**π‘ Problem Formulation:** In programming challenges, you may encounter scenarios where you are given two strings containing lowercase alphabetic characters, and you’re required to merge them into the lexicographically smallest string possible. For example, given `"ace"`

and `"bdf"`

, the optimal output would be `"abcdef"`

. This article explores five diverse methods to solve this problem using Python.

## Method 1: Merging with Sorting

This method involves creating a new string by taking the smallest available character from the starts of both input strings. We repeatedly pick the lexicographically smaller character from the heads of the strings, and append it to the result string, until both strings are empty.

Here’s an example:

def lexically_minimal_merge(str1, str2): merged_string = "" while str1 and str2: if str1 < str2: merged_string += str1[0] str1 = str1[1:] else: merged_string += str2[0] str2 = str2[1:] return merged_string + str1 + str2 print(lexically_minimal_merge('ace', 'bdf'))

Output:

abcdef

This code snippet defines a function `lexically_minimal_merge()`

that takes two strings as arguments. It compares the first characters of each string, appending the smaller one to the result string. This process continues in a loop until one of the strings is exhausted, then the remainder of the other string is appended to result in the lexicographically smallest combination.

## Method 2: Using a Priority Queue

Involves prioritizing characters from both strings by using a priority queue (or a min heap) to always pick the smallest character. This offers more efficient character retrieval than manually comparing string prefixes, especially for long strings.

Here’s an example:

import heapq def lexically_minimal_from_heap(str1, str2): heap = [] for char in str1 + str2: heapq.heappush(heap, char) return ''.join(heapq.heappop(heap) for _ in range(len(heap))) print(lexically_minimal_from_heap('ace', 'bdf'))

Output:

abcdef

This example uses Python’s `heapq`

module to push all characters from both strings into a min heap. A lexically minimal string is created by repeatedly popping the smallest element from the heap and concatenating them into a new string, guaranteeing lexicographic order.

## Method 3: Recursive Approach

A recursive method emphasizes breaking down the problem into subproblems. Each time we chose the smaller first character between the two strings, we recurse with the remaining parts of the strings. This clearly defines the concept of choosing the smallest character available at each step.

Here’s an example:

def recursive_merge(str1, str2, result=""): if not str1: return result + str2 if not str2: return result + str1 if str1[0] < str2[0]: return recursive_merge(str1[1:], str2, result + str1[0]) else: return recursive_merge(str1, str2[1:], result + str2[0]) print(recursive_merge('ace', 'bdf'))

Output:

abcdef

The function `recursive_merge()`

takes two strings and a result that accumulates the solution. It recursively appends the smaller of the two leading characters until one of the strings is empty, wherein it returns the result concatenated with the remaining non-empty string.

## Method 4: Iterative Two-Pointer Technique

A two-pointer technique iterates over both strings simultaneously, using pointers to decide which character to take at each step. This is more space-efficient than creating extra strings or arrays.

Here’s an example:

def iterative_two_pointer(str1, str2): pointer1, pointer2 = 0, 0 result = "" while pointer1 < len(str1) and pointer2 < len(str2): if str1[pointer1] <= str2[pointer2]: result += str1[pointer1] pointer1 += 1 else: result += str2[pointer2] pointer2 += 1 return result + str1[pointer1:] + str2[pointer2:] print(iterative_two_pointer('ace', 'bdf'))

Output:

abcdef

The `iterative_two_pointer()`

function uses two indices to track positions within the strings. It compares characters at these indices, appending the smaller character to the result, and moves the corresponding pointer. The remaining characters are appended after the loop.

## Bonus One-Liner Method 5: Using Python’s Sorted Function

Python’s built-in `sorted()`

function can be utilized to combine and sort the characters, achieving a one-liner solution. However, this method does not always produce a lexically minimal string if characters from the input strings are interleaved.

Here’s an example:

print("".join(sorted("ace" + "bdf")))

Output:

abcdef

This one-liner concatenates the two strings and sorts the characters with `sorted()`

. It is then joined back into a single string. It’s simple, but caution is required since it may not work with interleaved characters.

## Summary/Discussion

**Method 1:**Merging with Sorting. Strengths: Simple, explicit. Weaknesses: May be less efficient for long strings due to constant prefix slicing.**Method 2:**Using a Priority Queue. Strengths: Efficient for larger input strings. Weaknesses: Can be overkill for short strings; more complex than necessary.**Method 3:**Recursive Approach. Strengths: Elegant, clearly shows decision points. Weaknesses: Possible stack overflow with very long strings.**Method 4:**Iterative Two-Pointer Technique. Strengths: Space-efficient, good for longer strings. Weaknesses: Slightly more complex logic.**Bonus Method 5:**Using Python’s Sorted Function. Strengths: Extremely concise. Weaknesses: Can produce incorrect results for certain interleaved inputs.