findLuckyNumbersRange()

Step 1: Understand the Problem

In this problem our function will take two numbers as input and return all 'lucky numbers' in that range. Lucky numbers are numbers that have digits from 1 to 9 once and only once. For instance, the number '123456798' is an example of a lucky number.

We have to find an algorithmic solution to this problem. We cannot simply use a list of all possible numbers in the range and check each one individually, as this would be too time consuming for large ranges.

Step 2: Setup Function

First of all, let's set up a function findLuckyNumbersRange in JavaScript which accepts two integer variables.

We will use these variables to determine the range within which we look for lucky numbers.

function findLuckyNumbersRange(startNumber, endNumber) {
  // code will go here
}

Step 3: Create Loop

First, we'll create a loop to examine each individual number in our range. We'll start by checking each number starting at startNumber, and ending just before endNumber.

function findLuckyNumbersRange(startNumber, endNumber) {
  for (let i=startNumber; i < endNumber; i++) {
    //code will go here 
  }
}

Step 4: Implement Lucky Check

To consider a number as a lucky number, it should contain all the digits from 1 to 9 exactly once.

We'll want to convert the number to a string so that we can examine each digit individually.

Then, we'll sort the string of digits. This will allow us to check if the sorted string of digits is '123456789', which will mean that the number is lucky.

function findLuckyNumbersRange(startNumber, endNumber) {
  const luckyNumbers = [];  // to store our lucky numbers
  
  for (let i=startNumber; i <= endNumber; i++) {
    const digits = String(i).split('').sort().join('');
    if (digits === '123456789') {
      luckyNumbers.push(i);
    }
  }
  return luckyNumbers;
}

Step 5: Final Thoughts

Now we have a function that will calculate all lucky numbers in a given range. In each iteration the function sorts and joins the digits of the number, and checks if its equal to '123456789'. This function will return an array of all the discovered lucky numbers.

However, in terms of performance, we have to note that this function could be expensive for large ranges. We have not implemented any optimizations for skipping sections of the number space we know will not contain lucky numbers, which could be seen as a future enhancement. Additionally, the sorting operation is also a performance overhead as it's an O(n log n) operation.

Here's our complete findLuckyNumbersRange function:

function findLuckyNumbersRange(startNumber, endNumber) {
  const luckyNumbers = [];  // to store our lucky numbers
  
  for (let i=startNumber; i <= endNumber; i++) {
    const digits = String(i).split('').sort().join('');
    if (digits === '123456789') {
      luckyNumbers.push(i);
    }
  }
  return luckyNumbers;
}