249. Group Shifted Strings - Detailed Explanation

Problem Statement:

Given a list of strings, group all strings that belong to the same shifting sequence.

Two strings belong to the same shifting sequence if one can be shifted to become the other. Shifting a string means replacing each letter with its next letter in the alphabet (wrapping around from 'z' to 'a'). For example:

• "abc" can be shifted to "bcd", "cde", …, and finally "zab", and then back to "abc".

A group of shifted strings contains strings that share the same relative differences between consecutive characters. For example, "abc", "bcd", and "xyz" belong to the same group because if you compute the differences between adjacent letters (considering wrap-around modulo 26), they have the same pattern.

If a string has only one character, it forms its own group with all single-character strings.

Example Inputs and Outputs:

1. Example 1:

   • Input: ["abc", "bcd", "acef", "xyz", "az", "ba", "a", "z"]
   • Output (one possible grouping):

     [
       ["abc", "bcd", "xyz"],
       ["az", "ba"],
       ["acef"],
       ["a", "z"]
     ]
     

   • Explanation:
     • The strings "abc", "bcd", and "xyz" share the same shifting pattern.
     • "az" and "ba" both have the pattern where the difference between their two characters is 25 (when adjusted modulo 26).
     • "acef" forms a group by itself.
     • All single-letter strings group together.

2. Example 2:

   • Input: ["a"]
   • Output: [["a"]]
   • Explanation:
     A single letter always forms its own group (with other single-letter strings if present).

Hints for Solving the Problem:

1. Normalization via Shifting:
   Think about how you can represent each string by its shifting pattern. For instance, if you “normalize” a string by subtracting the ASCII value of its first character from every character (with wrap-around using modulo 26), then two strings with the same shifting pattern will have the same normalized form.

2. Consecutive Differences:
   Alternatively, you can compute the differences between consecutive characters (modulo 26). For example, for "abc" the differences are [1, 1] and for "bcd" they are also [1, 1].

3. Using a HashMap/Dictionary:
   Use a hash table to group strings by their computed key (either normalized form or the tuple of differences).

Approach: Grouping by Normalized Key

Step-by-Step Explanation:

1. Compute the Key for Each String:

   • For strings with length ≥ 2:
     Compute the differences between consecutive characters, adjusted with modulo 26. For instance, for string s, for each index (i) from 1 to len(s)-1, compute
     [ \text{diff}_i = (s[i] - s[i-1] + 26) \mod 26 ] Use the sequence of differences as the key. For example:
     • "abc" → differences: ([1, 1])
     • "bcd" → differences: ([1, 1])
     • "xyz" → differences: ([1, 1])
   • For strings of length 1:
     Use a fixed key (such as an empty tuple or a special marker) since all single-letter strings belong together.

2. Group Strings by the Computed Key:

   • Use a dictionary (hash map) where each key is the normalized form (or the tuple of differences) and the corresponding value is a list of strings having that key.

3. Return the Groups:

   • Finally, return all grouped lists as the answer.

Python Code: