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The above algorithm, find_element_recursively, works on a list and processes the elements one by one, starting from the leftmost position and moving toward the rightmost position until it finds the element that matches the target or reaches the end of the list.
What is the average-case time complexity of this algorithm?

Question

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The above algorithm, find_element_recursively, works on a list and processes the elements one by one, starting from the leftmost position and moving toward the rightmost position until it finds the element that matches the target or reaches the end of the list.
What is the average-case time complexity of this algorithm?

BlackTom AI · Accepted Answer

Question restatement: The algorithm find_element_recursively processes a list from left to right, examining elements one by one until it finds the target or reaches the end. What is its average-case time complexity?

Option 1: O(n^2). This would imply that the algorithm's work grows quadratically with the input size. However, a linear scan that touches each element at most a constant amount of times does not accumulate to a quadratic total in the average case. Therefore, this option overestimates the cost.

Option 2: O(log n). This indicates logarithmic growth, typical of divide-and-conquer approaches (like binary search) that halve the problem space each step. Our described method does not halve the list or skip large portions; it checks elements sequentially, so logarithmic behavior is not achieved here.

Option 3: O(1). Constant time would mean the algorithm finishes in the same amount of work regardless of list size, which is impossible for a sequential scan that may need to inspect multiple elements in the average case. Hence this is incorrect.

Option 4: O(n). This is the linear-time complexity that matches a single-pass, left-to-right scan where, on average, you might examine about half of the elements before finding the target (or reach the end if not present). Each element is visited a constant amount of work, and the number of visits scales linearly with n, which aligns with the described average behavior.

Summary reasoning: The algorithm performs a single pass over the list in the average case, inspecting elements until it finds the target or ends. This leads to linear growth with respect to the list size, i.e., O(n). The other options represent complexities that are inconsistent with a straightforward left-to-right element-by-element scan.

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