How is the time complexity of merging two sorted arrays to create one larger sorted array determined?

This is my solution to a problem that involves merging two arrays of integers sorted in ascending order into one larger sorted array. The function merges the arrays by comparing elements from each array and appending them to a new array in an orderly fashion.

function mergeSortedArrays(arr1, arr2) {
        // Check input
        if (arr1.length === 0 && arr2.length === 0) {
            return arr1;
        } else if (arr1.length === 0) {
            return arr2;
        } else if (arr2.length === 0) {
            return arr1;
        }
    
        // Initialize variables
        let i = 0, j = 0;
        const mergedArray = [];
    
        // Loop through & compare
        while (i < arr1.length && j < arr2.length) {
            if (arr1[i] <= arr2[j]) {
                mergedArray.push(arr1[i]);
                i++;
            } else {
                mergedArray.push(arr2[j]);
                j++;
            }
        }
    
        if (j < arr2.length) {
            while (j < arr2.length) {
                mergedArray.push(arr2[j]);
                j++;
            }
        } else if (i < arr1.length) {
            while (i < arr1.length) {
                mergedArray.push(arr1[i]);
                i++;
            }
        }
    
        return mergedArray; // O(1)
    }

The time complexity of this function has been a point of analysis as it's not solely bound by the length of either array but also influenced by the upper and lower bounds of each array. Further examination beyond linear time complexity, O(n), is required for a precise understanding of its operational efficiency.

Answer №1

The time complexity is O(m+n) where m represents the size of the first array and n represents the size of the second array.

Answer №2

During each iteration of the initial while loop, either i or j is incremented. The maximum number of times this loop can run is equal to the sum of the lengths of arr and arr2 minus 2, as exceeding this limit will result in breaking out of the testing conditions.

We can refer to the final states of i and j as k and l respectively, where one of them reaches its upper bound while the other remains below its upper bound. This implies that the initial while loop runs for a total of k + l times.

Following the first loop, there are two exclusive while loops, each starting at k or l and proceeding until it reaches its upper limit. These two loops will execute (arr.length - k) and (arr2.length - l) times respectively.

Therefore, the overall execution count is as follows: first while loop: (k + l) + second set of loops:

   if k != arr.length (which means l == arr2.length): (arr.length - k) 
   if l != arr2.length (which means k == arr.length): (arr2.length - l)

Adding up these counts gives us:

  if k != arr.length: (k + l) + (arr.length - k) = l + arr.length = arr2.length + arr.length
  if l != arr2.length: (k + l) + (arr2.length - l) = k + arr2.length = arr.length + arr2.length

Both scenarios lead to a total execution count of arr.length + arr2.length, resulting in a complexity of O(m + n).

Answer №3

The time complexity of the algorithm can be expressed as O(arr1.length + arr2.length) because we are iterating through both arrays only once.

Answer №4

Considering the simplicity of handling just 2 arrays with a straightforward algorithm, and simultaneously iterating through both arr1 and arr2 on every index, the worst-case scenario does amount to O(n + m) where n represents the length of arr1 and m represents the length of arr2. However, it's important to acknowledge that this time complexity isn't solely reserved for the worst-case scenario but also accurately depicts the exact time required, denoted as Θ(m+n).

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