Problem Description

In this problem, we are given two strings s and t. Our task is to determine if the two strings are isomorphic. Isomorphic strings are strings where each character in one string can be consistently replaced with another character to result in the second string. It is crucial to note that in isomorphic strings, the replacement of characters is done in such a way that the sequence of characters is maintained. Also, no two distinct characters can map to the same character, but a character is allowed to remap to itself.

In simpler terms, the pattern of characters in both strings should align. If 'x' in string s is replaced by 'y' in string t, then all 'x's in s must correspond to 'y's in t, and no other character in s should map to 'y'.

Intuition

To determine if s and t are isomorphic, we establish a mapping for each character in s to its corresponding character in t. This can be done using two hash tables or arrays, here indexed by the character's ASCII code, ensuring no collision for character storage.

Each array d1 and d2 records the latest position (index + 1 to account for the zeroth position) where each character is encountered in s and t, respectively. We iterate the strings simultaneously and compare the mapping indices.

If for any character pair a from s and b from t, the mappings in d1 and d2 mismatch, this means a prior different character already created a different mapping, hence they are not isomorphic, and we return false.

If the mappings agree for every pair of characters (implying consistent mapping), we update both arrays d1 and d2 marking the current position with the iteration index. If we complete the traversal without conflicts, the strings are isomorphic, indicated by returning true.

The idea is based on the concept that two isomorphic strings must have characters appear in the same order; hence their last occurrence should be at corresponding positions in both strings. This method allows us to verify that condition efficiently.

Solution Approach

The solution employs two arrays d1 and d2 of size 256 each (since the ASCII character set size is 256). These arrays are used to record the last seen positions of characters from s and t respectively. Another perspective is to consider d1 and d2 as mappings that keep track of positions rather than characters. This is subtly different from traditional mappings of characters to characters and avoids issues with checking for unique mappings back and forth.

Here is the step-by-step implementation breakdown:

1. We initialize two arrays d1 and d2 with zeros, assuming that a character's ASCII value is a valid index. This suits our purpose since characters that have not been seen are initialized with 0 by default.

2. We iterate over the strings s and t together. In Python, the zip function is useful for this as it bundles the two iterables into a single iterable each pair of characters can be accessed together in the loop.

3. During each iteration, we convert the characters a and b from s and t into their respective ASCII values using Python's ord() function.

4. We compare the current values at indices a and b in arrays d1 and d2. If d1[a] is not equal to d2[b], it implies that either a was previously associated with a different character than b or b with a different character than a, which means that the characters a and b are part of inconsistent mappings—violation of isomorphic property—hence, we return false.

5. If the comparison does not lead to a discrepancy, we need to update d1[a] and d2[b] with the current position of iteration, indicated by i (start counting from 1 to correctly handle the zeroth case). This step essentially marks the new last occurrence of each character pair as we proceed through the strings.

6. If all character pairs pass the condition checks and we successfully reach the end of the iteration, it suggests that all mappings were consistent. Hence, we return true.

By utilizing arrays and checking indices, we avoid the need for complicated hash table operations, which can be more expensive in terms of memory and time.

Here's the implementation of the above algorithm:

class Solution:
    def isIsomorphic(self, s: str, t: str) -> bool:
        d1, d2 = [0] * 256, [0] * 256
        for i, (a, b) in enumerate(zip(s, t), 1):
            a, b = ord(a), ord(b)
            if d1[a] != d2[b]:
                return False
            d1[a] = d2[b] = i
        return True

In this approach, we use array indices to represent characters and array values to represent the last positions where characters appeared, comparing the evolution of these positions as a way to check for isomorphism.

Example Walkthrough

Let's use a small example to illustrate the solution approach. Suppose we have two strings s = "paper" and t = "title". We want to determine if they are isomorphic.

1. We initialize two empty arrays d1 and d2 of size 256 to store last seen positions of characters from s and t respectively, starting with all zeros.

2. Iterating over the strings s and t together, we start with the first characters p from s and t from t. We convert them to their ASCII values and get a = ord('p') and b = ord('t').

3. We check the current values at d1[a] and d2[b]. Since both are zero (indicating that we haven't encountered these characters before), we continue and update d1[a] and d2[b] with 1. Now d1[ord('p')] and d2[ord('t')] both hold the value 1.

4. Moving to the next characters, a from s and i from t, we again convert them to their ASCII values and get a = ord('a') and b = ord('i'). As before, d1[ord('a')] and d2[ord('i')] are both zero, so we update them to 2.

5. The following pairs are p and t again, and since we already have d1[ord('p')] = d2[ord('t')] = 1 from the previous steps, this reaffirms the mapping, and we update these indices to 3 now.

6. Next pairs are e from s and l from t. They yield a = ord('e') and b = ord('l'). Finding both d1[ord('e')] and d2[ord('l')] at zero, we update their indices to 4.

7. The last characters are r from s and e from t, converting to a = ord('r') and b = ord('e'). As d1[ord('r')] and d2[ord('e')] are both zero, we update them to 5.

8. Now that we've reached the end of both strings without finding any discrepancies in the mapping, this means our mapping procedure remained consistent throughout. Thus, we conclude that s = "paper" and t = "title" are isomorphic and return true.

The solution confirms that corresponding characters in both strings appear in the same order and can be mapped one-to-one, fulfilling the condition for the strings to be isomorphic.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n). This is because the code iterates through all the characters of the strings s and t once, where n is the length of the string s. Each iteration involves constant time operations like accessing and updating the value in arrays d1 and d2, and comparing the values.

The space complexity of the code is O(C), where C is the constant size of the character set used in the problem. In this case, C = 256, which covers all the extended ASCII characters. The space is taken by two arrays d1 and d2, each of which has 256 elements to accommodate all possible characters in the strings. Since the size of the character set does not scale with the size of the input, it is considered a constant space complexity.

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