Mathematical Analysis of Non Recursive Algorithms

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Mathematical Analysis of Non Recursive
Algorithms (Section 2.3)
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Non Recursive Algorithms
Complexity of Algorithm X
Algorithm X (n) i ? 5 j ?
while ( i lt ) k ?
Algorithm Y() for ()
while ()
return
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MaxElement
MaxElement(A1..n) maxval ? A1 for i ? 2
to n do if Ai gt maxval
maxval ? Ai return maxval
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UniqueElement
UniqueElement(A1..n) for i ? 1 to n-1 do
for j ? i1 to n do if Ai
Aj return false return true
5
MatrixMultiplication
UniqueElement(A1..n,1..n, B1..n,1..n) for i
? 1 to n do for j ? 1 to n do
Ci,j ?0 for k ? 1 to n do
Ci,j ? Ci,j Ai,k
Bk,j
return C
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Recursive Algorithms
Algorithm X (n, ) if (n 0) return value
while ( i lt )
return Algorithm X(n1) // where n1 lt n
Complexity of Algorithm X T(n)
//n is the size of the input
T(n) f(n) T(n1) T(0) 1
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Solving a Recursive Equation

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

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SumElement

• SumElement(A1..n, i)
• If (i n) then return An
• Else
• return Ai sumElement(A,i1)