leetcode 1320. Minimum Distance to Type a Word Using Two Fingers (dp O(N) Solution)

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:

2 <= word.length <= 300
Each word[i] is an English uppercase letter.

設dis(a,b)爲a移動到b的花費。對於只考慮一個手指的情況，很容易得到sum。這時考慮第二個手指移動能夠節省的費用save，用sum-save即可得到最優值。
考慮now移動到to
dp[now]，表示第二個手指頭到當前位置能夠達到的最大節省。
$\sum_{0}^{26} dp[now] = max(dp[now],dp[i] + dis(now,to)-dis(i,to))$
代碼如下:

class Solution {
public:
    int getdis(const int &val1,const int &val2){
        const int b = 6;
        int x1 = val1/b;
        int y1 = val1%b;
        int x2 = val2/b;
        int y2 = val2%b;
        int dis = abs(x1-x2)+abs(y1-y2);
        return dis;
    }
    
    int minimumDistance(string word) {
        vector<int> dp(27,0);
        int len = word.size();
        int sum = 0,save = 0;
        for(int i=0;i<len-1;++i){
            int now = word[i]-'A';
            int to = word[i+1]-'A';
            sum += getdis(now,to);
            for(int j=0;j<26;++j){
                dp[now] = max(dp[now],dp[j]+(getdis(now,to)-getdis(j,to)));
            }
            save = max(save,dp[now]);
        }
        return sum-save;
    }
};