Time complexity of median of medians algorithm

Question

Hello I am taking Introduction to algorithm class this semeseter. However I have some problem in calculating time complexity of median of medians algorithm (here). I'm wondering how to get T(n)<=10cn from T(n)<=T(0.2n)+T(0.7n)+cn..

I think I cannot apply mater theorem to the expression above and wikipedia says I should use induction but I don't know How..

Answer 1

It is using induction. Assume for less than or equal n we have T(n) <= 10*c*n (we know the base of induction is true for equal or less than T(10) as they are constant and we can use large enough value for c). Now we want to prove this for T(n + 1). We know T(n + 1) <= T(0.2(n + 1)) + T(0.7( n + 1)) + c(n+1). From the assumption of the induction as 0.2(n + 1) and 0.7(n+1) is less than n (for n > 10), T(0.2(n + 1)) <= 10*c*0.2(n + 1) and T(0.7(n + 1)) <= 10*c*0.7(n + 1). Therefore, T(n+1) <= 2*c(n+1) + 7*c(n+1) + n*c = 10*c*n. The proof is completed!