Longest palindromic substring

思路： 这道题比较难，有一个解释很好：

Brute force solution, O(N³):
The obvious brute force solution is to pick all possible starting and ending positions for a substring, and verify if it is a palindrome. There are a total of C(N, 2) such substrings (excluding the trivial solution where a character itself is a palindrome).

Since verifying each substring takes O(N) time, the run time complexity is O(N³).

Dynamic programming solution, O(N²) time and O(N²) space:
To improve over the brute force solution from a DP approach, first think how we can avoid unnecessary re-computation in validating palindromes. Consider the case “ababa”. If we already knew that “bab” is a palindrome, it is obvious that “ababa” must be a palindrome since the two left and right end letters are the same.

Stated more formally below:

Define P[ i, j ] ← true iff the substring S_i … S_j is a palindrome, otherwise false.

Therefore,

P[ i, j ] ← ( P[ i+1, j-1 ] and S_i = S_j )

The base cases are:

P[ i, i ] ← true
P[ i, i+1 ] ← ( S_i = S_i+1 )

 

 

 

代码：

class Solution {
public:
    string longestPalindrome(string s) {
        // Start typing your C/C++ solution below
        // DO NOT write int main() function
        string result;
        
        int n = s.length();
        int longestBegin = 0;
        int maxLen = 1;
        vector<vector<bool> > table(1000, vector<bool>(1000, false));
        
        for (int i = 0; i < n; i++) 
            table[i][i] = true;
        
        for (int i = 0; i < n-1; i++){
            
            if (s[i] == s[i+1]) {
                
                table[i][i+1] = true;
                longestBegin = i;
                maxLen = 2;
            }
        }
        
        for (int len = 3; len <= n; len++)
            for (int i = 0; i < n-len+1; i++) {
                
                int j = i+len-1;
                if (s[i] == s[j] && table[i+1][j-1]) {
                    table[i][j] = true;
                    longestBegin = i;
                    maxLen = len;
                }
            }
        return s.substr(longestBegin, maxLen);
    }
};

 

A simpler approach, O(N2) time and O(1) space:
In fact, we could solve it in O(N2) time without any extra space.

We observe that a palindrome mirrors around its center. Therefore, a palindrome can be expanded from its center, and there are only 2N-1 such centers.

You might be asking why there are 2N-1 but not N centers? The reason is the center of a palindrome can be in between two letters. Such palindromes have even number of letters (such as “abba”) and its center are between the two ‘b’s.

Since expanding a palindrome around its center could take O(N) time, the overall complexity is O(N2).

 

class Solution {
public:
    
    string expandAroundCenter(string s, int c1, int c2) {
        
        int l = c1, r = c2;
        int n = s.length();
        
        while (l >= 0 && r <= n-1 && s[l] == s[r]) {
            
            l--;
            r++;
        }
        return s.substr(l+1, r-l-1);
    }
    
    
    string longestPalindrome(string s) {
        
        int n = s.length();
        
        if (n == 0) return "";
        string longest = s.substr(0, 1);  // a single char itself is a palindrome
        
        for (int i = 0; i < n-1; i++) {
            
            string p1 = expandAroundCenter(s, i, i);
            if (p1.length() > longest.length())
                longest = p1;
            string p2 = expandAroundCenter(s, i, i+1);
            if (p2.length() > longest.length())
                longest = p2;
        }
        
        return longest;
    }
};