Problem Description

The problem presents a method of abbreviating strings by replacing non-adjacent, non-empty substrings with the length of those substrings. The objective is to evaluate whether a given abbreviated string, abbr, correctly represents the abbreviation of another string, word. It's important to note that a valid abbreviation:

• replaces non-empty and non-adjacent substrings,
• does not have leading zeros in the numerical representations of the lengths,
• is not created by replacing adjacent substrings.

The goal is to determine if the given abbr represents a valid abbreviation of the word.

Here are some examples:

• "s10n" is a valid abbreviation of "substitution" because the substring of 10 characters, "ubstitutio", is replaced by its length, without leading zeros.
• "s55n" is not valid because it implies adjacent substrings were replaced, which is against the rules.

We are tasked with writing a function that takes word and abbr and returns True if abbr is a valid abbreviation of word, and False otherwise.

Intuition

The solution is implemented by simultaneously iterating through both word and abbr and checking for matches and valid abbreviations. The approach is as follows:

• Iterate over each character in the word using a pointer i and a pointer j for iterating over the abbr.
• If the current characters in word and abbr match, continue to the next characters of both strings.
• If the current character in abbr is a digit, it indicates an abbreviated part of word. We then:
  • Check for any leading zeros or if the number is '0', which are not valid.
  • Parse the complete number indicating the length of the abbreviated substring.
  • Skip ahead in word by the length of the abbreviated part.
• After the loop:
  • If i is at the end of word and j is at the end of abbr, the abbreviation is valid, and we return True.
  • If either i is not at the end of word or j is not at the end of abbr, the abbreviation is not valid, and we return False.

We leverage the facts that letters must match exactly and numbers must accurately represent the length of the skipped substrings in the word.

This approach efficiently determines if the abbreviation is valid by inspecting each character only once, resulting in an O(n) time complexity, where n is the length of the word.

Learn more about Two Pointers patterns.

Solution Approach

The solution uses a straightforward approach based on two-pointer iteration and string manipulation techniques without the need for additional data structures. Let's walk through the implementation and the reasoning behind it:

• Two integer pointers, i and j, are initialized to 0. These act as indexes to traverse word and abbr respectively.
• Two variables, m and n, hold the lengths of word and abbr. They help in checking whether we have reached the end of the strings during iteration.

The algorithm then enters a while-loop, with the condition that i is less than m, letting us loop until the end of word:

• The first if statement within the loop checks whether j has reached the end of abbr. If it has, then abbr cannot be a valid abbreviation of word because there are unmatched characters in word, so the function returns False.

• If word[i] is equal to abbr[j], we have a direct character match, and we advance both pointers.

• If word[i] is not equal to abbr[j], the algorithm checks for a number in abbr by entering another while-loop which continues as long as abbr[k] is a digit. This is determined by the isdigit() method, effectively parsing an integer from the abbreviation.

• After exit from the inner while-loop, a substring t from abbr[j] to abbr[k] holds the number.

• The validity of the number is then checked:

  • It should be a digit and not have leading zeros.
  • It should not be 0 since an abbreviation can't represent a substring of 0 length.

• If t is valid, the length int(t) is added to pointer i to skip the matching characters in word. Pointer j is set to k to continue checking abbr after the number.

Finally, after the while-loop:

• The function returns True only if i == m and j == n, meaning that both the word and abbreviation were completely matched.

Overall, the solution uses an iterative approach and basic string operations without requiring additional memory, i.e., in O(1) space complexity, and time complexity is O(n), where n is the length of the word.

Example Walkthrough

Let's consider a small example to illustrate the solution approach described in the content above. Suppose we have the word "international" and the abbreviation "i11l". We want to determine if "i11l" is a correct abbreviation for "international".

Here's the step-by-step process of checking the validity of the abbreviation:

1. Initialize two pointer variables, i and j, to 0. They will iterate over the characters in "international" and "i11l", respectively.

2. Iterate while i < len("international").

   • Step 1: Both i and j point to 'i'. Since the characters match, we move i and j forward to 1.

   • Step 2: j now points to '1'. This signifies an abbreviation. Since '1' is a digit and not '0', there are no leading zeros, and so it is initially considered valid.

   • Step 3: We continue to move j forward to gather the complete number representing the length of the abbreviation. We find that j is '1' at index 1 and '1' at index 2. This represents the number 11.

   • Step 4: After parsing the complete number 11, we move the i pointer forward in "international" by 11 characters, landing it on 'l'.

   • Step 5: We set j to the index after the parsed number, which is 3, pointing to 'l'.

3. Now, both i and j point to 'l' again, which match, so we move both i and j forward by 1.

4. At this point, i == len("international") and j == len("i11l"), indicating that we have reached the end of both the word and the abbreviation. Since all characters have been successfully matched, and the numerical abbreviations have been valid, we return True.

So, "i11l" is a valid abbreviation for "international".

Solution Implementation

Time and Space Complexity

Time Complexity

The provided algorithm iterates over each character of the word and abbr strings once. On each iteration, while checking if a part of abbr is a number, it may iterate through consecutive digits. However, each character in both strings is looked at mostly once because, after processing the digits, the index jumps from the beginning to the end of the numerical sequence.

Therefore, the time complexity of the algorithm can be represented as O(m + n), where m is the length of word and n is the length of abbr.

Space Complexity

The algorithm uses a constant amount of extra space for variables i, j, k, t, m, and n. It doesn't utilize any additional structures dependent on the input sizes.

Thus, the space complexity is O(1), which represents constant space.

Learn more about how to find time and space complexity quickly using problem constraints.