How Quick Sort shows different Time Complexity in different cases with examples
Example: analysis of quick sort
T(n) = 2*T(n/2) + n // T(n/2) = 2*T(n/4) + (n/2)
= 2*[ 2*T(n/4) + n/2 ] + n
= 22*T(n/4) + n + n
= 22*T(n/4) + 2n // T(n/4) = 2*T(n/8) + (n/4)
= 22*[ 2*T(n/8) + (n/4) ] + 2n
= 23*T(n/8) + 22*(n/4) + 2n
= 23*T(n/8) + n + 2n
= 23*T(n/8) + 3n
= 24*T(n/16) + 4n
and so on....
= 2k*T(n/(2k)) + k*n // Keep going until: n/(2k) = 1 <==> n = 2k
= 2k*T(1) + k*n
= 2k*1 + k*n
= 2k + k*n // n = 2k
= n + k*n
= n + (lg(n))*n
= n*( lg(n) + 1 )
~= n*lg(n))