Problem Statement

Given a list of integers, determine the count of numbers for which there exists another number in the list that is greater by exactly one unit.

In other words, for each number x in the list, if x + 1 also exists in the list, then x is considered for the count.

Examples:

1. Example 1:

   • Input: [4, 3, 1, 5, 6]
   • Expected Output: 3
   • Justification: The numbers 4, 3, and 5 have 5, 4, and 6 respectively in the list, which are greater by exactly one unit.

2. Example 2:

   • Input: [7, 8, 9, 10]
   • Expected Output: 3
   • Justification: The numbers 7, 8, and 9 have 8, 9, and 10 respectively in the list, which are greater by exactly one unit.

3. Example 3:

   • Input: [11, 13, 15, 16]
   • Expected Output: 1
   • Justification: Only the number 15 has 16 in the list, which is greater by exactly one unit.

Constraints:

• 1 <= arr.length <= 1000
• 0 <= arr[i] <= 1000

Solution

To solve this problem, the first step is to add all elements of the given array into a HashSet. This allows for constant time complexity (O(1)) checks for the presence of any number. After populating the HashSet, iterate through the array once more. During this iteration, for each element, check if its direct successor (the element itself plus one) is present in the HashSet. If the successor is found, increment a counter. This counter will keep track of the total number of elements that have their direct successors within the array. The final value of this counter, after completing the iteration over the array, will be the solution to this problem.

1. Hashset Creation: Begin by initializing an empty hashset. Traverse the list and insert each number into the hashset. This will allow for constant-time lookups to check the existence of any number in the list.

2. Counting Valid Numbers: Traverse the list again. For each number x, check if x + 1 exists in the hashset. If it does, increment a counter. This counter will keep track of all the numbers that satisfy the given condition.

3. Result: After traversing the entire list, the counter will hold the total count of numbers for which there exists another number greater by exactly one unit.

4. Return: Return the counter as the final result.

This approach is efficient because it leverages the constant-time lookup property of hashset, ensuring that the algorithm runs in linear time.

Algorithm Walkthrough:

Given the input list [4, 3, 1, 5, 6]:

• Initialize an empty hashset.
• Insert each number from the list into the hashset: {4, 3, 1, 5, 6}.
• Initialize a counter to 0.
• Traverse the list:
  • For 4, check if 5 exists in the hashset. It does, so increment the counter.
  • For 3, check if 4 exists in the hashset. It does, so increment the counter.
  • For 1, check if 2 exists in the hashset. It doesn't, so move on.
  • For 5, check if 6 exists in the hashset. It does, so increment the counter.
  • For 6, check again if 7 exists in the hashset. It still doesn't, so move on.
• The counter now holds the value 3, which is the final result.

Here is the visual representation of the algorithm: