Question

The method we usually use to look up an entry in a phone book is not exactly the same as a binary search because, when using a phone book, we don’t always go to the midpoint of the sublist being searched. Instead, we estimate the position of the target based on the alphabetical position of the first letter of the person’s last name. For example, when we are looking up a number for "Smith," we first look toward the middle of the second half of the phone book, instead of in the middle of the entire book. Suggest a modification of the binary search algorithm that emulates this strategy for a list of names. Is its computational complexity any better than that of the standard binary search?

Answer 1

See explaination

Step-by-step explanation:

Given a sorted array of n uniformly distributed values arr[], write a function to search for a particular element x in the array.

Linear Search finds the element in O(n) time, Jump Search takes O(√ n) time and Binary Search takes O(log n) time.

The Interpolation Search is an improvement over Binary Search for instances, where the values in a sorted array are uniformly distributed. Binary Search always goes to the middle element to check. On the other hand, interpolation search may go to different locations according to the value of the key being searched. For example, if the value of the key is closer to the last element, the interpolation search is likely to start search toward the end side.

Algorithm

The rest of the Interpolation algorithm is the same except the above partition logic.

Step1: In a loop, calculate the value of “pos” using the probe position formula.

Step2: If it is a match, return the index of the item, and exit.

Step3: If the item is less than arr[pos], calculate the probe position of the left sub-array. Otherwise, calculate the same in the right sub-array.

Step4: Repeat until a match is found or the sub-array reduces to zero.

Time Complexity: If elements are uniformly distributed, then O (log log n)). In the worst case, it can take up to O(n).

Auxiliary Space: O(1)

For the case of alphabets(not integer values), it is considered that words are arranged in lexographic oreder for Interpolation search algorithm beacuse we need sorted and uniformly distributed vales in an array.