514. Freedom Trail

514. Freedom Trail

Difficulty: Hard

Topics: Array, Dynamic Programming, Depth-First Search, Breadth-First Search

In the video game Fallout 4, the quest "Road to Freedom" requires players to reach a metal dial called the "Freedom Trail Ring" and use the dial to spell a specific keyword to open the door.

Given a string ring that represents the code engraved on the outer ring and another string key that represents the keyword that needs to be spelled, return the minimum number of steps to spell all the characters in the keyword.

Initially, the first character of the ring is aligned at the "12:00" direction. You should spell all the characters in key one by one by rotating ring clockwise or anticlockwise to make each character of the string key aligned at the "12:00" direction and then by pressing the center button.

At the stage of rotating the ring to spell the key character key[i]:

1. You can rotate the ring clockwise or anticlockwise by one place, which counts as one step. The final purpose of the rotation is to align one of ring's characters at the "12:00" direction, where this character must equal key[i].
2. If the character key[i] has been aligned at the "12:00" direction, press the center button to spell, which also counts as one step. After the pressing, you could begin to spell the next character in the key (next stage). Otherwise, you have finished all the spelling.

Example 1:

• Input: ring = "godding", key = "gd"
• Output: 4
• Explanation:

For the first key character 'g', since it is already in place, we just need 1 step to spell this character.
For the second key character 'd', we need to rotate the ring "godding" anticlockwise by two steps to make it become "ddinggo".
Also, we need 1 more step for spelling.
So the final output is 4.

Example 2:

• Input: ring = "godding", key = "godding"
• Output: 13

Constraints:

• 1 <= ring.length, key.length <= 100.
• ring and key consist of only lower case English letters.
• It is guaranteed that key could always be spelled by rotating ring.

Solution:

Here's a breakdown of how we approach it:

Problem Understanding

We are given two strings:

• ring: the circular ring of characters.
• key: the word we need to spell by rotating the ring.

For each character in key, we rotate the ring to align the character at the top (index 0), and this rotation can happen either clockwise or anticlockwise. The goal is to find the minimum number of rotations (steps) required to spell all the characters of key.

Steps for the Solution

1. Character Positions:

   • First, we precompute the positions of each character in ring. This allows us to efficiently look up all positions where a given character appears.

2. Dynamic Programming Table:

   • Let dp[i][j] represent the minimum number of steps required to spell the first i characters of key, ending with key[i-1] aligned at position j in ring.

3. Base Case:

   • For the first character of key, we initialize the DP table by calculating the steps needed to rotate ring to align key[0] with index 0.

4. State Transition:

   • For each character in key, we iterate through all positions in ring where this character occurs. For each such position, we compute the minimum steps to move from every possible position of the previous character in key. The rotation can happen clockwise or anticlockwise, and we take the minimum steps needed for this transition.

5. Final Answer:

   • After processing all characters in key, the answer will be the minimum number of steps to spell the last character of key from any possible position in ring.

Let's implement this solution in PHP: 514. Freedom Trail

<?php
class Solution {

    /**
     * @param String $ring
     * @param String $key
     * @return Integer
     */
    function findRotateSteps($ring, $key) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
    }
}

// Test cases
$solution = new Solution();
echo $solution->findRotateSteps("godding", "gd") . "\n"; // Output: 4
echo $solution->findRotateSteps("godding", "godding") . "\n"; // Output: 13
?>

Explanation:

1. Preprocessing:

   • We create an array $char_positions where the key is a character, and the value is an array of all indices in ring where this character occurs. This helps in efficiently finding positions of each character when needed.

2. Dynamic Programming Table Initialization:

   • For the first character of key, we initialize the DP table by calculating how many steps are required to move the first character of key to the top position (0 index of ring).

3. DP Table Filling:

   • For each character in key, we compute the minimum number of steps required to move from the previous character's positions to the current character's positions, updating the DP table with the minimum steps.

4. Result Extraction:

   • After processing all characters in key, the result is the minimum number of steps found in the last row of the DP table.

Time Complexity

• Precomputing the character positions takes O(n), where n is the length of ring.
• The DP solution iterates through each character in key and for each character, it checks all possible positions in ring, resulting in O(k * m^2) complexity, where k is the length of key and m is the number of positions of each character in ring.

Thus, the solution efficiently handles the problem's constraints and gives the correct minimum steps to spell the key.

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