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Data Structures and Algorithms in Java
6th Edition
ISBN: 9781119278023
Author: Michael T. Goodrich; Roberto Tamassia; Michael H. Goldwasser
Publisher: Wiley Global Education US
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Textbook Question
Chapter 4, Problem 50C
Consider the Fibonacci function, F(n) (see Proposition 4.20). Show by induction that F(n) is Ω((3/2)n).
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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):
Let 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)).
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)),
Chapter 4 Solutions
Data Structures and Algorithms in Java
Ch. 4 - Prob. 1RCh. 4 - The number of operations executed by algorithms A...Ch. 4 - The number of operations executed by algorithms A...Ch. 4 - Prob. 4RCh. 4 - Prob. 5RCh. 4 - Prob. 6RCh. 4 - Prob. 7RCh. 4 - Prob. 8RCh. 4 - Prob. 9RCh. 4 - Prob. 10R
Ch. 4 - Prob. 11RCh. 4 - Prob. 12RCh. 4 - Prob. 13RCh. 4 - Prob. 14RCh. 4 - Prob. 15RCh. 4 - Prob. 16RCh. 4 - Prob. 17RCh. 4 - Prob. 18RCh. 4 - Prob. 19RCh. 4 - Prob. 20RCh. 4 - Prob. 21RCh. 4 - Prob. 22RCh. 4 - Show that 2n+1 is O(2n).Ch. 4 - Prob. 24RCh. 4 - Prob. 25RCh. 4 - Prob. 26RCh. 4 - Prob. 27RCh. 4 - Prob. 28RCh. 4 - Prob. 29RCh. 4 - Prob. 30RCh. 4 - Prob. 31RCh. 4 - Prob. 32RCh. 4 - Prob. 33RCh. 4 - Prob. 34RCh. 4 - Prob. 35CCh. 4 - Prob. 36CCh. 4 - Prob. 37CCh. 4 - Prob. 38CCh. 4 - Prob. 39CCh. 4 - Prob. 40CCh. 4 - Prob. 41CCh. 4 - Prob. 42CCh. 4 - Prob. 43CCh. 4 - Draw a visual justification of Proposition 4.3...Ch. 4 - Prob. 45CCh. 4 - Prob. 46CCh. 4 - Communication security is extremely important in...Ch. 4 - Al says he can prove that all sheep in a flock are...Ch. 4 - Consider the following justification that the...Ch. 4 - Consider the Fibonacci function, F(n) (see...Ch. 4 - Prob. 51CCh. 4 - Prob. 52CCh. 4 - Prob. 53CCh. 4 - Prob. 54CCh. 4 - An evil king has n bottles of wine, and a spy has...Ch. 4 - Prob. 56CCh. 4 - Prob. 57CCh. 4 - Prob. 58CCh. 4 - Prob. 59CCh. 4 - Prob. 60PCh. 4 - Prob. 61PCh. 4 - Perform an experimental analysis to test the...Ch. 4 - Prob. 63P
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- 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_forwardLet f(n) and g(n) be asymptotically positive functions. Prove or disprove following. f(n) + g(n) = q(min(f(n), g(n))).arrow_forward3. Prove by induction that T(n) = 2T (n/2) + cn is O(n logn).arrow_forward
- 3.1-1 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_forward1. Let f(n) and g(n) be asymptotically positive functions. Prove or disprove the follow- ing conjectures: (a) f(n) + g(n) = 0(min(f(n), g(n))). (b) f(n) + w(f(n)) = ©(f(n)).arrow_forward(b) Prove: max(f(n), g(n)) E O(f(n)g(n)), i.e., s(n) E O(f(n)g(n)). Assume for all natural n, f(n) > 1 and g(n) > 1. Let s(n) = max(f(n), g(n)).arrow_forward
- Mathematical Induction: Binet's formula is a closed form expression for Fibonacci numbers. Prove that binet(n) =fib(n). Hint: observe that p? = p +1 and ² = b + 1. function fib(n) is function binet(n) is match n with let case 0 → 0 2 case 1 1 otherwise in L fib(n – 1) + fib(n – 2) V5arrow_forwardDetermine 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_forwardShow that f (n) is O(g(n)) if and only if g(n) is Ω( f (n)).arrow_forward
- * Prove that for any constants c, c', log.(n) = 0(log. (n)).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_forwardLet f(n) and g(n) be positive functions over the natural numbers. For each of the following claims either prove formally that the claim is correct, or disprove it by giving a counter example. a) f(n) is e(f(n/2)) . b) f(n) + g(n) is E(min(f(n),g(n)). c) f(n) + g(n) is E(max(f(n),g(n)). d) if f(n)f(n) is O(n“) then f(n) is O(n).arrow_forward
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