动态规划最常见的习题 （最长公共子串、最长公共子序列、最短编辑距离）

（1）理论部分：

 

 

（2）习题：

　　最长公共子串：

package month7.dp;

//https://www.nowcoder.com/questionTerminal/181a1a71c7574266ad07f9739f791506
public class 最长公共子串 {
    public static void main(String[] args) {
        最长公共子串 s = new 最长公共子串();
        String str1 = "ABABCGGG";
        String str2 = "BABCAH";
        System.out.println(s.Longest_Common_Substring(str1, str2));
    }
//将二维数组table[i][j]用来记录具有这样特点的子串——结尾同时也为为串x1x2⋯xi与y1y2⋯yj的结尾——的长度
    public int Longest_Common_Substring(String str1, String str2) {
        int len1 = str1.length();
        int len2 = str2.length();
        int result = 0;
        int num = 0;
        int[][] table = new int[len1 + 1][len2 + 1];
        
        for (int i = 0; i <= len1; i++) {
            table[i][0] = 0;
        }
        for (int j = 0; j <= len2; j++) {
            table[0][j] = 0;
        }
        for (int i = 1; i <= len1; i++) {
            for (int j = 1; j <= len2; j++) {
                char c1 = str1.charAt(i-1);
                char c2 = str2.charAt(j-1);
                if (c1 != c2) {
                    table[i][j] = 0;
                } else { // c1 == c2
                    table[i][j] = table[i - 1][j - 1] + 1;
                    if (result < table[i][j]) {
                        result = table[i][j];
                        num = i;
                        System.out.println("-----" + num + "--");

                    }
                }
            }
        }
        System.out.println(str1.substring(num - result , num ));
        return result;
    }

    public int Longest_Common_Substring2(String str1, String str2) {
        int len1 = str1.length();
        int len2 = str2.length();
        int result = 0;
        int num = 0;
        int[][] table = new int[len1 + 1][len2 + 1];
        for (int i = 0; i <= len1; i++) {
            for (int j = 0; j <= len2; j++) {
                if (i == 0 || j == 0) {
                    table[i][j] = 0;
                } else if (str1.charAt(i - 1) != str2.charAt(j - 1)) {
                    table[i][j] = 0;
                } else { // c1 == c2
                    table[i][j] = table[i - 1][j - 1] + 1;
                    if (result < table[i][j]) {
                        result = table[i][j];
                        num = i;
                        System.out.println("-----" + num + "--");

                    }
                }
            }
        }
        System.out.println(str1.substring(num - result , num ));
        return result;
    }

}

　　最长公共子序列：

package month7.dp;

public class 最长公共子序列 {
    int[][] b = new int[10][10];
    public static void main(String[] args) {
        最长公共子序列 s = new 最长公共子序列();
//        String str1 ="ABCD";
//        String str2 = "EACB";
        String str1 ="AGGTHAB";
        String str2 = "GXTXAYB";
        System.out.println(s.Longest_Common_Subsequence(str1, str2));
        s.print_lsc(str1.length(), str2.length(), str1);
        
    }
    //用二维数组c[i][j]记录串x1x2⋯xi与y1y2⋯yj的LCS长度
    public int Longest_Common_Subsequence(String str1,String str2){
        int len1 = str1.length();
        int len2 = str2.length();
        int[][] table = new int[str1.length()+1][str2.length()+1];
//        int[][] b = new int[str1.length()+1][str2.length()+1];
        for(int i=0;i<=len1;i++){
            for(int j=0;j<=len2;j++){
                if(i==0 || j == 0){
                    table[i][j] = 0;
                }else if(str1.charAt(i-1) != str2.charAt(j-1)){
                    if(table[i-1][j] > table[i][j-1]){
                        table[i][j] = table[i-1][j];
                        b[i][j] = 1;
                    }else{
                        table[i][j] = table[i][j-1];
                        b[i][j] = 2;
                    }
//                    table[i][j] = Math.max(table[i-1][j], table[i][j-1]);
                }else{ // Xi = Yj
                    table[i][j] = table[i-1][j-1] +1;
                    b[i][j] = 3;
                }
            }
        }
        int commomLength =  table[len1][len2];
        return commomLength;
    }
    public void print_lsc(int i,int j,String str1){
        if(i ==0 || j == 0)
            return ;
        if(b[i][j] == 3){
            print_lsc(i-1, j-1, str1);
            System.out.print(str1.charAt(i-1));
        }else if(b[i][j] == 2){
            print_lsc(i, j-1, str1);
        }else if(b[i][j] == 1){
            print_lsc(i-1, j, str1);
        }
    }
}

　　最短编辑距离

package month7.dp;

//https://www.cnblogs.com/masterlibin/p/5785092.html
public class 最短编辑距离 {
    public static void main(String[] args) {
        最短编辑距离 s = new 最短编辑距离();

    }

    public int editionDistance(String word1, String word2) {
        if (word1 == null || "".equals(word1)) {
            return word2.length();
        }
        if (word2 == null || "".equals(word1)) {
            return word1.length();
        }
        int len1 = word1.length();
        int len2 = word2.length();
        int[][] table = new int[len1 + 1][len2 + 1];
        for (int i = 0; i <= len1; i++) {
            table[i][0] = i;
        }
        for (int j = 0; j <= len2; j++) {
            table[0][j] = j;
        }
        for (int i = 1; i <= len1; i++) {
            for (int j = 1; j <= len2; j++) {
                char c1 = word1.charAt(i - 1);
                char c2 = word2.charAt(j - 1);
                if (c1 == c2) {
                    table[i][j] = table[i - 1][j - 1];
                } else {
                    int add = table[i][j - 1] + 1;
                    int delate = table[i - 1][j] + 1;
                    int edition = table[i - 1][j - 1] + 1;
                    table[i][j] = Math.min(add, Math.min(delate, edition));
                }
            }
        }
        return table[len1][len2];
    }

}