What Is Count Inversions In An Array?

While looking for your favorite cereal in the grocery shop, having a sorted shelf is a must. However, how do you know that cereal is sorted alphabetically? You can observe the naming on the shelf!

But would that be possible if there are more than 10,000 types of cereals on that shelf? Not really!

Similarly while dealing with huge arrays, you need to check how far an array is from getting sorted completely.

The only way to know how long it will take to sort an array accurately is through the count inversions.

The count inversions may vary with different operations like an addition to an array, removal of elements, merging of arrays, and much more.

Even if you try to remove duplicates from a string or an array, the parameter for sorting the arrays get changed.

But what do the count inversions mean and indicate?

This article revolves around this concept in detail along with approaches to resolve this problem statement. We will also discuss the methods by which you can print the count inversions of an array with accuracy and efficiency.

Let’s discuss the step-by-step procedures and begin with understanding what are count inversions and what they indicate.

What do count inversions indicate?

Count inversions indicate in an array, how many turns it will take to sort the array as per the appropriate criterion. Simply put, if the array is completely and accurately sorted the count inversion becomes one.

However, on the same hand if the array is sorted in reverse (in decreasing order) the count inversions are found to be maximum.

Let’s understand this with an example and consider array A. Elements in array A are {8, 4,3,2}.

Now as you can see, the elements are placed in decreasing Or descending order. Hence, the inversions are maximum in this case.

The inversions of the given array become

(8, 4)
(4,3)
(3, 2)
(8, 3)
(8, 2)
(4, 2)

Thus, the total number of inversions for this sorted array will be 6.

Let’s now consider an array where the elements are not completely reverse-sorted. P[]= {1, 4,2,6}

In this case, the first and last elements are at the right positions whereas the middle two elements are not sorted accurately. Hence, in this case, you will find the following inversion:

(4, 2)

It indicates that the array is only one swap far from being accurately sorted as per the given conditions.

Now that you have understood how the count inversions are used and why we need them, we shall now move on to the methods to find the count inversions.

Approaches to finding count inversions in an array

There are two different methods through which you can easily find the count inversions of a given array. These methods are:

Brute force method
Merge sort method

Both methods are highly accurate for finding the count inversions however, the use of each method depends on different parameters such as the size of an array. Let’s understand both methods in detail with their working algorithm.

Brute force method

The brute force method of counting the inversions is quite simple and efficient. In this method, the approach is to simply consider every pair of arrays that is possible for a given set.

Now, each pair will be evaluated if it fulfills the condition of your code. In case it is found to be true, the increment is counted and hence, the final result is calculated.

Following are the steps of the algorithm on which this method is executed:

The first step of this algorithm is to initialize the value of the “count” Variable as zero. Insert count= 0
Now, you will have to run a loop from the initial value i.e. 0 to N. Where N is the number of elements in an array.
Inside this loop, another nested loop will run from i+1 to N. This will make sure all the pairs are made accurately
After this, you will have to describe the condition in the next step. For both, the loops keep the same condition as A[i]>A[j], increment counted
This condition will ensure whenever a succeeding element is larger than the preceding one, the count value will increase
Finally, print the count and get the value of count inversions

Since the time complexity in this method is higher, it is highly useful for smaller arrays. However, if you have a larger array, you will have to adopt a new approach. Let’s discuss the second approach.

Merge sort method

The merge sort algorithm is widely used while dealing with huge amounts of data in an array. The approach of this method is kept similar to that of the merge sort algorithm.

In this method, the array is divided into two parts. The left and right half of an array is checked for inversions and at the end, both the inversions are merged to get the final output.

The algorithm of this method can be described in the following steps:

The first step is to divide the array from the middle into two halves. The left and right halves of the array will be divided in the primary step
Now, simply count the inversions of an array for the left and right halves individually. Also, make sure you consider all the elements found during the merging of the array
Next, you will have to call this function continuously unless there is only 1 element is present in respective halves of the array
After that, count the number of inversions in the array through two pointer technique for accuracy
Finally, add the value of mid-1 to the inversion count variable if the array is sorted completely

Winding up

Once you learn how to count inversions through different methods, it is easy to merge and sort huge size of arrays. Also, you can remove duplicates from a string to observe the difference in count inversion for an array.