Java Program to Find the Missing Element in a Given Permutation

The task is to determine the missing number from the given sequence. The array will contain all numbers in this range except for one.

Approaches to Find the Missing Number

Naive Approach: Using Hashing

This approach involves creating an auxiliary array (hashArray) to keep track of the frequency of each element in the input array. The element with a frequency of 0 in this auxiliary array is the missing number.

Algorithm

1. Create an auxiliary array (hashArray) of size N+1 initialized to zero.
2. Iterate over the input array nums[] and increment the corresponding index in the auxiliary array.
3. The index with a value of 0 in the auxiliary array is the missing number.

Let's implement the above algorithm in a Java program.

Output:

 
4   

Explanation

In the above program, the frequency of each number in the input array is stored in an auxiliary array (hash) of size N+1. We increment the value at the matching index in the hash array as we iterate through the input array.

We first fill in the frequency array and then search the hash array for the index that doesn't change to zero. Since it implies that the matching number in the range [1, N] did not appear in the input array, this index denotes the missing number.

Efficient Approach: Using Sum Formula

This approach leverages the formula for the sum of the first N natural numbers:

By subtracting the sum of the array elements from this total, we will get the missing number.

Output:

 
4   

Explanation

First, the formula is used to get the total sum for the range [1, N]. Next, the input array's total element count is calculated. The value that should have been in the array is not represented by the difference between these two sums that provide the missing number.

Efficient Approach: Using XOR Operation

When a number is XORed with itself, the result is 0, and when a number is XORed with 0, it gives the number itself.

Output:

 
4   

Explanation

To obtain xorTotal, first XOR every number from 1 to N. Get xorArray by XORing every element in the input array. You may find the missing number by XORing xorTotal and xorArray. All other digits cancel out, leaving only the missing number.

Conclusion

The naive approach involves using an auxiliary array, while more efficient methods use the sum formula or XOR operation. The latter approaches are preferable for their O(1) space complexity and straightforward implementation, making them suitable for larger datasets.