Mathematical Analysis of Non- recursive Algorithm PREPARED BY, DEEPA. B, AP/ CSE VANITHA. P, AP/ CSE

Steps in mathematical analysis of non-recursive algorithms Decide on parameter n indicating input size Identify algorithm‘s basic operation Determine worst, average, and best case for input of size n Set up summation for C(n) reflecting algorithm‘s loop structure Simplify summation using standard formulas

Example: Selection sort Input: An array A[0..n-1] Output: Array A[0..n-1] sorted in ascending order for i ← 0 to n-2 do min ← i for j = i + 1 to n – 1 do if A[j] < A[min] min ← j swap A[i] and A[min]

basic operation: comparison Inner loop: n-1 S(i) = Σ 1 = (n-1) – (i + 1) + 1 = n – 1 - i j = i+1 Outer loop: n-2 n-2 Op = Σ S(i) = Σ (n – 1 – i) = Σ (n – 1) - Σ i i = 0 i = 0 Basic formula: n Σ i = n(n+1) / 2 i = 0 Op = (n – 1 )(n -1 ) – (n-2)(n-1)/2 = (n – 1) [2(n – 1) – (n – 2)] / 2 = =(n – 1) n / 2 = O(n2)

Time efficiency of nonrecursive algorithms General Plan for Analysis Decide on parameter n indicating input size Identify algorithm’s basic operation Determine worst, average, and best cases for input of size n Set up a sum for the number of times the basic operation is executed Simplify the sum using standard formulas and rules

Useful summation formulas and rules  l  i  n 1 = 1+1+…+1 = n - l + 1 In particular,  l  i  n 1 = n = n   (n)  1  i  n i = 1+2+…+n = n(n+1)/2  n 2 /2   (n 2 )  1  i  n i 2 = …+n 2 = n(n+1)(2n+1)/6  n 3 /3   (n 3 )  0  i  n a i = 1 + a +…+ a n = (a n+1 - 1)/(a - 1) for any a  1 In particular,  0  i  n 2 i = …+ 2 n = 2 n   (2 n )  (a i ± b i ) =  a i ±  b i  ca i = c  a i  l  i  u a i =  l  i  m a i +  m+1  i  u a i

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