Solve, in big-θ, the following recurrence relationT(n) = 4 * T(n/4), where n = 4
T(n) = c, where n = 1
for a constant c.

Question

Solve, in big-θ, the following recurrence relationT(n) = 4 * T(n/4), where n >= 4
T(n) = c, where n = 1
for a constant c.

BlackTom AI · Accepted Answer

For recurrence T(n) = 4 T(n/4) with T(1)=c, apply the divide-and-conquer (Master theorem / recursion-tree) idea: here a = 4 and b = 4, so n^{log_b a} = n^{log_4 4} = n^1 = n. Therefore the solution is Θ(n). Below is the detailed justification.

1) Relevant concepts
- Master Theorem for recurrences of form T(n) = a T(n/b) + f(n). When f(n) is absent (or constant per leaf), the growth is driven by n^{log_b a}.
- Recursion-tree interpretation: count number of subproblems at each level and their sizes, sum costs across levels (here only base costs contribute).

2) Step-by-step reasoning (recursion-tree)
- Level 0: 1 problem of size n.
- Level 1: a = 4 problems, each of size n/4.
- Level i: 4^i problems, each of size n/4^i.
- Recursion stops when size = 1: n/4^L = 1 => 4^L = n => L = log_4 n.

3) Intermediate calculations
- Number of leaves = 4^L = 4^{log_4 n} = n (because b^{log_b n} = n).
- Each leaf cost is T(1) = c, so total leaf cost = c * n = Θ(n).
- There is no additional non-recursive work at internal nodes in the recurrence as given, so internal levels contribute 0 (or constant times smaller). Thus total T(n) = Θ(n).

4) Why the correct answer is correct
The dominant contribution comes from the n leaves each costing c, giving linear total cost. Master theorem: n^{log_b a} = n, so T(n) = Θ(n).

5) Why other options are incorrect (brief)
- Θ(1), Θ(log n): too small; cost grows with n.
- Θ(n log n): would require an extra Θ(n) work per level across log levels (not present).
- Θ(n^2), Θ(n^3), Θ(2^n), Θ(n!): much larger than the actual linear cost.

6) Conclusion
The recurrence solves to Θ(n), which matches the provided answer Θ(n).

Solve, in big-θ, the following recurrence relationT(n) = 4 * T(n/4), where n >= 4 T(n) = c, where n = 1 for a constant c.单项选择题

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