Example 4: Consider the following version of the Fibonacci sequence starting from Fo= 0 and defined by: Fo=0 F₁ = 1 } Fibonacci Sequence Fn+2Fn+1+ F₁; n ≥ 0: Prove the following identity, for any fixed k ≥ 1 and all n ≥ 0. Fn+kFkFn+1 + Fx-1Fn- The Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Assertion: Fn+k=FxFn+1 + Fk-1Fn: for any fixed k ≥ 1 and all n ≥ 0 k-1, n=0 F₁ F₁F₁ + FoFo = 1 k=1, n=1 F₁₂ F₁F₂ + FoF₁ = ? 1 = 1*1 + 0*1 = 1 k-5, n=4 F, F5F5 + F4F4 = 5*5 +3*3 = 34 Assumption for pair (k,n: k≥ 1, n ≥ 0) assertion is valid. Is this identity true for n+1, k+1? Fn+k+2= Fk+1Fn+2 + FkFn+1 ?? True??

Which of the following is the solution using the Master Theorem for the recurrence relation T(n)=4T(n/2) + log n A) 0 (n²) B) 0(n² log n) C) 0 (2") D) 0 (n!) E) 0 (n log n)

The runtime complexity, T(n), of the three following recurrence relation solved by Master's Theorem) are T(n) = 6T(n/3) + n² logn T(n) = 64T(n/8)- n² log n T(n) = 4T(n/2) + n/logn (The solution for the three relations are respectively given by) A: (nlogn), 9(n²), (n² logn) B: 9(n²), (n² logn), (n log n) C: (n²logn), Master's Theorem does not apply, 9(n²) D: 0(n²), Master's Theorem does not apply, 9(n² log n) E: 0(n² logn), 9(n²), Master's Theorem does not apply F: 9(n²), (n² logn), Master's Theorem does not apply

Find the order of growth for solutions of the following recurrences using master theorem. 1. T(n) = 4T(n/2) + n, T(1) = 1 2. T(n) = 4T(n/2) + n^2, T(1) = 1 3. T(n) = 4T(n/2) + n^3, T(1) = 1

find recurrence relations that yield the following sequences, answer a and b

Let ap = 1, an = an-1 + 3 be the recurrence relation of algorithm A and T(1) = 1, T(n) = T(n/5)+n be the recurrence relation of algorithm B. 1. Find the solution for both relations. Use any method.

Determine the strongest asymptotic relation between the functions f(n)= 2log and g(n)-(logn), i.e., whether f(n) = o(g(n)), f(n) = O(g(n)), f(n) = (g(n)), f(n) = w(g(n)), or f(n) = (g(n)).

Explain the complete work !

Assume that void some_fct (int X); has time complexity Ⓒ(X) for (k 0; k <= N; k k+1) { some fct (k); // } Final answer: TCfor-k = O(_ =