Generating desired pairs within a range using JavaScript

We are required to write a JavaScript function that takes in a number n. Our function should generate an array containing the pairs of integers [a, b] that satisfy the following conditions:

0 <= a <= b <= n

This means we need to find all possible pairs where the first number is less than or equal to the second number, and both numbers are within the range from 0 to n (inclusive).

Algorithm Approach

The solution uses nested loops where:

• The outer loop iterates from 0 to n (for the first element 'a')
• The inner loop starts from 'i' and goes to n (for the second element 'b')
• This ensures a ? b condition is always maintained

Example

Following is the complete implementation:

const num = 4;

const findPairs = (n = 1) => {
    const arr = [];
    for(let i = 0; i 

[
  [ 0, 0 ],
  [ 0, 1 ],
  [ 0, 2 ],
  [ 0, 3 ],
  [ 0, 4 ],
  [ 1, 1 ],
  [ 1, 2 ],
  [ 1, 3 ],
  [ 1, 4 ],
  [ 2, 2 ],
  [ 2, 3 ],
  [ 2, 4 ],
  [ 3, 3 ],
  [ 3, 4 ],
  [ 4, 4 ]
]


Optimized Version

Here's a more concise version using array literals directly:

const findPairsOptimized = (n = 1) => {
    const result = [];
    for(let i = 0; i 

[
  [ 0, 0 ],
  [ 0, 1 ],
  [ 0, 2 ],
  [ 0, 3 ],
  [ 1, 1 ],
  [ 1, 2 ],
  [ 1, 3 ],
  [ 2, 2 ],
  [ 2, 3 ],
  [ 3, 3 ]
]


How It Works

For n = 4, the algorithm generates:

• i = 0: pairs [0,0], [0,1], [0,2], [0,3], [0,4]
• i = 1: pairs [1,1], [1,2], [1,3], [1,4]
• i = 2: pairs [2,2], [2,3], [2,4]
• i = 3: pairs [3,3], [3,4]
• i = 4: pair [4,4]

The total number of pairs generated for any n is (n+1)(n+2)/2, which represents the number of combinations with repetition allowed.

Conclusion

This nested loop approach efficiently generates all valid pairs where a ? b within the given range. The time complexity is O(n²) and space complexity is O(n²) for storing the result.