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 3.2, Problem 8E
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Let 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)),
Give asymptotic upper and lower bounds for each of the following recurrences. Justify your
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Determine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)
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Introduction to Algorithms
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- Show that: logn = O(n)arrow_forwardIf f(n) belongs to θ( g(n)) then it belongs to O(g(n)) True or False?arrow_forwardQuestion 3) Use the master theorem to give an asymptotic tight bound for the following recurrences. Tell me the values of a, b, the case from the master theorem that applies (and why), and the asymptotic tight bound. 3a) T(n) = : 2T (n/4) + n 3b) T(n) = 16T(n/4) + (√√n)³arrow_forward
- 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 )arrow_forwardLet f (n) and g(n) be positive functions (for any n they give positive values) and f (n) = O(g(n)).Prove or disprove the following statement:arrow_forwardLet, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an−1 + an−2 + n2, express an in terms of n.arrow_forward
- Solve the following recurrence exactly. t(n) = n if n = 0, 1, 2 or 3 t(n-1) + t(n-3) + t(n-4) otherwise Express your answer as simple as possible using the Θ notation.arrow_forwardShow by induction that T(n) ≤ 3n for all n ≥ 1arrow_forwardSolve the recurrence by using repeated substitution. Show the work. T(n) = T(n-1) + narrow_forward
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