What is average case time complexity of binary search n is the number of elements in the list order of Nlogn order of logN order of n order of n2?

In any computational system, the search is one of the most critical functionalities to develop. Search techniques are used in file retrievals, indexing, and many other applications. There are many search techniques available. One of which is the binary search technique.

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A binary search algorithm works on the idea of neglecting half of the list on every iteration. It keeps on splitting the list until it finds the value it is looking for in a given list. A binary search algorithm is a quick upgrade to a simple linear search algorithm. 

This article will discuss areas like binary search algorithm average complexity and binary search worse case along with giving a brief idea of binary search algorithm first, along with best and worse case complexity of binary search algorithm.

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Working of a Binary Search Algorithm

The first thing to note is that a binary search algorithm always works on a sorted list. Hence the first logical step is to sort the list provided. After sorting, the median of the list is checked with the desired value.

• If the desired value is equal to the central index’s worth, then the index is returned as an answer. 

• If the target value is lower than the central index’s deal of the list, then the list’s right side is ignored. 

• If the desired value is greater than the central index’s value, then the left half is discarded. 

• The process is then repeated on shorted lists until the target value is found. 

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Example #1

Let us look at the algorithm with an example. Assume there is a list with the following numbers:

1, 15, 23, 7, 6, 14, 8, 3, 27

Let us take the desired value as 27. The total number of elements in the list is 9. 

The first step is to sort the list. After sorting, the list would look something like this:

1, 3, 6, 7, 8, 14, 15, 23, 27

As the number of elements in the list is nine, the central index would be at five. The value at index five is 8. The desired value, 27, is compared with the value 8. First, check whether the value is equal to 8 or not. If yes, return index and exit. 

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As 27 is greater than 8, we would ignore the left part and only traverse the list’s right side. The new list to traverse is:

14, 15, 23, 27

Note: In practice, the list is not truncated. Only the observation is narrowed. So, the “new list” should not be confused as making a new list or shortening the original one. Although it could be implemented with a new list, there are two problems. First, there will be a memory overhead. Each new list will increase the space complexity. And second, the original indexes need to be tracked on each iteration.

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The new central index can be taken as the second or third element, depending on the implementation. Here, we will consider the third element as central. The value 23 is compared with value 27. As the value is greater than the central value, we will discard the left half. 

The list to traverse is:

27

As the list contains only a single element, it is considered to be the central element. Hence, we compare the desired value with 27. As they match, we return the index value of 27 in the original list. 

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Example #2

In the same list, let us assume the desired value to be 2. 

First, the central value eight is compared with 2. As the desired value is smaller than the central value, we narrow our focus down to the list’s left-hand side. 

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The new traversal will consist of:

1, 3, 6, 7

Let us take the central element as the second element. The desired value two is compared with 3. As the value is still smaller, we again narrow the focus down to the list’s left-hand side. 

The new traversal will consist of:

1

As the traversing list has only one element, the value is directly compared to the remaining element. We see that the values do not match. Hence, we break out of the loop with an error message: value not found. 

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Time and Space complexity

People often do not have an understanding of binary search worst case and best case. The time complexity of the binary search algorithm is O(log n). The best-case time complexity would be O(1) when the central index would directly match the desired value. Binary search worst case differs from that. The worst-case scenario could be the values at either extremity of the list or values not in the list. 

In the worse case complexity of the binary search, the values are present in such a way that either they are at the extremity of the list or are not present in the list at all. Below is a brief description of how to find worse case complexity of the binary search. 

The equation T(n)= T(n/2)+1 is known as the recurrence relation for binary search. 

To perform binary search time complexity analysis, we apply the master theorem to the equation and get O(log n).

Worse case complexity of the binary search is often easier to compute but carries the drawback of being too much pessimistic. 

On the other hand, another type of binary search time complexity analysis, which is binary search algorithm average complexity, is a rarely chosen measure. As it is harder to compute and requires an in-depth knowledge of how much input has been distributed, people tend to avoid binary search algorithm average complexity. 

Below are the basic steps to performing Binary Search.

1. Find the mid element of the whole array, as it would be the search key.
2. Look at whether or not the search key is equivalent to the item in the middle of the interval and return an index of that search key. 
3. If the value of the middle item in the interval is more than the search key, reduce the interval’s lower half.
4. If the opposite, then lower the upper half.
5. Repeat from point 2, until the value is found or the interval gets empty.

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The space complexity of the binary search algorithm depends on the implementation of the algorithm. There are two ways of implementing it:

1. Iterative method
2. Recursive method

Both methods are quite the same, with two differences in implementation. First, there is no loop in the recursive method. Second, rather than passing the new values to the next iteration of the loop, it passes them to the next recursion. In the iterative method, the iterations can be controlled through the looping conditions, while in the recursive method, the maximum and minimum are used as the boundary condition. 

In the iterative method, the space complexity would be O(1). While in the recursive method, the space complexity would be O(log n). 

Benefits 

• A binary search algorithm is a fairly simple search algorithm to implement. 

• It is a significant improvement over linear search and performs almost the same in comparison to some of the harder to implement search algorithms.

• The binary search algorithm breaks the list down in half on every iteration, rather than sequentially combing through the list. On large lists, this method can be really useful.

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Conclusion

A binary search algorithm is a widely used algorithm in the computational domain. It is a fat and accurate search algorithm that can work well on both big and small datasets. A binary search algorithm is a simple and reliable algorithm to implement. With time and space analysis, the benefits of using this particular technique are evident. 

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