Goals

Analysis of Nonrecurisve Algorithms

• Let's look at Find Max Min Average
  • As we discussed, line 6 would be a fine basic operation.
  • So T(n) = Σ_i=0^n-1 1 = n-1 - 0 +1 = n
• Let's do the same with Selection Sort
  • array size for n.
  • Line 4 for the basic operation

  • T(n) = Σi=0n-2 Σj=i+1 n-1 1
         = Σi=0n-2 n-1 - (i+1) + 1
         = Σi=0n-2 n-i-1
         = Σi=0n-2 n - 1 Σi=0n-2 i
         = (n-1) Σi=0n-2 1 - (n-2)(n-1)/2
         = (n-1) (n-2 -0+1) - (n-2)(n-1)/2
         = (n-1)[(n-1) -(n-2)/2]
         = (n-1)[n-1/2n -1+1]
         = (n-1)(1/2n) 
         = (n-1)(n)/2
         = 1/2 n2 - 1/2 n
         ∈ O(n2)
             

  • Observe, we could shift the indexes by 1 to simplify some of the math.
    • And that would correspond to the algorithm in FORTRAN
  • Another way we could have looked at this one is more intuitive
    • The first time, loop 3-5 would execute n-1 times.
    • The second time it would execute n-2 times.
    • ...
    • The last time it would execute 1 time.

    • T(n) = n-1 + n-2 + n-3 + ... + 2 + 1
           = $Sigma;i=1n-1 i
           = (n-1)(n)/2
               

• Matrix Multiplication

  • MATRIX_MULTIPLY(A,B)
    // both are nxn
    
     for i ← 0 to n-1 do
         for j ← 0 to n-1 do
             C[i,j] ← 0
             for k ← 0 to n-1  do  
                  C[i,j] ← C[i,j] + A[i,k]*B[k,j]
     return C
    
    	

  • Same questions
    • Does it work?
    • How does it work?
    • Is it correct?
    • Critical Operation?
    • Measure of the size of the input?

    • T(n) = Σ0n-1 Σ0n-1 Σ0n-1 1
           = Σ0n-1 Σ0n-1 n 
           = Σ0n-1 n Σ0n-1 1 
           = Σ0n-1 n2 
           = n2 Σ0>n-1 1
           = n3
                    

    • Is this best, worst, or average?