Ready for an adventure in sorting algorithms? We will solve two fun problems: "Find the K-th Ordinal Number in a List" and "Count the Number of Flips in a List". These tasks portray situations where we need clever system design. Let's employ Quick Sort and Merge Sort to find efficient solutions. Buckle up!

Picture an array of numbers and a number k. Your mission is to discover the k-th smallest number in that array. k starts from 1, so when k = 1, we seek the smallest number; when k = 2, we want the second smallest, and onwards.

The first solution might involve scanning and shrinking the array by removing the smallest number until you reach the k-th smallest. But this method, while straightforward, has a time complexity of $O(n^2)$ due to continuous array rewriting.

An efficient plan might be to sort the array and then directly select the k-th number:

JavaScript
Copy
1inputArray.sort((a, b) => a - b);
2return inputArray[k - 1];

This method has a better time complexity - $O(n \log n)$. But can we do even better? Quick Sort thinks so.

Quick Sort can provide an optimal solution. We’ll divide the array into two parts using a pivot: the left side contains numbers less than the pivot, while the right side has all greater numbers.

If the pivot's position equals k, that's our answer! If not, we repeat the process on the necessary partition.

It's coding time! Let’s make a function for partitioning in JavaScript.

JavaScript
Copy
1function partition(arr, low, high) {
2  let pivot = arr[low];
3  let i = low;
4
5  for (let j = low + 1; j <= high; j++) {
6    if(arr[j] <= pivot) {
7      i++;
8      [arr[i], arr[j]] = [arr[j], arr[i]];
9    }
10  }
11
12  [arr[i], arr[low]] = [arr[low], arr[i]];
13  return i;
14}

Now we use our partition function in the main logic. If the pivot's position equals k, return the pivot. Otherwise, check the appropriate partition!

JavaScript
Copy
1function findKthSmallest(numbers, k) {
2  if (!numbers || numbers.length < k) return Number.MIN_SAFE_INTEGER;
3  return kthSmallest(numbers, 0, numbers.length - 1, k);
4}
5
6function kthSmallest(arr, start, end, k) {
7  if (k > 0 && k <= end - start + 1) {
8    let pos = partition(arr, start, end);
9    if (pos - start === k - 1) {
10      return arr[pos];
11    }
12    if (pos - start > k - 1) {
13      return kthSmallest(arr, start, pos - 1, k);
14    }
15    return kthSmallest(arr, pos + 1, end, k - pos + start - 1);
16  }
17  return Number.MAX_SAFE_INTEGER;
18}
19
20console.log(findKthSmallest([1, 7, 2, 4, 2, 1, 6], 5));  // Prints 4

Number.MIN_SAFE_INTEGER is returned by findKthSmallest if the input array numbers is empty or has fewer elements than k. This could represent a case where the k smallest value doesn't exist.

Number.MAX_SAFE_INTEGER is used in the kthSmallest function if k is either less than 1 or greater than the length of the portion of the array being considered. This could mean that there's an error in the k parameter passed, or the array doesn't have enough elements.

These extreme values are used because they are unlikely to be a valid result in any normal situation, making it easy to spot when something has gone wrong. It's a convention to return values which clearly indicate error situations, aiding in debugging and error handling.

Problem two presents an array; your challenge is counting the flips or inversions. An inversion is a pair of numbers with the higher one coming first.

For instance, in numbers = [4, 2, 1, 3], there are four inversions: (4, 2), (4, 1), (4, 3), and (2, 1).

Merge Sort can efficiently solve this. It sorts an array in $O(n \log n)$ by splitting it, sorting the halves, and merging them. During this process, we can count the inversions.

When merging, a smaller number on the right than a number on the left reveals an inversion. And it's not just one: there are as many inversions as the remaining numbers on the left.

We begin with a mergeAndCount helper function in JavaScript:

JavaScript
Copy
1function mergeAndCount(arr1, arr2) {
2  let i = 0;
3  let j = 0;
4  let merged = [];
5  let inversions = 0;
6
7  while (i < arr1.length || j < arr2.length) {
8    if (i === arr1.length) {
9      merged.push(arr2[j]);
10      j++;
11    } else if (j === arr2.length) {
12      merged.push(arr1[i]);
13      i++;
14    } else if (arr1[i] <= arr2[j]) {
15      merged.push(arr1[i]);
16      i++;
17    } else {
18      merged.push(arr2[j]);
19      j++;
20      inversions += (arr1.length - i);
21    }
22  }
23  return [merged, inversions];
24}

Finally, we develop countInversions, aided by mergeAndCount:

JavaScript
Copy
1function countInversions(arr) {
2  if (arr.length === 1) return [arr, 0];
3
4  const middle = Math.floor(arr.length / 2);
5  const [left, leftInversions] = countInversions(arr.slice(0, middle));
6  const [right, rightInversions] = countInversions(arr.slice(middle));
7  const [merged, mergeInversions] = mergeAndCount(left, right);
8
9  return [merged, leftInversions + rightInversions + mergeInversions];
10}
11
12console.log(countInversions([4, 2, 1, 3])); // Prints [ [ 1, 2, 3, 4 ], 4 ]

Counting inversions at each merge and summing creates an efficient way to count total inversions.

We've dived into Quick Sort and Merge Sort, applying them to two exciting tasks, starting from simple solutions to more efficient ones and building them in JavaScript. Now it's your turn!