5. Longest Palindromic Substring （DP）

Given a string S, find the longest palindromic substring in S. You may assume that the maximum length of S is 1000, and there exists one unique longest palindromic substring.

思路：如果不用动态规划，在两个for循环的情况下，还得依次比较i，j间的每个字符，O(n3)。使用动态规划，O(n2)

char* longestPalindrome(char* s) {
    int n = strlen(s);
    int max = 1;
    int pStart = 0;

    bool flag[n][n];
    for(int i = 0; i < n; i++){
        flag[i][i] = true;
        for(int j = i+1; j < n; j++){
            flag[i][j] = false;
        }
    }
    
    for(int j = 1; j < n; j++){ //when iterate j, we should already iterated j-1, 可以理解成j之前已排序好=>用插入排序的顺序遍历
        for(int i = 0; i < j; i++){ 
            if(s[i]==s[j]){
                flag[i][j] = (j==i+1)?true:flag[i+1][j-1];
              
                if(flag[i][j] && j-i+1 > max){
                    
                    max = j-i+1;
                    pStart = i;
                }
            }
        }
    }

    s[pStart+max]='';
    return &s[pStart];
}

方法II：KMP+动态规划，时间复杂度在最好情况下达到O(n)

首先在字符串的每个字符间加上#号。For example: S = “abaaba”, T = “#a#b#a#a#b#a#”。这样所有的回文数都是奇数，以便通过i的对应位置i’获得p[i]

P[i]存储以i为中心的最长回文的长度。For example: 

T = # a # b # a # a # b # a #

P = 0 1 0 3 0 1 6 1 0 3 0 1 0

下面我们说明如何计算P[i]。

假设我们已经处理了C位置（中心位置），它的最长回文数是abcbabcba，L指向它左侧位置，R指向它右侧位置。

现在我们要处理i位置。

if P[ i' ] ≤ R – i,

then P[ i ] ← P[ i' ] 那是因为在L到R范围内，i'的左侧与i的右侧相同，i'的右侧与i的左侧相同，i'左侧与右侧相同 =>i左侧与右侧相同。

else P[ i ] ≥ P[ i' ]. (Which we have to expand past the right edge (R) to find P[ i ].

If the palindrome centered at i does expand past R, we update C to i, (the center of this new palindrome), and extend R to the new palindrome’s right edge.

char* preProcess(char* s) {
    int n = strlen(s);
    if (n == 0) return "^$";
    char* ret = malloc(sizeof(char)*(n*2+4));
    char* pRet = ret;
    *pRet++ = '^'; //开始符^
    for (int i = 0; i < n; i++){
      *pRet++ = '#';
      *pRet++ = s[i];
    }
    *pRet++ = '#';
    *pRet = '$';//结束符$
    return ret;
}
 
char* longestPalindrome(char* s) {
    char* T = preProcess(s);
    int n = strlen(T);
    int P[n]; 
    int C = 0, R = 0;
    char* ret;
    for (int i = 1; i < n-1; i++) {
      int i_mirror = 2*C-i; // equals to i_mirror = C - (i-C)
    
      //if p[i_mirror] < R-i: set p[i] to p[i_mirror]
      if(R>i){
          if(P[i_mirror] <= R-i){
              P[i] = P[i_mirror];
          }
          else P[i] = R-i;
      }
      else P[i] = 0;
      
      //else: Attempt to expand palindrome centered at i
      while (T[i + 1 + P[i]] == T[i - 1 - P[i]]) //因为有哨兵^$所以不用担心越界; +1, -1检查下一个元素是否相等，若相等，扩大p[i]
        P[i]++;
      
      //if the palindrome centered at i does expand past R
      if (i + P[i] > R) {
        C = i;
        R = i + P[i];
      }
    }
 
    // Find the maximum element in P.
    int maxLen = 0;
    int centerIndex = 0;
    for (int i = 1; i < n-1; i++) {
      if (P[i] > maxLen) {
        maxLen = P[i];
        centerIndex = i;
      }
    }
    
    ret = malloc(sizeof(char)*maxLen+1);
    strncpy(ret, s+(centerIndex - 1 - maxLen)/2, maxLen);
    ret[maxLen] = '';
    return ret;
  }