🧩 Lexicographically Smallest String After Applying Operations

Difficulty: Medium
Topics: String Manipulation, BFS, Math, Simulation

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📝 Problem Statement

You are given a string s of even length consisting of digits from 0 to 9, and two integers a and b.

You can apply either of the following two operations any number of times and in any order on s:

1. ➕ Add Operation:
   Add a to all odd indices of s (0-indexed).
   Digits post 9 are cycled back to 0.

   Example:
   If s = "3456" and a = 5,
   then s becomes "3951".

2. 🔄 Rotate Operation:
   Rotate s to the right by b positions.

   Example:
   If s = "3456" and b = 1,
   then s becomes "6345".

Your task is to return the lexicographically smallest string you can obtain by applying the above operations any number of times on s.

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📘 Lexicographical Order Definition

A string a is lexicographically smaller than a string b (of the same length) if in the first position where they differ,
the character in a is smaller than the character in b.

Example:
"0158" is lexicographically smaller than "0190"
because they differ at the third character ('5' < '9').

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💡 Examples

Example 1:

Input:  s = "5525", a = 9, b = 2
Output: "2050"

Explanation: We can apply the following operations:

Step	Operation	Result
1	Start	5525
2	Rotate	2555
3	Add	2454
4	Add	2353
5	Rotate	5323
6	Add	5222
7	Add	5121
8	Rotate	2151
9	Add	2050 ✅

There is no way to obtain a string smaller than "2050".

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Example 2:

Input:  s = "74", a = 5, b = 1
Output: "24"

Explanation:

Start  → "74"
Rotate → "47"
Add    → "42"
Rotate → "24"

There is no smaller string than "24".

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Example 3:

Input:  s = "0011", a = 4, b = 2
Output: "0011"

Explanation:
No sequence of operations can give a smaller string than "0011".

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⚙️ Constraints

• 2 <= s.length <= 100
• s.length is even
• s consists of digits from 0 to 9 only
• 1 <= a <= 9
• 1 <= b <= s.length - 1

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