Largest Merge Of Two Strings

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Greedy AlgorithmsStringsTwo Pointers

You are given two strings word1 and word2. You want to construct a string merge in the following way: while either word1 or word2 are non-empty, choose one of the following options:

If word1 is non-empty, append the first character in word1 to merge and delete it from word1.
- For example, if word1 = "abc" and merge = "dv", then after choosing this operation, word1 = "bc" and merge = "dva".
If word2 is non-empty, append the first character in word2 to merge and delete it from word2.
- For example, if word2 = "abc" and merge = "", then after choosing this operation, word2 = "bc" and merge = "a".

Return the lexicographically largest merge you can construct.

A string a is lexicographically larger than a string b (of the same length) if in the first position where a and b differ, a has a character strictly larger than the corresponding character in b. For example, "abcd" is lexicographically larger than "abcc" because the first position they differ is at the fourth character, and d is greater than c.

Example 1:

Input: word1 = "cabaa", word2 = "bcaaa"
Output: "cbcabaaaaa"
Explanation: One way to get the lexicographically largest merge is:
- Take from word1: merge = "c", word1 = "abaa", word2 = "bcaaa"
- Take from word2: merge = "cb", word1 = "abaa", word2 = "caaa"
- Take from word2: merge = "cbc", word1 = "abaa", word2 = "aaa"
- Take from word1: merge = "cbca", word1 = "baa", word2 = "aaa"
- Take from word1: merge = "cbcab", word1 = "aa", word2 = "aaa"
- Append the remaining 5 a's from word1 and word2 at the end of merge.

Example 2:

Input: word1 = "abcabc", word2 = "abdcaba"
Output: "abdcabcabcaba"

Constraints:

1 <= word1.length, word2.length <= 3000
word1 and word2 consist only of lowercase English letters.

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the lengths of word1 and word2?
Can word1 or word2 be empty strings, and if so, what should the output be?
Are word1 and word2 guaranteed to contain only lowercase English letters?
If the prefixes of word1 and word2 are identical, how deep should I look to break the tie lexicographically?
In the case of a tie, is it acceptable to choose *any* lexicographically larger character from either string, or is there a preferred order if the characters are equal for a long prefix?

Brute Force Solution

Approach

The brute force method means trying every possible combination of merging two strings to find the largest possible merged string. We explore all ways of picking characters from each input string to build a potential result.

Here's how the algorithm would work step-by-step:

Consider the first character of both strings. We have two choices: pick the first character from the first string, or pick the first character from the second string.
For each choice, we then look at the remaining characters in both strings and again, decide whether to take from the first or the second string.
Keep building new merged strings by trying all possible combinations of characters taken from the two input strings until both strings are completely used up.
As we build each merged string, we compare it to all the other merged strings we've generated so far.
The 'largest' string according to the normal dictionary order is the one we keep track of. We compare each new merged string with our currently 'largest' string and update if necessary.
After exploring every possible combination of picking characters from the two strings, the string we have tracked as the 'largest' is our answer.

Code Implementation

def largest_merge_of_strings_brute_force(string1, string2):
    largest_merged_string = ""

    def merge_recursive(index_1, index_2, current_merged_string):
        nonlocal largest_merged_string

        # If both strings are exhausted, update largest string
        if index_1 == len(string1) and index_2 == len(string2):
            if current_merged_string > largest_merged_string:
                largest_merged_string = current_merged_string
            return

        # Try taking from string1 if it's not exhausted
        if index_1 < len(string1):
            merge_recursive(index_1 + 1, index_2, current_merged_string + string1[index_1])

        # Try taking from string2 if it's not exhausted
        if index_2 < len(string2):
            merge_recursive(index_1, index_2 + 1, current_merged_string + string2[index_2])

    # Begin recursive exploration to find the largest merge
    merge_recursive(0, 0, "")

    return largest_merged_string

Big(O) Analysis

Time Complexity

O(2^(m+n)) – The brute force approach explores every possible combination of characters from the two input strings word1 and word2, with lengths m and n respectively. At each step, we have two choices: take a character from word1 or word2. This branching continues until both strings are exhausted. Therefore, the number of possible merges grows exponentially with the combined length of the two strings, leading to 2^(m+n) possibilities. The Big O time complexity is thus O(2^(m+n)).

