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Mathematical Analysis of Non Recursive Algorithms (Section 2.3)

Mathematical Analysis of Non Recursive Algorithms (Section 2.3). O(1). O(f(n)). O(g(n)). O(t(n)). O(1). Non Recursive Algorithms. Complexity of Algorithm X =. Algorithm X (n) i  5 j  … while ( i < …) …

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Mathematical Analysis of Non Recursive Algorithms (Section 2.3)

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  1. Mathematical Analysis of Non Recursive Algorithms (Section 2.3)

  2. O(1) O(f(n)) O(g(n)) O(t(n)) O(1) Non Recursive Algorithms Complexity of Algorithm X = Algorithm X (n) i  5 j  … while ( i < …) … k  Algorithm Y(…) … for (…) while (…) … return …

  3. MaxElement MaxElement(A[1..n]) maxval  A[1] for i  2 to n do if A[i] > maxval maxval  A[i] return maxval

  4. UniqueElement UniqueElement(A[1..n]) for i  1 to n-1 do for j  i+1 to n do{ if A[i] = A[j] return false} return true

  5. MatrixMultiplication UniqueElement(A[1..n,1..n], B[1..n,1..n]) for i  1 to n do for j  1 to n do C[i,j] 0 for k  1 to n do C[i,j]  C[i,j] + A[i,k] *B[k,j] return C

  6. termination f(n) Recursive Algorithms Algorithm X (n, …) if (n = 0) return value while ( i < …) … return Algorithm X(n1) // where n1 < n Complexity of Algorithm X = T(n) //n is the size of the input T(n) = f(n) + T(n1) T(0) = 1

  7. Solving a Recursive Equation • Make a few evaluations of T(n) for a few values n1, n2, n3 according to the recursive call • Deduce a pattern for T(n) • Compute the number of times, k, the recursive call is made until the termination condition (e.g.,. T(0)) • Use 1 and 3 for obtaining a final equation T(n) • Solve the equation T(n)

  8. SumElement(A[1..n], i) If (i = n) then return A[n] Else return A[i] + sumElement(A,i+1) SumElement

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