The master method is a formula for solving recurrence relations of the form:
T(n) = aT(n/b) + f(n), where, n = size of input a = number of subproblems in the recursion n/b = size of each subproblem. All subproblems are assumed to have the same size. f(n) = cost of the work done outside the recursive call, which includes the cost of dividing the problem and cost of merging the solutions Here, a ≥ 1 and b > 1 are constants, and f(n) is an asymptotically positive function.
An asymptotically positive function means that for a sufficiently large value of n, we have
f(n) > 0.
The master theorem is used in calculating the time complexity of recurrence relations (divide and conquer algorithms) in a simple and quick way.
a ≥ 1 and
b > 1 are constants and
f(n) is an asymptotically positive function, then the time complexity of a recursive relation is given by
T(n) = aT(n/b) + f(n) where, T(n) has the following asymptotic bounds: 1. If f(n) = O(nlogb a-ϵ), then T(n) = Θ(nlogb a). 2. If f(n) = Θ(nlogb a), then T(n) = Θ(nlogb a * log n). 3. If f(n) = Ω(nlogb a+ϵ), then T(n) = Θ(f(n)). ϵ > 0 is a constant.
Each of the above conditions can be interpreted as:
- If the cost of solving the sub-problems at each level increases by a certain factor, the value of
f(n)will become polynomially smaller than
nlogb a. Thus, the time complexity is oppressed by the cost of the last level ie.
- If the cost of solving the sub-problem at each level is nearly equal, then the value of
nlogb a. Thus, the time complexity will be
f(n)times the total number of levels ie.
nlogb a * log n
- If the cost of solving the subproblems at each level decreases by a certain factor, the value of
f(n)will become polynomially larger than
nlogb a. Thus, the time complexity is oppressed by the cost of
Solved Example of Master Theorem
T(n) = 3T(n/2) + n2 Here, a = 3 n/b = n/2 f(n) = n2 logb a = log2 3 ≈ 1.58 < 2 ie. f(n) < nlogb a+ϵ , where, ϵ is a constant. Case 3 implies here. Thus, T(n) = f(n) = Θ(n2)
Master Theorem Limitations
The master theorem cannot be used if:
- T(n) is not monotone. eg.
T(n) = sin n
f(n)is not a polynomial. eg.
f(n) = 2n
- a is not a constant. eg.
a = 2n
a < 1