Rearrange Characters to Make Target String

You are given two 0-indexed strings s and target. You can take some letters from s and rearrange them to form new strings. Return the maximum number of copies of target that can be formed by taking letters from s and rearranging them.

Example 1:

Input: s = "ilovecodingonleetcode", target = "code"
Output: 2

Example 2:

Input: s = "abcba", target = "abc"
Output: 1

Example 3:

Input: s = "abbaccaddaeea", target = "aaaaa"
Output: 1

Constraints:

• 1 <= s.length <= 100
• 1 <= target.length <= 10
• s and target consist of lowercase English letters.

Write a function to solve this problem efficiently. Explain the time and space complexity of your solution. Also, discuss any edge cases that might arise and how to handle them. Provide code implementation in Python.

def max_number_of_targets(s: str, target: str) -> int:
    """Calculates the maximum number of copies of target that can be formed from s.

    Args:
        s: The source string.
        target: The target string.

    Returns:
        The maximum number of copies of target that can be formed.
    """
    s_counts = {}
    target_counts = {}

    for char in s:
        s_counts[char] = s_counts.get(char, 0) + 1

    for char in target:
        target_counts[char] = target_counts.get(char, 0) + 1

    ans = float('inf')
    for char, count in target_counts.items():
        if char not in s_counts:
            return 0
        ans = min(ans, s_counts[char] // count)

    return ans

# Example Usage
s = "ilovecodingonleetcode"
target = "code"
print(f"Maximum number of '{target}' in '{s}': {max_number_of_targets(s, target)}")

s = "abcba"
target = "abc"
print(f"Maximum number of '{target}' in '{s}': {max_number_of_targets(s, target)}")

s = "abbaccaddaeea"
target = "aaaaa"
print(f"Maximum number of '{target}' in '{s}': {max_number_of_targets(s, target)}")

Naive Approach

The most straightforward approach would be to iterate through all possible sub-sequences of s, checking if they can be rearranged to form target. However, this is highly inefficient due to the exponential number of possible sub-sequences.

Optimal Approach

The optimal approach involves counting the frequency of each character in both s and target. Then, for each character in target, we determine how many times target can be formed based on the availability of that character in s. The minimum of these values will be the maximum number of target strings that can be formed.

Big(O) Run-time Analysis

The run-time complexity is dominated by the counting of characters in both strings. Let $n$ be the length of string s and $m$ be the length of string target. The algorithm iterates through s once to count character frequencies, which takes $O(n)$ time. Similarly, it iterates through target once to count character frequencies, which takes $O(m)$ time. Finally, it iterates through the character counts of target, which in the worst case is $O(m)$. Thus, the overall time complexity is $O(n + m)$.

Big(O) Space Usage Analysis

The space complexity is determined by the storage required for the character counts. In the worst case, s_counts and target_counts can store frequencies for all unique characters in s and target respectively. In the problem constraints, it mentions that s and target consist of lowercase English letters, which means there are at most 26 unique characters. Therefore, the space complexity is $O(1)$ because it's bounded by a constant number of characters regardless of the input size.

Edge Cases

1. Empty Strings: If either s or target is an empty string, the function should handle it gracefully. If target is empty, any number of copies can be made (theoretically infinite or limited by s). If s is empty but target is not, zero copies can be made.
2. Target Contains Characters Not in Source: If target contains characters that are not present in s, the function should return 0 because it's impossible to form even a single copy of target.
3. Large Input Strings: For extremely large input strings, the character counting might become memory-intensive. However, given the constraints, this is not a significant concern.
4. Target String Longer Than Source String: If the target string is longer than the source string, and contains characters from the source string, then it is still possible to return 0, as you can not create the target string from the source. An empty source string is handled in #1.