Question

I am given this recurrence relation:

T (n) = T (n − a) + T (a) + cn

C > 0 , a >= 1 ..

my problem is with T (a) , I don’t understand how you can “recurs” a constant??

Like, if am trying to build a recurrence tree, I would go by doing this:

T (n) =>  cn            =>         cn
         /  \                    /    \
      T(a)  T(n - a)           ca      c*(n-a)
                             /    \     /     \
                            ??    ??  T(n-2a)  T(a)

You see what I mean? What does T(a) represent??

Any resource will be much appreciated. Thanks.

OR, think of it iteratively:

T (n) = T (n − a) + T (a) + cn
T (n) = T (n -2a) + T (a) + ????

Answer 1

So you have:

T(n) = T(n-a) + T(a) + cn

What is T(n-a)? Simply take n-a as your input:

T(n-a) = T((n-a)-a) + T(a) + c(n-a)

Now what is T(a)? Similarly, take a as an input:

T(a) = T(a-a) + T(a) + ca

Combining them, you obtain:

T(n) = ( T((n-a)-a) + T(a) + c(n-a) )+ ( T(a-a) + T(a) + ca ) + cn
     = T(n-2a) + T(a) + c(n-a) + T(0) + T(a) + ca + cn
     = T(n-2a) + 2T(a) + T(0) + c((n-a) + a + n)

Now depending on your base case, T(0) probably is some constant.
Hope that helps.