What is the recurrence relation for the linear search recursive algorithm?OptionsT(n+1)+c2T(n-1)+cT(n-2)+cT(n-1)+c

Break Down the Problem

1. We need to identify the recursive nature of the linear search algorithm.
2. We must analyze how many operations are performed in each recursive call and find the relationship that defines its complexity.

Relevant Concepts

1. A linear search algorithm recursively checks each element of a list until the desired element is found or the list is exhausted.
2. If we denote $T(n)$ as the time complexity for a search in a list of size $n$, the relation can describe the work done for each recursive call.

Analysis and Detail

1. When searching a list of size $n$:

   • The algorithm checks the first element, which takes a constant time $c$.
   • It then proceeds to search the remaining $n-1$ elements.

   Therefore, the recurrence relation can be described as: $T(n) = T(n-1) + c$ where $c$ is the constant time taken to compare the first element.

Verify and Summarize

1. We check the options provided:
   • Option 1: $T(n+1) + c$ - This doesn't follow the recursive nature of linear search.
   • Option 2: $T(n-1) + c$ - This correctly represents the logic as we reduce the size by 1 for each search until we find the element or exhaust the list.
   • Option 3: $T(n-2) + c$ - This skips an element, which isn't correct for linear search.
   • Option 4: $T(n-1) + c$ - This correctly describes the linear search.

Final Answer

The correct recurrence relation for the linear search recursive algorithm is: $T(n) = T(n-1) + c$