Amihai Savir & Ilya Mirsky - 2013

Foundations of Data Structures

Practical Session #3Recurrence

08/04/2013

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Solution Methods

• Substitution methodGuess the form of the solution and prove it by induction.

• Iteration methodConvert the recurrence into a summation and solve it

• Master methodNext practical session…

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Question 0- warm up

Find the most tight asymptotic relation between the following pairs:

1. )

Θ

ΘO

ΘΘ

ΩΩOΘ

Ω

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Question 1

Prove that using the substitution method.

SolutionThe form of the solution is already given, so we don't have to guess it. We only need to prove that , i.e., that .

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Question 1 cont’d

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Question 2

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Question 3

fact(n) {if (n == 0) return 1return n * fact(n – 1)

}

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Question 4

Fibonacci series is defined as follows:

Find an iterative algorithm and a recursive one for computing element number in Fibonacci series, i.e., Fibonacci(n).Analyze the running-time of each of the algorithms.

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Question 4 cont’d

recFib(n) { if (n ≤ 1) return n else return recFib(n-1) + recFib(n-2)}

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Question 4 cont’d

In the same manner:

The series ends when

To summarize:

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Question 4 cont’d

iterFib (n) { allocate f[0..n] f[0] = 0 f[1] = 1 for (i=2 ; i ≤ n ; i++) f[i] = f[i-1] + f[i-2] return f[n] }

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Question 5public static int findMax(int [] arr) }

return FindMax(arr,0, arr.length-1);{public static int findMax( int a[], int left, int right )}

int middle;int max_l, max_r;if ( left == right )

return a[left];else }

middle = (left + right) / 2;max_l = FindMax( a, left, middle); max_r = FindMax( a, middle+1, right); return Math.max(max_l,max_r);

{{

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Question 5 cont’d