Recr_4
Identify the recurrence relation for the binary_search function described below, which recursively searches for a value in a sorted list.

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BlackTom AI · Accepted Answer

To begin, restate what is being asked: identify the recurrence relation for a binary_search function that recursively searches a sorted list.
Option 1 evaluates the cost as 2T(n-1) + Θ(1). This would correspond to two recursive calls on size n-1, which is not how binary search operates; binary search halves the problem size, not processes two nearly full-sized subproblems. Therefore this option mischaracterizes the branching factor.
Option 2 proposes T(n-1) + Θ(n). Here the non-recursive term is Θ(n), which implies linear work outside the recursion, and the subproblem size is n-1. Binary search does constant work outside the recursive call (besides the division step) and reduces the problem by half, not by one. This makes the relation inconsistent with the algorithm's behavior.
Option 3 uses 2T(n/2) + Θ(1). Although it halves the size, it includes two recursive calls at each step. That would mirror a divide-and-conquer algorithm that splits into two halves and solves both, such as certain tree traversals, but binary search only proceeds down a single half after a comparison, so it should have a single T(n/2) term, not two. This option overestimates the recursion.
Option 4 presents T(n) = Θ(1) if n < 2; T(n/2) + Θ(1) otherwise. This matches the standard binary search pattern: constant work for base cases, plus a single recursive call on half the input, plus a constant amount of work for the comparison and boundary calculations. The halving step reflects the divide-and-conquer nature correctly, and the non-recursive cost is constant (Θ(1)).
Option 5 writes T(n) = Θ(1) if n < 2; T(n/2) + Θ(n) otherwise. While this includes a halved problem size, the non-recursive term Θ(n) is too large for binary search, since the only fixed amount of work aside from the recursion is a constant-time comparison and index arithmetic, not linear work in n.
Option 6 uses T(n-1) + Θ(1), which again reduces the problem by one element each time, matching a linear search rather than binary search. This contradicts the halving strategy that defines binary search.
In summary, the only option that faithfully mirrors binary search behavior—single recursive call on half the input with constant extra work—is Option 4. The other choices either double-count work through multiple recurrences, or introduce non-constant non-recursive costs, or reduce the problem size by one rather than by half. All of these misalignments point to those options as incorrect.

Recr_4 Identify the recurrence relation for the binary_search function described below, which recursively searches for a value in a sorted list.

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