TheLongestPalindromicSubstring.md

How to find The Longest Palindromic Substring

Author: labuladong

Translator: Lrc123

Palindrome questions are very common in the interview, this article provides some insights about palindromic problem.

To specific :

A palindrome is a word, number, phrase, or other sequence of characters which reads the same backward as forward, such as madam, racecar. reference

For example: aba and abba are both palindromic, because they are symetric strings, that you can read each of them in reversed order, and you can just get a same string.

Notice: palindrome string could be in either odd length or even length, a good solution would be double pointers. Next, I'll show you how doulbe pointers work in a real leetcode problem.

string longestPalindrome(string s) {}

1. Thinking

Given a string s, find the longest palindromic substring in s.

A very interesting pespective: 1. Reversing s in to s' 2. Finding the longest common substring.

For instance, a string abacd, a reversed version is dcaba, and the longest common string is aba, seemingly perfect.

However, it would be wrong when we apply to aacxycaa, which a reversed version would be aacyxcaa, then the longest common substring turns out to be aac. But, what we need should be aa.

Although this way has its faults, we can still get some inspirations that we can transform a problem seemingly hard into another simpler problem that we can understand easier.

Now, the double pointers

Core idea: start a scanner from the mid point of the string we represent the idea into pseudo code:

for 0 <= i < len(s):
    find a palindrome that set s[i] as its mid point
    update the answer

When the length of string is even, for instance: abba, the code above would not work.

So, a better version here :

for 0 <= i < len(s):
    find a palindrome that set s[i] as its mid point
    find a palindrome that set s[i] and s[i + 1] as its mid point
    update the answer

PS: you may encounter some problems like : outofIndex error. Don't worry, we'll fix them later.

2. Implementation

a function implementation:

string palindrome(string& s, int l, int r) {
    // avoid outOfIndex error
    while (l >= 0 && r < s.size()
            && s[l] == s[r]) {
        // scanning toward both directions
        l--; r++;
    }
    // return a palindrome that set s[l] and s[r] as mid point
    return s.substr(l + 1, r - l - 1);
}

Why we need both pointer l and pointer r? In this way, we can handle palindrome strings in odd and even length

for 0 <= i < len(s):
    # find a palindrome that set s[i] as its mid 
    palindrome(s, i, i)
    # find a palindrome that set s[i] and s[i + 1] as its mid  
    palindrome(s, i, i + 1)
    update the answer

Completed code solution:

string longestPalindrome(string s) {
    string res;
    for (int i = 0; i < s.size(); i++) {
        // find a palindrome that set s[i] as its mid 
        string s1 = palindrome(s, i, i);
        // find a palindrome that set s[i] and s[i + 1] as its mid  
        string s2 = palindrome(s, i, i + 1);
        // res = longest(res, s1, s2)
        res = res.size() > s1.size() ? res : s1;
        res = res.size() > s2.size() ? res : s2;
    }
    return res;
}

Thus, this leetcode problem is solved. Now, we get:

Time complexity: O(N^2)

Space complexity: O(1)

By the way, a dynamic programming approach can also work in this problem in a same time complexity. However, we need at least O(N^2) spaces to store DP table. Therefore, in this problem, dp approach is not the best solution.

In addition, Manacher's Algorithm requires only O(N) time complexity. You readers can search it through the Internet by your own interests. It should be very interesting.

Stick to original high-quality articles, and strive to make clear the algorithm problems. Welcome to follow my Wechat official account "labuladong" for the latest articles.