Group Shifted Strings

Given a list of strings, we want to group together strings that are "shift sequences" of each other. A shift sequence means that each character in a string can be shifted to the next character in a circular manner (e.g., 'a' -> 'b', 'b' -> 'c', ..., 'z' -> 'a'). This challenge tests your ability to identify patterns and group similar data structures.

Problem Description

You are given a list of strings, strings. Each string consists only of lowercase English letters.

The task is to group all the strings that belong to the same shifting sequence. Two strings are in the same shifting sequence if:

They have the same length.
The difference between the ASCII values of consecutive characters in the first string is the same as the difference between the ASCII values of consecutive characters in the second string. This difference should be calculated cyclically, meaning that shifting from 'z' to 'a' results in a difference of 1.

For example, "abc" and "bcd" are in the same shifting sequence because:

'b' - 'a' = 1
'c' - 'b' = 1

And for "bcd":

'c' - 'b' = 1
'd' - 'c' = 1

Also, "az" and "ba" are in the same shifting sequence:

'z' - 'a' = 25 (or -1 cyclically, which is equivalent to 25 modulo 26)
'a' - 'z' = 1 (cyclically)

For "ba":

'a' - 'b' = -1 (or 25 cyclically)

The output should be a list of lists, where each inner list contains strings that belong to the same shifting sequence. The order of the inner lists and the order of strings within each inner list does not matter.

Edge Cases:

Empty input list.
Input list containing only one string.
Strings with length 1. All strings of length 1 are considered to be in the same shifting sequence.
Strings that wrap around, like "az" and "za".

Examples

Example 1:

Input: ["abc", "bcd", "acef", "xyz", "az", "ba", "a", "z"]
Output: [["abc", "bcd", "xyz"], ["az", "ba"], ["acef"], ["a", "z"]]
Explanation:
- "abc", "bcd", and "xyz" all have a difference of 1 between consecutive characters.
- "az" and "ba" have a cyclical difference of 25 (or -1) between consecutive characters.
- "acef" forms its own group as its pattern is unique.
- "a" and "z" are single-character strings, which all belong to the same shifting sequence.

Example 2:

Input: ["a"]
Output: [["a"]]
Explanation: A single string forms its own group.

Example 3:

Input: ["ab","ba"]
Output: [["ab"],["ba"]]
Explanation: The difference between 'b' and 'a' is 1. The difference between 'a' and 'b' is -1 (or 25 cyclically). These are not the same patterns.

Example 4:

Input: ["yz", "za"]
Output: [["yz", "za"]]
Explanation:
- For "yz": 'z' - 'y' = 1.
- For "za": 'a' - 'z' = 1 (cyclically).
Both have a shift of 1 between consecutive characters.

Constraints

1 <= strings.length <= 1000
1 <= strings[i].length <= 100
strings[i] consists of only lowercase English letters.

Notes

To calculate the cyclic difference between two characters c1 and c2 (where c2 follows c1 in the sequence), you can use the formula: (ASCII(c2) - ASCII(c1) + 26) % 26. This ensures the difference is always a positive value between 0 and 25.
A good approach would be to define a canonical representation for each shifting sequence. Strings with the same canonical representation belong to the same group.
Consider how you might handle strings of length 1.