5.Longest Palindromic Substring (String; DP, KMP)

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I：动态规划，遍历到i的时候要保证之前的元素都已经计算过状态，所以遍历顺序同插入排序，时间复杂度O(n2)

class Solution {
public:
    string longestPalindrome(string s) {
      int len = s.length();
      if(len==) return s;

      bool dp[len][len]={false};
      int maxLen=;
      int start=;

      //initialize
      for(int i = ; i < len; i++){
          dp[i][i]=true;
      }

      for(int i=; i<len; i++){
          for(int j = ; j<i; j++){
              if(s[i]==s[j] && (j==i- || dp[j+][i-])){
                  dp[j][i]=true;
                  if(i-j+ > maxLen){
                      maxLen=i-j+;
                      start=j;
                  }
              }
          }
      }

      return s.substr(start, maxLen);
    }
};

思路II：KMP，一种字符串匹配方法。

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。

p[i]必定>=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。具体地，

if P[ i' ] ≤ R – i,

then P[ i ] ← P[ i' ]

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.

 

时间复杂度分析：

In each step, there are two possibilities. 

• If P[ i ] ≤ R – i, we set P[ i ] to P[ i' ] which takes exactly one step.
• Otherwise we attempt to change the palindrome’s center to i by expanding it starting at the right edge, R. Extending R (the inner while loop) takes at most a total of N steps, and positioning and testing each centers take a total of N steps too. Therefore, this algorithm guarantees to finish in at most 2*N steps, giving a linear time solution.

那么总共时间复杂度最坏是O(n2)，最好是O(n)

class Solution {
public:
  string preProcess(string s) {
    int n = s.length();
    if (n == ) return "^$";
    string ret = "^"; //开始符^
    for (int i = ; i < n; i++)
      ret += "#" + s.substr(i, );

    ret += "#$"; //结束符$
    return ret;
  }

  string longestPalindrome(string s) {
    string T = preProcess(s);
    int n = T.length();
    int *P = new int[n]; //状态数组长度等于原来字符串的长度，不用给#计算状态
    int C = , R = ;
    for (int i = ; i < n-; i++) {
      int i_mirror = *C-i; // equals to i_mirror = C - (i-C)

      //if p[i_mirror] < R-i: set p[i] to p[i_mirror]
      P[i] = (R > i) ? min(R-i, P[i_mirror]) : ;

      //else: Attempt to expand palindrome centered at i
      while (T[i +  + P[i]] == T[i -  - 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 = ;
    int centerIndex = ;
    for (int i = ; i < n-; i++) {
      if (P[i] > maxLen) {
        maxLen = P[i];
        centerIndex = i;
      }
    }
    delete[] P;

    return s.substr((centerIndex -  - maxLen)/, maxLen);
  }
};