How can two sorted lists be joined to maintain sorted order in the resulting array?

Question Analysis

This question is asking about merging two pre-sorted lists into a single list that retains the sorted order. This is a common problem with applications in various algorithms, notably in the merge step of the Merge Sort algorithm. It is important to recognize that because both lists are already sorted, we can employ a more efficient method than simply concatenating and then sorting. The goal is to take advantage of the existing order to minimize the number of comparisons and operations needed.

Answer

To merge two sorted lists while maintaining the sorted order, you can use a two-pointer technique. Here's a step-by-step explanation of how to do this:

1. Initialize Pointers: Start with two pointers, i and j, initialized to the first element of each list (list1 and list2). Also, prepare an empty list result to store the merged output.

2. Traverse Both Lists:

   • Compare the elements pointed to by i and j.
   • Append the smaller element to the result list.
   • Move the pointer of the list from which the element was taken.

3. Handle Remaining Elements:

   • Once one of the lists is exhausted, append the remaining elements of the other list directly to result. Since the lists are sorted, the remaining elements are already in order.

4. Time Complexity: This algorithm runs in O(n + m) time, where n and m are the lengths of the two lists, respectively. This is because each element from both lists is processed exactly once.

Here's a concise Python implementation:

def merge_sorted_lists(list1, list2):
    i, j = 0, 0
    result = []

    # Traverse both lists and append the smaller element to the result
    while i < len(list1) and j < len(list2):
        if list1[i] < list2[j]:
            result.append(list1[i])
            i += 1
        else:
            result.append(list2[j])
            j += 1

    # Append remaining elements of list1, if any
    while i < len(list1):
        result.append(list1[i])
        i += 1

    # Append remaining elements of list2, if any
    while j < len(list2):
        result.append(list2[j])
        j += 1

    return result

# Example usage:
list1 = [1, 3, 5]
list2 = [2, 4, 6]
merged_list = merge_sorted_lists(list1, list2)
print(merged_list)  # Output: [1, 2, 3, 4, 5, 6]

By following these steps, you can efficiently merge two sorted lists while maintaining their order.