Which of the following algorithm structures would lead to the recurrence equation T(n) = T(n-1)+O(1), for the worst-case time cost on data of size n?

Question

BlackTom AI · Accepted Answer

A recurrence T(n) = T(n-1) + O(1) indicates each recursive step reduces the problem size by exactly one and performs only constant additional work beyond the recursive call.

1. Relevant knowledge points and concepts
- Recurrence relations describe recursive time cost: T(n) in terms of smaller inputs.
- T(n) = T(n-1) + O(1) arises when one recursive call is made on size n-1 and the non-recursive work per call is bounded by a constant.
- Typical examples: linear search, a recursive factorial or sum routine that does one constant-time operation per level.
- Solving such recurrences yields Theta(n) time.

2. Step-by-step reasoning
- Model the constant extra work as c for some c > 0 and base case T(1) = d.
- Write T(n) = T(n-1) + c.

3. Intermediate calculation steps
- Expand:
  T(n) = T(n-1) + c
       = (T(n-2) + c) + c = T(n-2) + 2c
       = ... = T(1) + (n-1)c
- Substitute base case: T(n) = d + (n-1)c = cn + (d - c).
- Therefore T(n) ∈ Theta(n).

4. Why the correct answer is correct
- The described algorithm (“Using constant effort (independent of size), reduce the amount of data by one, solve the reduced problem recursively, and then do constant work to adjust the result…”) exactly matches the recurrence’s structure: one recursive call on n-1 and constant non-recursive work, producing T(n) = T(n-1) + O(1).

5. Why other possible structures would be incorrect
- Reducing by a constant fraction (e.g., n/2) yields T(n) = T(n/2) + O(1) → Theta(log n).
- Multiple recursive calls (e.g., two calls on ~n/2) give T(n) = 2T(n/2) + ... → Theta(n) or Theta(n log n) depending on work.
- Doing linear extra work per call would give T(n) = T(n-1) + O(n) → Theta(n^2).
Thus those patterns do not match T(n) = T(n-1) + O(1).

6. Conclusion
The provided answer precisely describes the algorithmic structure that leads to T(n) = T(n-1) + O(1); hence the final answer matches the given correct answer.

Which of the following algorithm structures would lead to the recurrence equation T(n) = T(n-1)+O(1), for the worst-case time cost on data of size n?Single choice

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