1625\. Lexicographically Smallest String After Applying Operations

Medium

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`:

*   Add `a` to all odd indices of `s` **(0-indexed)**. Digits post `9` are cycled back to `0`. For example, if `s = "3456"` and `a = 5`, `s` becomes `"3951"`.
*   Rotate `s` to the right by `b` positions. For example, if `s = "3456"` and `b = 1`, `s` becomes `"6345"`.

Return _the **lexicographically smallest** string you can obtain by applying the above operations any number of times on_ `s`.

A string `a` is lexicographically smaller than a string `b` (of the same length) if in the first position where `a` and `b` differ, string `a` has a letter that appears earlier in the alphabet than the corresponding letter in `b`. For example, `"0158"` is lexicographically smaller than `"0190"` because the first position they differ is at the third letter, and `'5'` comes before `'9'`.

**Example 1:**

**Input:** s = "5525", a = 9, b = 2

**Output:** "2050"

**Explanation:** We can apply the following operations: 

Start: "5525" 

Rotate: "2555" 

Add: "2454" 

Add: "2353" 

Rotate: "5323" 

Add: "5222" 

Add: "5121" 

Rotate: "2151" 

Add: "2050" 

There is no way to obtain a string that is lexicographically smaller then "2050".

**Example 2:**

**Input:** s = "74", a = 5, b = 1

**Output:** "24"

**Explanation:** We can apply the following operations:

Start: "74" 

Rotate: "47"

Add: "42"

Rotate: "24"

There is no way to obtain a string that is lexicographically smaller then "24".

**Example 3:**

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

**Output:** "0011"

**Explanation:** There are no sequence of operations that will give us a lexicographically smaller string than "0011".

**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`