Space Complexity

O(2^N) – The brute force approach described explores all possible merge combinations. At each step, we have two choices (take from the first string or the second), leading to a branching factor of 2. The depth of this decision tree is at most N, where N is the combined length of the two input strings. Therefore, the number of possible merged strings generated and compared is 2^N. In the worst case, we're keeping all these generated merged strings in memory to compare, driving up the auxiliary space. Finally, the space complexity is O(2^N) because we store all intermediate merged string results, each with length at most N, leading to an upper bound dominated by the exponential growth of possible strings.

Optimal Solution

Approach

We want to create the biggest possible string by combining two other strings. At each step, we'll pick the 'larger' of the two string beginnings to add to our result, always choosing the one that leads to a lexicographically bigger final string. This avoids exploring less promising combinations.

Here's how the algorithm would work step-by-step:

Compare the beginnings of the two strings we're merging.
If one string's beginning part is 'bigger' than the other, add that 'bigger' part to our result string and remove it from its original string.
If the beginning parts are the same, we need to look further into both strings to see which one would lead to a bigger final result string. Effectively, perform a lexicographical comparison of the remainder of the strings.
Keep doing this until both initial strings are empty.
The result string we've built is the largest possible merge.

Code Implementation

def largestMerge(word1, word2):
    merged_string = ""
    string_one_index = 0
    string_two_index = 0

    while string_one_index < len(word1) or string_two_index < len(word2):
        # If word1's current part is larger, add it to the result.
        if word1[string_one_index:] > word2[string_two_index:]:

            merged_string += word1[string_one_index]
            string_one_index += 1

        # If word2's current part is larger (or equal), add it to the result.
        else:

            merged_string += word2[string_two_index]
            string_two_index += 1

    #After exhausting both strings, return the merged result.
    return merged_string

Big(O) Analysis

Time Complexity

O(n^2) – The algorithm iterates until both input strings, whose combined length is n, are empty. In each iteration, a lexicographical comparison is performed. In the worst-case scenario, when the prefixes of the two strings are identical for an extended length, a comparison of the remaining substrings is needed. This substring comparison can take O(n) time in the worst case. Since we potentially do this comparison for each of the n characters in the combined strings, the overall time complexity is O(n * n), or O(n^2).

Space Complexity

O(N) – The algorithm constructs a new string, the merged string, whose maximum length is the sum of the lengths of the two input strings. In the worst case, the merged string will have a length of N, where N represents the combined length of the two input strings. Therefore, the auxiliary space required to store this merged string scales linearly with the input size N. Other variables used are constant space (e.g., indices), so the dominant space complexity is O(N).

Edge Cases

Case	How to Handle
word1 or word2 is null	Return the non-null string or an empty string if both are null.
word1 or word2 is empty	Return the non-empty string or an empty string if both are empty.
word1 and word2 are identical strings	The algorithm should pick either string sequentially since they are lexicographically equal; can pick either.
word1 and word2 are very long strings (e.g., 10^5 characters)	The solution should have a linear time complexity in terms of the total number of characters to avoid timeouts.
word1 and word2 contain only the same character (e.g., 'aaaa' and 'bbb')	The algorithm should correctly pick from the string with the lexicographically larger character.
Prefix of word1 is equal to prefix of word2, but different characters follow.	The algorithm should continue comparing characters until the prefixes differ to determine the lexicographically larger string.
One word is a prefix of the other (e.g., word1 = 'abc', word2 = 'abcd')	The algorithm should handle the prefix situation gracefully, ensuring the longer string's suffix is appended.
Memory constraints when handling extremely large strings.	Ensure the chosen language efficiently handles string concatenation without excessive memory allocation.

word1 or word2 is null

How to Handle:

Return the non-null string or an empty string if both are null.

word1 or word2 is empty

How to Handle:

Return the non-empty string or an empty string if both are empty.

word1 and word2 are identical strings

How to Handle:

The algorithm should pick either string sequentially since they are lexicographically equal; can pick either.

word1 and word2 are very long strings (e.g., 10^5 characters)

How to Handle:

The solution should have a linear time complexity in terms of the total number of characters to avoid timeouts.

word1 and word2 contain only the same character (e.g., 'aaaa' and 'bbb')

How to Handle:

The algorithm should correctly pick from the string with the lexicographically larger character.

Prefix of word1 is equal to prefix of word2, but different characters follow.

How to Handle:

The algorithm should continue comparing characters until the prefixes differ to determine the lexicographically larger string.

One word is a prefix of the other (e.g., word1 = 'abc', word2 = 'abcd')

How to Handle:

The algorithm should handle the prefix situation gracefully, ensuring the longer string's suffix is appended.

Memory constraints when handling extremely large strings.

How to Handle:

Ensure the chosen language efficiently handles string concatenation without excessive memory allocation.