Problem Description

In this problem, you're given three strings s1, s2, and baseStr, all of which are of the same length. Characters at corresponding positions in s1 and s2 are considered equivalent, which means s1[i] is equivalent to s2[i]. This equivalence relationship is reflexive, symmetric, and transitive; that means if a is equivalent to b, then b is equivalent to a (symmetry), and if a is equivalent to b, b is equivalent to c, then a is equivalent to c (transitivity). Using these equivalences, your goal is to transform baseStr into its lexicographically smallest equivalent string.

An example provided is with s1 = "abc" and s2 = "cde", which means a == c, b == d, and c == e. For the baseStr = "eed", both "acd" and "aab" are equivalent strings, but "aab" is the lexicographically smaller one.

Intuition

To find the lexicographically smallest string equivalent to baseStr, we can leverage a classic data structure known as the Union-Find or Disjoint Set Union (DSU). This structure helps efficiently handle equivalence relationships by finding and uniting equivalent components.

1. Initialize a disjoint set where each character initially belongs to its own set, which is indexed from 0 to 25 (mapping to 'a' through 'z').

2. Iterate over s1 and s2 simultaneously and, using the find() function, locate the root parent of the characters at the current indices in the Union-Find. If these roots are different, we unite the sets by making the root with a smaller index the parent of the root with a larger index. This step builds up the equivalence classes.

3. For each character in baseStr, replace it with the smallest equivalent character in its equivalence class. We find the smallest equivalent character by finding the root parent of the set the character belongs to.

4. After processing all characters of baseStr, the resulting string is the lexicographically smallest equivalent string.

The provided solution code follows the above approach using Union-Find to efficiently transform baseStr into its lexicographically smallest form respecting the equivalency relations defined by s1 and s2.

Learn more about Union Find patterns.

Solution Approach

The solution uses the Union-Find algorithm to group equivalent characters together. Each character maps to an integer from 0 to 25, corresponding to 'a' through 'z'. The find function and an array p are used to manage the parent relationships of these integers, which is key to understanding the connected components (equivalence classes) of characters.

The implementation steps are as follows:

1. Parent Array Initialization: An array p of size 26 is initialized, where each index represents a unique character from 'a' to 'z'. Initially, each character is its own parent: p[i] = i.

   p = list(range(26))

2. Find Function: The find function takes an integer x and finds the root parent in the Union-Find structure. This is done by recursively finding the parent until p[x] == x. Upon finding the root parent, path compression is performed by setting p[x] to the root parent to flatten the structure, making future queries faster.

   def find(x):
       if p[x] != x:
           p[x] = find(p[x])
       return p[x]

3. Union Steps: The main loop iterates over the indices of s1 and s2, and for each pair of characters, it calculates their corresponding integer representations. Using the find function, it locates the root parents of each character's set. To maintain lexicographical order, it unions the sets by setting the parent of the higher index to the parent of the lower index.

   for i in range(len(s1)):
       a, b = ord(s1[i]) - ord('a'), ord(s2[i]) - ord('a')
       pa, pb = find(a), find(b)
       if pa < pb:
           p[pb] = pa
       else:
           p[pa] = pb

4. Building the Result String: Lastly, for each character in baseStr, it is replaced by the lexicographically smallest equivalent character. This is achieved by finding the root parent of each character's set and converting it back to the character representation.

   res = []
   for a in baseStr:
       a = ord(a) - ord('a')
       res.append(chr(find(a) + ord('a')))
   return ''.join(res)

By following these steps, the solution re-encodes baseStr into its smallest lexicographical form according to the equivalences defined by s1 and s2, and returns the result as a new string.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Suppose we have the following input:

• s1 = "ab"
• s2 = "bc"
• baseStr = "adb"

The mapping arising from s1 and s2 means a == b and b == c. Since the equivalence relationship is transitive, we can also deduce that a == c.

Using the Union-Find algorithm, we'll follow the steps to find the lexicographically smallest equivalent string of baseStr.

1. Parent Array Initialization: We initialize the array p where p[i] = i for i = 0 to 25. For simplicity, assume the array is indexed starting at 0 for 'a', 1 for 'b', etc.

   p = [0, 1, 2, ... 25]

2. Find Function: We define a find function that will be used to find the root parent of a character's set in our Union-Find structure, ensuring that we apply path compression along the way.

3. Union Steps: We iterate over s1 and s2 to build the equivalency classes:

   For i = 0 (considering a and b):

   • a maps to 0, and b maps to 1.
   • find(0) returns 0, and find(1) returns 1.
   • Since 0 < 1, we set p[1] to 0.

   Now p looks like this: [0, 0, 2, ... 25].

   For i = 1 (considering b and c):

   • b maps to 1, and c maps to 2.
   • find(1) returns 0 (after path compression), and find(2) returns 2.
   • Since 0 < 2, we set p[2] to 0.

   After this step, our p array reflects the transitive closure: [0, 0, 0, ... 25].

4. Building the Result String: We construct the lexicographically smallest equivalent string for baseStr, "adb":

   • For a (0), find(0) returns 0, so a remains a.
   • For d (3), find(3) returns 3, so d remains d.
   • For b (1), find(1) now returns 0 due to path compression, so b is replaced by a.

   After this process, baseStr is transformed into "aad".

As a result, the smallest equivalent string we can form from the base string "adb" following the equivalence relationships defined by s1 and s2 is "aad".

Solution Implementation

Time and Space Complexity

The given code represents a solution to find the smallest equivalent string based on the character mappings described by s1 and s2, and then applies this mapping to baseStr.

Time Complexity:

The time complexity of the code is derived from the following operations:

• A loop that iterates through s1 and s2 strings (both are of equal length): for i in range(len(s1)).
• The find function, which is a path compression technique in the Union-Find algorithm. The amortized time complexity for each call to find is O(α(n)), where α(n) is the Inverse Ackermann function. It is nearly constant and very slow-growing, basically considered a constant for practical purposes.
• Iterating through each character in baseStr and performing the find operation.

Since we run the find operation for each character in s1/s2 and then again for each character in baseStr, the total time complexity is O(n + m * α(n)), where n is the length of s1/s2 and m is the length of baseStr. The term α(n) can usually be omitted when discussing practical time complexity, leading to O(n + m).

Space Complexity:

The space complexity is determined by:

• The parent array p of fixed size 26, which corresponds to the 26 possible characters: O(26) or O(1) since it's a constant size.
• The res list that will eventually contain every character in baseStr: O(m), where m is the length of baseStr.

Therefore, the space complexity is O(m + 1) or simply O(m) since the term O(1) is negated when compared to non-constant space usage.

In conclusion, the time complexity of the provided code is O(n + m) and the space complexity is O(m).

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