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Chapter 9: Problem 10

The maximum number of comparisons that a binary search function will make when searching for a value in a 2,000 -clement array is ___________.

Short Answer

Expert verified
Answer: The maximum number of comparisons is approximately 11.

Step by step solution

01

Understanding Binary Search

Binary search is a search algorithm that is used to find the position of a target value within a sorted array. It works by repeatedly dividing the search interval in half, and comparing the middle element to the target value. If the middle element is equal to the target value, the search ends successfully. Otherwise, if the target value is less than the middle element, the search continues in the left half of the array. If the target value is greater than the middle element, the search continues in the right half of the array.
02

Find the Maximum Number of Comparisons

In the worst-case scenario, each comparison will divide the array in half until only one element remains. The maximum number of comparisons made can be found by finding the number of times the array can be divided in half before only one element remains. We can represent this as follows: n = 2000 elements Our goal is to find x such that after dividing the array by 2 for x times, we're left with only 1 element: $$ \frac{2000}{2^x} = 1 $$ We can solve for x by taking the logarithm base 2 of both sides of the equation: $$ x = \log_{2}(2000) $$
03

Calculate the Result

Now, we need to calculate the value of x: $$ x = \log_{2}(2000) \approx 10.97 $$ Since we cannot perform a fraction of a comparison, we need to round up to the nearest whole number to ensure that all elements are compared. This gives us: $$ x \approx 11 $$
04

Conclusion

The maximum number of comparisons that a binary search function will make when searching for a value in a 2,000-element array is approximately 11.

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Most popular questions from this chapter

If an array is sorted in ______________ order, the values are stored from highest to lowest.

If an array is sorted in ____________ order, the values are stored from lowest to highest.

Team up with two to three other students and jointly decide how you would organize, order, and locate the data used in the following application. Be prepared to present your group's design to the rest of the class. The program to be developed is a menu-driven program that will keep track of parking tickets issued by the village that is hiring you. When a ticket is issued the program must be able to accept and store the following information: ticket number, officer number, vehicle license plate state and number, location, violation code (this indicates which parking law was violated), and date and time written. The program must store information on the amount of the fine associated with each violation code. When a ticket is paid the program must be able to accept and store the information that it has been paid, the amount of the payment, and the date the payment was received. The program must be able to accept inquiries such as displaying the entire ticket record when a ticket number is entered. The program must also be able to produce the following reports: A list of all tickets issued on a specific date, ordered by ticket number A list of all tickets for which payment was received on a specific date and the total amount of money collected that day A report of all tickets issued in a one-month period, ordered by officer number, with a count of how many tickets each officer wrote A report of all tickets that have not yet been paid, or for which payment received was less than payment due, ordered by vehicle license number

The ___________ search algorithm steps sequentially through an array, comparing each item with the search value.

Which sort, bubble sort or selection sort, would require fewer passes to sort a set of data that is already in the desired order?
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