We use --------------------- to verify that our solution is correct. We use ------------------ to discover the answer/solution.

Question

BlackTom AI · Accepted Answer

Recursion tree provides a level-by-level expansion of the recurrence so you can sum costs and confirm a guessed closed form; repeated substitution (algebraic unrolling) is the technique used to expose the pattern and thereby discover the solution.

1. Relevant knowledge points and concepts
- Recurrence relation: an equation defining T(n) in terms of smaller inputs.
- Substitution method (unrolling): repeatedly substitute the recurrence into itself to reveal the pattern; this helps form a conjectured closed form.
- Recursion tree: a visual expansion of the recurrence into levels; summing costs across levels gives a concrete verification of the conjectured bound.
- Master theorem: a shortcut for common divide-and-conquer recurrences but not the same as the pattern-discovery/unrolling process.

2. Complete reasoning process (step-by-step using a concrete example)
Take T(n) = 2T(n/2) + n.
- Substitution (unrolling) to discover: 
  T(n) = 2T(n/2) + n
       = 2[2T(n/4) + n/2] + n
       = 4T(n/4) + 2n
       = 4[2T(n/8) + n/4] + 2n
       = 8T(n/8) + 3n
  After k levels: T(n) = 2^k T(n/2^k) + k n.
  Stop when n/2^k = 1 => k = log_2 n. Hence T(n) = n T(1) + n log_2 n = Theta(n log n). This unrolling discovers the n log n pattern.

3. Intermediate steps shown above (the repeated substitutions and solving for k).

4. Why the correct answer is correct
- The substitution/unrolling step is how the pattern (the n log n form) is discovered. The recursion tree then visualizes each level: level i contributes n, there are log n levels, so summing the tree confirms the n log n result. Thus, recursion tree is used to verify, and the substitution method is used to discover.

5. Why other options are incorrect (briefly)
- Option 1: "substitution method, the recurrence relation" — a recurrence relation is the problem statement, not a technique to discover solutions.
- Option 2: "substitution method, master theorem" — the master theorem is a direct solver for many recurrences but it is not the substitution/unrolling discovery method; also it’s usually used for quick solution, not the stepwise discovery named here.
- Option 4: "master theorem, the substitution method" — master theorem is for deriving bounds quickly, but the question pairs verification vs discovery; the recursion tree (visual summation) is the standard verification tool named in the correct choice.

6. Conclusion
Therefore the correct pairing is: "the recursion tree , the substitution method", which matches the provided answer.

We use --------------------- to verify that our solution is correct. We use ------------------ to discover the answer/solution.单项选择题

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