Minimum Distance to Type a Word Using Two Fingers

You have a keyboard layout as shown above in the XY plane, where each English uppercase letter is located at some coordinate, for example, the letter A is located at coordinate (0,0), the letter B is located at coordinate (0,1), the letter P is located at coordinate (2,3) and the letter Z is located at coordinate (4,1).

Given the string word, return the minimum total distance to type such string using only two fingers. The distance between coordinates (x1,y1) and (x2,y2) is |x1 - x2| + |y1 - y2|. 

Note that the initial positions of your two fingers are considered free so don't count towards your total distance, also your two fingers do not have to start at the first letter or the first two letters.

 

Example 1:

Input: word = "CAKE"
Output: 3
Explanation: 
Using two fingers, one optimal way to type "CAKE" is: 
Finger 1 on letter 'C' -> cost = 0 
Finger 1 on letter 'A' -> cost = Distance from letter 'C' to letter 'A' = 2 
Finger 2 on letter 'K' -> cost = 0 
Finger 2 on letter 'E' -> cost = Distance from letter 'K' to letter 'E' = 1 
Total distance = 3
Example 2:

Input: word = "HAPPY"
Output: 6
Explanation: 
Using two fingers, one optimal way to type "HAPPY" is:
Finger 1 on letter 'H' -> cost = 0
Finger 1 on letter 'A' -> cost = Distance from letter 'H' to letter 'A' = 2
Finger 2 on letter 'P' -> cost = 0
Finger 2 on letter 'P' -> cost = Distance from letter 'P' to letter 'P' = 0
Finger 1 on letter 'Y' -> cost = Distance from letter 'A' to letter 'Y' = 4
Total distance = 6
Example 3:

Input: word = "NEW"
Output: 3
Example 4:

Input: word = "YEAR"
Output: 7
 

Constraints:


Solution:

class Solution {
    public int minimumDistance(String word) {
        int n = word.length();
        Integer[][][] memo = new Integer[27][27][n];
        return helper(word, 0, null, null, memo);
    }
    
    private int helper(String word, int pos, Character c1, Character c2, Integer[][][] memo) {
        if (pos >= word.length()) return 0;
        int idx1 = c1 == null ? 0 : c1 - 'A' + 1;
        int idx2 = c2 == null ? 0 : c2 - 'A' + 1;
        if (memo[idx1][idx2][pos] == null) {
            Character to = word.charAt(pos);
            memo[idx1][idx2][pos] = Math.min(
                cost(c1, to) + helper(word, pos + 1, to, c2, memo),
                cost(c2, to) + helper(word, pos + 1, c1, to, memo)                      
            );
        }
        return memo[idx1][idx2][pos];
    }
    
    private int cost(Character from, Character to) {
        if (from == null || to == null) return 0;
        int a = from - 'A';
        int b = to - 'A';
        return Math.abs(a / 6 - b / 6) + Math.abs(a % 6 - b % 6);
    }
}