Recurrence & Master Theorem Visualizer

Recurrence & Master Theorem

Explore recurrences of the form T(n) = aT(n/b) + f(n), compare recursion-tree levels, and classify the recurrence using the Master Theorem.

How to use

• Set a (number of subproblems, a ≥ 1), b (shrink factor, b > 1), k (the non-recursive work is f(n) = n^k) and an example input size n, then press Analyze Recurrence.
• The left panel shows the critical exponent log_ba, the Master Theorem case and the predicted bound.
• The recursion tree lists one row per level i: aⁱ subproblems (small squares, at most 30 drawn; “…” means more), each of size n/bⁱ, and the total work on that level aⁱ·(n/bⁱ)^k. The bar compares level work with the heaviest level. At most 10 levels are drawn.

Master Theorem

Compare f(n) = n^k with n^log_ba:

• Case 1 (k < log_ba): recursive work dominates; level work grows towards the leaves; T(n) = Θ(n^log_ba).
• Case 2 (k = log_ba): every level contributes the same work; T(n) = Θ(n^k log n).
• Case 3 (k > log_ba): the root dominates; level work shrinks towards the leaves; T(n) = Θ(n^k). Case 3 needs the regularity condition a·f(n/b) ≤ c·f(n) for some c < 1; for f(n) = n^k it always holds with c = a/b^k.

Examples