Master theorem

MASTER THEOREM

Let a ≥ 1 and b > 1 be constants, let f(n) be a function, and let T(n) be defined on the nonnegative integers by the recurrence

T(n) = aT(n/b) + f (n),

where we interpret n/b to mean either ⌊n/b⌋ or ⌈n/b⌉. Then T(n) can be bounded asymptotically as follows.

1. If f (n) = O(n^{log_b a - ε}) for some constant ε > 0, then T(n) = Θ(n^{log_b a}).
2. If f (n) = Θ(n^{log _b a}), then T(n) = Θ(n^{log_b a} lg n).
3. If f (n) = Ω(n^{log_b a + ε}) for some constant ε > 0, and if af(n/b) ≤ cf(n) for some constant c < 1 and all sufficiently large n, then T(n) = Ω(f (n)).