Question

The average number of comparisons performed by linear search to find an item in an array of N elements is _________.

Short answer

Answer: The average number of comparisons needed to locate an item in an array of N elements using linear search is (N+1)/2.

Step-by-step solution

Understand Linear Search

Linear search is a simple searching algorithm that starts with the first element in the array and compares each element in the array to the target item until the item is found, or we reach the end of the array. The performance of linear search depends on the position of the target item in the array.

Analyze the Best, Average, and Worst Cases

Best Case: The best case scenario occurs when the target item is the first element in the array. In this case, only one comparison is needed, so the best case is 1. Average Case: In the average case, the target item can be found anywhere in the array. To calculate the number of comparisons in the average case, we sum the number of comparisons for each possible position and divide by the total number of positions (N). The sum of integers from 1 to N is given by N*(N+1)/2, so the average case is (N*(N+1)/2) / N, which simplifies to (N+1)/2. Worst Case: The worst case scenario occurs when the target item is the last element in the array, or the item is not in the array at all. In both cases, we need to compare all elements in the array to confirm the item is not there. So the worst case is N.

Calculate the Average of the Best, Average, and Worst Cases

To find the overall average, we simply take the average of the best, average, and worst cases. Since we are assuming the target elements distribution is uniform, we use the average case (N+1)/2 as our desired outcome.

Fill in the Blank

Based on the analysis of linear search in an array of N elements, the average number of comparisons is (N+1)/2.