Master Theorem

Master theorem provides a solution in asymptotic terms to solve time complexity problem of most divide and conquer algorithms.

Recurrence relations of the form:

T(n) = a T(n/b) + f(n) where a >= 1 and b > 1

Case 1:

f(n) = O(n^c) where c < log_b a

Then: T(n) = Θ(n^{log_b a})

Case 2:

f(n) = Θ(n^c log^kn) where c = log_b a
Then: T(n) = Θ(n^c log^k+1n)

Case 3:

f(n) = Ω(n^c) where c > log_b a

a f(n/b) <= k f(n) for some constant k < 1 and sufficiently large n

Then: T(n) = Θ(f(n)) 

Examples

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 1. T(n) = 2 T(n/2) + n²

 a = 2, b = 2, f(n) = n²  -> c = 2 > log_b a

And 2 (n² / 4) <= k n², choosing k = 1/2

2. Binary Search

T(n) = T(n/2) + O(1)

T(n) = O(log n)

3. Merge Sort

T(n) = 2 T(n/2) + O(n)

T(n) = O(n log n)

Notes

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There are some conditions where we cannot apply master theorem.

1. a < 1 or b < 1

2. f(n) is not positive

3. f(n) = n / (log n)

Because 1/(log n) < n^ε for any constant ε > 0.