Data Structures and Algorithms in Java
6th Edition
ISBN: 9781118771334
Author: Michael T. Goodrich
Publisher: WILEY
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Chapter 4, Problem 16R
Explanation of Solution
Given:
It is given that if
Big-oh notation:
In big-oh notation, let “f” and “g” be functions from the integers or the real numbers to the real numbers. It means that
Proof:
Assume that
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Suppose that f(n) is O(g(n)), g(n) is O(h(n)), h(n) = l(n) + k(n) where l(n) is O(f(n)/2).
Which of the following may not hold.
O k² (n) is N(f(n))
O f(n) + g(n) is E(h(n))
O 2f(n) is N(e(n))
O e(n)h(n) is O(g² (n) + f(n))
Suppose that f (n) = 0(g(n)) and f(n) = 0(h(n)), then it is ( always / sometimes / never ) the case that g(n) = 0(h(n)).
Prove or disprove that if f₁ (n) = N(9₁ (n)) and f2 (n) = N(92 (n)), then
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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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- Show that T(n) = O(n²) where T(n) = 1 + 2 + 3+...+narrow_forwardTrue/False? Let f(n)=O(g(n)) and g(n)=O(T(n)). Is this true that f(n)=O(T(n))? Prove your claim and justify formally (using the formal definitions of asymptotic notations).arrow_forwardGiven T1(n)=O(f(n)) and T2(n)=O(g(n)) . Find T1(n).T2(n) and proof the statement is truearrow_forward
- f(n) = 1 if n is odd f(n) = n if n is even 1. Prove/disprove that f(n) is O(n) (Big-Oh) 2. Prove/disprove that f(n) is Omega (n) 3. Prove/disprove that f(n) is Theta (n)arrow_forwardf(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):arrow_forward11. Show that lower order terms don't matter in the O() notation by showing that if g(n) = O(f(n)) then g(n) + f(n) = O(f(n)).arrow_forward
- Prove the following: a. [(a mod n) - (b mod n)] mod n = (a - b) mod n b. [(a mod n) * (b mod n)] mod n = (a * b) mod narrow_forwardwe are assuming f(n) and g(n) are asymptotically positive functions. Prove/ disproveeach of the following.arrow_forwardAre any of the following implications always true? Prove or give a counter-example.a) f(n) = Θ(g(n)) ⇒ f(n) = cg(n) + o(g(n)), for some real constant c > 0. (Here small o) b) f(n) = Θ(g(n)) ⇒ f(n) = cg(n) + O(g(n)), for some real constant c > 0.(here big O) Please complete botharrow_forward
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