Review master theorem

For recurrent relation with the following format:

(T(n) = a T(frac{n}{b}) + f(n))

Let (c=log_b a) be the critical exponent. Master theorem compares the relative growth of (f(n)) and (n^{c}), specifically:

• if (exists epsilon gt 0), (f(n) in O(n^{c-epsilon})), which means (f(n)) is upper bounded by (n^c), then (T(n) in Theta(n^c));
• if (exists k geq 0, f(n) in Theta(n^c log^{k}n)), which means (f(n)) grows in the same order as (n^c log^k n), then, (f(n) in Theta(n^c log^{k+1}n))
• if (exists epsilon gt 0), (f(n) in Omega(n^{c+epsilon})) and (exists c<1, a f(frac{n}{b}) < cf(n)) for all sufficiently large (n), then (T(n) in Theta(f(n)))

Some examples:

1. (T(n) = 3T(n/2) + n^2)
2. (T(n) = 4T(n/2) + n^2)
3. (T(n) = 10T(n/3) + n^2)
4. (T(n) = 2T(n-1) + 1) Tower of Hanio
5. (T(n) = T(sqrt{n}) + 1)
6. (T(n) = 2T(n/2) + nlog n)

Relation 4) and 5) cannot be solved by master theorem, but could be solved by iterated substitution.

This is a test of markdown

(T(n) = O(n) + frac{1}{n} sum_{k=1}^{n}T(n-k))

iterated substitution: