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 3.1, Problem 1E
Program Plan Intro
To prove that the expression
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Let 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:
Let f(n) and g(n) be asymptotically positive functions. Prove or disprove following. f(n) + g(n) = q(min(f(n), g(n))).
f(n) = O(f(n)g(n))
Indicate whether the below is true or false. Explain
your reasoning.
For all functions f(n) and g(n):
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Introduction to Algorithms
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- 6. Let f(n) and g(n) be non-negative functions. Show that: max(f(n), g(n)) = 0(f(n) + g(n)).arrow_forwardLet f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).arrow_forwardShow that if f (n) and g(n) are monotonically increasing functions, then so are the functions f (n) / C g(n) and f g(n) and if f (n) and g(n) are in addition non-negative, then f (n)/g(n) is monotonically increasing.arrow_forward
- use summation to analyze the running time (i.e. T(n)) of these functions and able to find some simple function f(n) such that T(n) = Θ(f(n)). show the steps, pleasearrow_forward3.1-1 Let f(n) and g(n) be asymptotically nonnegative functions. Using the basic defi- nition of -notation, prove that max(f(n), g(n)) = Ⓒ(f(n) + g(n)).arrow_forwardThe Legendre Polynomials are a sequence of polynomials with applications in numerical analysis. They can be defined by the following recurrence relation: for any natural number n > 1. Po(x) = 1, P₁(x) = x, Pn(x) = − ((2n − 1)x Pn-1(x) — (n − 1) Pn-2(x)), n Write a function P(n,x) that returns the value of the nth Legendre polynomial evaluated at the point x. Hint: It may be helpful to define P(n,x) recursively.arrow_forward
- 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.)arrow_forwardGive an example of a function in n that is in O(√n) but not in Ω(√n). Briefly explainarrow_forwardDefine a function S : Z+ → Z+ as follows. For each positive integer n, S(n) = the sum of the positive divisors of n. Find S(17)arrow_forward
- Find f(n) and big Oarrow_forwardLet f(n) y g(n) two positive asymptotic functions. Prove or disprove the following conjectures: a) f(n) = O(g(n)) implies 2f(n) = O(29(n)). b) f(n) = O(g(n)) implies g(n) = N(f(n)). c) g(n) = O((g(n))²).arrow_forwardGiven f(n) ∈ Θ(n), prove that f(n) ∈ O(n²).arrow_forward
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