2851. String Transformation - Detailed Explanation

Problem Statement

You are given two strings, s and t, both consisting of lowercase English letters and having the same length. In one operation, you may choose a character in s and replace all of its occurrences with another character that is not currently present in s. The goal is to determine the minimum number of operations required to transform s into t. If it is impossible to perform such a transformation under these rules, return -1.

Example Inputs & Outputs

• Example 1:

  • Input:
    s = "aabcc"
    t = "ccdee"
  • Output: 3
  • Explanation:
    One valid sequence of operations is:
    1. Replace all 'c' in s with 'd' (assuming 'd' is not in s yet), so s becomes "aabdd".
    2. Replace all 'a' with 'c', so s becomes "ccbdd".
    3. Replace all 'b' with 'e', so s becomes "ccdee".

• Example 2:

  • Input:
    s = "leetcode"
    t = "codeleet"
  • Output: -1
  • Explanation:
    It is impossible to transform "leetcode" into "codeleet" because the required character mappings conflict; that is, at least one character in s would have to be mapped to two different target characters.

• Example 3:

  • Input:
    s = "ab"
    t = "ba"
  • Output: 3
  • Explanation:
    A transformation sequence could be:
    1. Replace 'a' with a temporary character (say, 'c'), so s becomes "cb".
    2. Replace 'b' with 'a', so s becomes "ca".
    3. Replace 'c' with 'b', so s becomes "ba". (Note that direct replacement is not allowed when the target character already exists in s, so an intermediate “dummy” character is needed to break the cycle.)

Constraints

• The strings s and t have the same length.
• Both strings contain only lowercase English letters.
• The size of the strings is usually limited (for example, up to 26) because the transformation rules are defined over the 26 lowercase letters.

Hints

1. Mapping Consistency:
   First, go through s and t character by character. If a character in s is supposed to transform into different characters in t (at different positions), then the transformation is impossible—return -1.

2. Represent as a Graph:
   Think of each transformation (from a character in s to the corresponding character in t) as a directed edge in a graph. If the mapping is “a → b” and “b → c” and so on, you might have chains or even cycles. Cycles require an extra temporary transformation to break.

3. Order of Operations:
   Once you have built the mapping, the number of operations is at least the number of direct character mappings (i.e. where the source and target differ). However, if a cycle exists (for example, a cycle like a → b, b → a), you must perform one extra operation to resolve it.

Approach 1: Direct Mapping with Cycle Detection

Step-by-Step Walkthrough

1. Build the Mapping:

   • Iterate over the indices of s and t.
   • For each position, if the character in s differs from the character in t:
     • If a mapping already exists for that character and it does not match the current target, return -1 (inconsistent mapping).
     • Otherwise, record the mapping (e.g. from 'a' to 'c').

2. Count Direct Transformation Operations:

   • Each mapping where the source character is different from the target contributes at least one operation.
   • For characters that already match (i.e. s[i] == t[i]), no operation is needed.

3. Detect and Handle Cycles:

   • Use a graph or DFS (depth-first search) to detect cycles in the mapping.
   • For every cycle detected (for example, 'a' → 'b' and 'b' → 'a' forms a cycle), you need to add one extra operation. The reason is that you cannot directly convert within a cycle because replacing one character might overwrite the target that is needed later.

4. Return the Total Count:

   • The total number of operations is the sum of the direct mappings plus the extra operations required for each cycle.

Visual Example

Consider the transformation:

• s = "ab"
• t = "ba"

Mapping:

• 'a' must transform to 'b'
• 'b' must transform to 'a'

This forms a cycle:

• Directly replacing 'a' with 'b' would cause both positions to have 'b', making it impossible to later get 'a' from the original 'b'.
• To resolve this, you can introduce a temporary character (say 'c'):
  1. Replace all 'a' with 'c' (temporary move).
  2. Replace all 'b' with 'a'.
  3. Replace all 'c' with 'b'.
• The operations here count as 3.

Approach 2: Graph-Based Transformation Order

Step-by-Step Walkthrough

1. Graph Construction:

   • Create a directed graph where each node is a character.
   • Add an edge from character x to character y if there is a required transformation from x (in s) to y (in t) and x ≠ y.

2. Cycle Detection:

   • Traverse the graph using DFS (or another cycle-detection algorithm) to check for cycles.
   • Each detected cycle indicates that you cannot perform the transformation in a single move for that cycle; an extra move is required to “break” the cycle.

3. Compute the Answer:

   • Count the number of mappings (i.e. the number of edges) which gives the number of direct operations.
   • Add one extra operation for each cycle detected.

Visual Example

For the transformation:

• s = "aabcc"
• t = "ccdee"

Mappings derived:

• 'a' → 'c'
• 'b' → 'd'
• 'c' → 'e'

Graph edges:

• a → c, b → d, c → e

There are no cycles in this graph, so the minimum operations equal the number of mappings: 3.

Code Examples

Python Code