Question

Here is a list of seven names: Sherman, Jane, Ted, Elise, Raul, Maki, John Search this list for each name in turn, using sequential soserch and courting the number of comparisons for each name. Now take the seven comparison counts and find their average. Did you get a number that you expected? Why?

Short answer

The average number of comparisons is 4.

Step-by-step solution

Understanding Sequential Search

Sequential search involves checking each item in the list one by one until the desired item is found. It's also known as linear search and is used here to find each name in the list.

Search for Sherman

Begin searching from the first name in the list. Since "Sherman" is the first name, it takes 1 comparison to find him.

Search for Jane

Start again from the first name. "Sherman" is not "Jane", so move to the second name. "Jane" is the second name, so it takes 2 comparisons to find her.

Search for Ted

Start from the first name. "Sherman" and "Jane" are not "Ted", so check the third name. "Ted" is the name found third, concluding the search with 3 comparisons.

Search for Elise

Check names sequentially from the first. "Sherman", "Jane", "Ted" are not "Elise", but the fourth name is. Thus, finding "Elise" takes 4 comparisons.

Search for Raul

Begin at the list's start, checking each name. Not "Sherman", "Jane", "Ted", or "Elise" but "Raul" is the fifth name, needing 5 comparisons.

Search for Maki

Search from the top of the list again. "Sherman, "Jane", "Ted", "Elise", and "Raul" are not "Maki", but "Maki" is next, requiring 6 comparisons.

Search for John

Sequentially check through all names; none match till the last one. "John" is last, resulting in 7 comparisons to find him.

Calculate the Average Number of Comparisons

Now average the number of comparisons done for each name by adding the counts (1+2+3+4+5+6+7 = 28) and dividing by the total number of names (7). The average is \(\frac{28}{7} = 4\).

Interpret the Results

The average number of comparisons is 4. This is what you might expect, as on average, in a list of 7 items, it would take checking about half of them to find a given item.

Key concepts

Understanding Linear Search

Linear search, also known as sequential search, is a straightforward search algorithm used to find a specific element within a list. It works by examining each element in the list, one by one, until the desired element is found or the list ends.

• The search begins at the start of the list and moves sequentially towards the end.
• If the search key matches an element, the search is successful, and the algorithm stops.
• If the search reaches the end of the list without finding a match, it concludes that the element is not present.

This method is simple and doesn’t require the list to be sorted, making it versatile for various situations. However, its simplicity comes at the cost of efficiency, especially when dealing with large datasets.
Linear search becomes less ideal for larger collections since it can potentially examine every element, resulting in a time complexity of O(n), where n is the number of elements in the list.

Algorithm Analysis in Linear Search

Algorithm analysis focuses on the performance of algorithms and their efficiency in terms of time and space complexity. In linear search, the primary measure of efficiency is the number of comparisons needed to find the target element.

• Best Case: The target element is the first in the list, requiring only a single comparison.
• Worst Case: The target element is the last, or not at all in the list, necessitating comparison with every element (n comparisons).
• Average Case: Typically, the target element might be found halfway through the list. For a list of n elements, this would require about n/2 comparisons, hence the average time complexity is O(n).

This analysis helps in understanding when and where linear search is an appropriate choice, typically in small or unsorted datasets where simplicity is preferred over speed.

The Essentials of Comparison Counting

Comparison counting in a linear search refers to the act of tallying the number of comparisons made by the algorithm to locate a desired element. This concept is critical for evaluating an algorithm's performance.

• Each comparison is checked against the target element during the search.
• The total number of comparisons gives an idea of the search's overall efficiency.

In the given exercise, each name search resulted in a different count of comparisons:- "Sherman": 1 comparison- "Jane": 2 comparisons- ... down to "John": 7 comparisonsBy summing these counts and calculating their average (\(\frac{28}{7} = 4\)), we get a sense of the expected number of checks for each search operation. This average matches our expectations, indicating roughly half the elements need checking, thus aligning with the average-case analysis of linear search.