1625. Lexicographically Smallest String After Applying Operations

1625. Lexicographically Smallest String After Applying Operations

Difficulty: Medium

Topics: String, Depth-First Search, Breadth-First Search, Enumeration, Weekly Contest 211

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 than "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 than "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

Hint:

1. Since the length of s is even, the total number of possible sequences is at most 10 * 10 * s.length.
2. You can generate all possible sequences and take their minimum.
3. Keep track of already generated sequences so they are not processed again.

Solution:

We need to find the lexicographically smallest string by applying two operations:

1. Add a to all odd indices (0-indexed), with digits wrapping around (mod 10)
2. Rotate the string right by b positions

Since the string length is at most 100, I can use BFS to explore all possible states.

Approach

1. Use BFS to explore all reachable string states
2. Keep track of visited strings to avoid cycles
3. For each string, generate two new strings:
   • One by applying the "add to odd indices" operation
   • One by applying the rotation operation
4. Track the lexicographically smallest string found

Let's implement this solution in PHP: 1625. Lexicographically Smallest String After Applying Operations

<?php
/**
 * @param String $s
 * @param Integer $a
 * @param Integer $b
 * @return String
 */
function findLexSmallestString($s, $a, $b) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo findLexSmallestString("5525", 9, 2) . "\n"; // Output: "2050"
echo findLexSmallestString("74", 5, 1) . "\n";   // Output: "24"
echo findLexSmallestString("0011", 4, 2) . "\n"; // Output: "0011"
?>

Explanation:

1. Initialization: Start with the original string s in both the queue and visited set.
2. BFS Loop: Process each string in the queue:
   • Compare with current best string
   • Generate two new strings by applying both operations
   • Add new strings to queue if not visited before
3. Operation 1 (Add to odd indices): For each odd index in the string, add a to the digit and take modulo 10 to handle wrapping.
4. Operation 2 (Rotation): Rotate the string right by b positions by concatenating the last b characters with the first n-b characters.
5. Termination: When the queue is empty, return the best string found.

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