DERIVING PARAMETERIZED TIME COMPLEXITY FUNCTIONS:

Example Algorithm #1:

    read n                        --
    sum = 0                        |
    i = 1                          |
    while (i <= n)            --   |
        sum = sum + P1(n) --   |   |     T(n) = 3n + nT(P1) + 5
        i = i + 1         --  --   |
    print sum              |   |  --
                   2 + T(P1)   |   | 
                               |   |
            n(2 + T(P1) + 1) + 1   |
            = 3n + nT(P1) +1       |
                                   |
                (3n + NT(P1) +1) + 4
                = 3n + nT(P1) + 5


Example Algorithm #2: 

    read n                                   --
    sum = 0                                   |
    i = 1                                     |
    while (i <= n)                       --   |
        j = P1(n)                    --   |   |
        while (j <= n)           --   |   |   |    T(n) = 3n^2 + n^2T(P2) + 
            sum = sum +       --  |   |   |   |             4n + nT(P1) + 5
               (i * (j + 1)) + |  |   |   |   |
	       P2(n)           |  |   |   |   |
            j = j + 1         -- --   |   |   |
        i = i + 1              |  |  --  --   |
    print sum                  |  |   |   |  --
                       2 + T(P2)  |   |   |   |
                                  |   |   |   |
               n(2 + T(P2) + 1) + 1   |   |   |
               = 3n + nT(P2) + 1      |   |   |
                                      |   |   |
          (3n + nT(P2) + 1) + 2 + T(P1)   |   |
          = 3n + nT(P2) + 3 + T(P1)       |   |
                                          |   |
       n((3n + nT(P2) + 3 + T(P1)) + 1) + 1   |
       = 3n^2 + n^2T(P2) + 4n + nT(P1) + 1    |
                                              |
        (3n^2 + n^2T(P2) + 4n + nT(P1) + 1) + 4 
         = 3n^2 + n^2T(P2) + 4n + nT(P1) + 5