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 8E
Program Plan Intro
To describe the definitions for
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f(n) = 2", g(n) = 2.01".
v.
Define a function
S: Z+ → Z+
as follows. For each positive integer n,
S(n) =
the sum of the positive divisors of n.
1.) S(13) =
2.)S (5) =
Analyze the running time (i.e. T(n)) of these functions. You
should be able to find some simple function f(n) such that T(n)
O(f(n)). You should show your work and rigorously justify your an-
1.
swer.
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Introduction to Algorithms
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- Find t(4)(n) for the function t(n)=5n−1/5+7n8/5.arrow_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_forwardShow that if f(x) and g(x) are functions from the set of real numbers to the set of real numbers, then f(x) is O(g(x)) if and only if g(x) is Ω(f(x)).arrow_forward
- Find (f o g) and (g o f) , where f (x) =x2+1 and g(x) = x + 2, are functions from R to R.arrow_forwardSimplify the given function using K map f(a,b,c,d)= ∑m (1, 2, 4, 11, 13, 14, 15) + d (0, 5, 7, 8, 10)arrow_forwardMinimize the following boolean function- F(A, B, C, D) = {m(0, 1, 3, 5, 7, 8, 9, 11, 13, 15)arrow_forward
- Let X= (-3,-2,-1,0, 1}. Let f: X→ N by f(x) = [7z+11: The range of the function isarrow_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_forward3.1-8 We can extend our notation to the case of two parameters n and m that can go to infinity independently at different rates. For a given function g(n, m), we denote by O(g(n, m)) the set of functions O(g(n,m)) = {f(n,m): there exist positive constants c, no, and mo such that 0≤ f(n,m) ≤ cg(n,m) for all n ≥ no or m≥ mo}. Give corresponding definitions for (g(n, m)) and (g(n, m)).arrow_forward
- Let N = {0, 1, 2, . . .} be the set of Natural Numbers. Given an n ∈ N, which of the followingconditions are necessary, and which of these conditions are sufficient, for the Natural Number,n, to be a factor of 10.(a) 1 is a factor of n.(b) 1 is a factor of 2n.(c) −n is a factor of 10.(d) 10 is a multiple of n.(e) n is divisible by 2.(f) n^2 is divisible of 5.(g) n = 10.arrow_forwardSimplify the following Boolean functions by means of K-map:a) f(A,B,C) =∑m(0,2,3,4,6)b) f(A,B,C,D) = ∑ m(0,2,3,7,8,10,12,13)c) F(X,Y,Z) = Y’Z+YZ+X’Y’Z’arrow_forwardGiven the function F(A, B, C, D) = Σm(0, 1, 3, 5, 7, 8, 9, 11, 13, 15)arrow_forward
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