrence, find the growth rate of T(n):T(n) = 4T(n/2) + 6n3 with T(1) = Θ(1)arrow_forwardIf g(n) = O(f(n)),by using the definition of Big-Θ, prove that f(n) + g(n) = Θ(f(n)).arrow_forwardLet T(n) be defined by the first-order linear recurrence T(n) = 2T(n-1) +8 Suppose it is given that T(2) = c. Compute T(0) by iterating backwards and express your answer in terms of c. T(0) =arrow_forward
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