# The Longest Palindromic Sub-string

## Trying to solve a problem on LeetCode

I saw this problem on LeetCode couple of years ago: Longest Palindromic Substring

I used to browse problems on LeetCode every time before I was seeking for a new job. But, you know, it's pretty relaxing. I mean, if I found that i couldn't solve a problem easily, I just gave up. No pressure.

Take this *Longest Palindromic Substring* for example, I met it a few times before, I knew there is an algorithm called *Manacher* can solve this. But I've never worked it out even though I knew about the algorithm. Let me try again today.

The first idea came into my head was, find all the palindromic sub-strings and check their lengths. It sounds like crazy but it's actually a viable plan. To make it work, we need to check each character and find out the palindromic sub-string that centered with this character. We can store the "radius" of the palindromic sub-string for each character in the original string, then find out the longest one. So the radius obvious can be 1 or more.

For example, for the character **d** in string

```
a b c d c b x
```

the radius is 3 (d, c, b). For character **b**, its radius is 1 (only including **b** itself).

The problem is, for a super long string, calculating the radius for each palindromic string is not efficient. We have to find a way to avoid always accumulating the radius from 1.

The Manacher's algorithm is based on this idea.

### Diagram 1

- We are going to find out the longest palindromic sub-string in a string
`s`

- We go through each character from left to right, and put the radius of the palindromic sub-string (which centered with the current character) into an array
`P`

. - So far the radius (on the right) of the palindromic sub-string with the centre character
`s[idx]`

can reach the position`maxRight`

, which is the most right position among all the palindromic sub-strings. - At the moment we are at the character
`s[i]`

. We are going to find the radius of**Pi**and save it in`P[i]`

.

Before we start, we need to handle this issue first: if a palindromic string has even number of characters, the centre of it is a space between two characters

If a string has odd number of character(s), the centre is a character:

So the first step of this Manacher algorithm is, eliminating this ambiguity by inserting an arbitrary character (that doesn't appear in the string), say "**#**", between every two characters of the string:

### Diagram 2

- When i < maxRight, the whole Pi or part of Pi has a "mirror" on the other side of
`idx`

. This "mirror" is**Pj**. Its centre character is s[j].- If
`P[j]`

is relatively small - the whole**Pj**is included in**Pidx**(See Diagram 1): We find the initial value of`P[i]`

. That is,`P[i] = P[j]`

. Then with`i`

as the centre, we expand**Pi**until it's not palindromic any more or reach the end of`s`

. - If
`P[j]`

is relatively large - part of**Pj**is outside**Pidx**(See Diagram 2): The initial value of`P[i]`

is`maxRight - i`

. Then with`i`

as the centre, we expand**Pi**until it's not palindromic any more or reach the end of`s`

.

- If
- When i > maxRight, it means there is no history record can be used to calculate
**Pi**, so`P[i]`

will start from 1.

So the critical part of this algorithm: what is the initial value of `P[i]`

? The answer is:

```
P[i] = min(P[j], maxRight - i);
```

And here's how to get `P[j]`

: From Diagram 1 we can easily get that `idx - j = i - idx`

, so `j = 2 * idx -i`

.

Now we know the position of the longest palindromic sub-string centre and its radius, so we can get its start position and then work out what this sub-string is. The following code is what I submitted to Leetcode, I'm happy with the 12ms runtime and 10 MB memory used.

```
class Solution {
public:
string longestPalindrome(string s) {
string retStr = s;
int newLen = s.length() * 2 + 1;
// First step: Insert "#" to make the string size an odd number
for (int i = 0; i < newLen; i += 2)
{
retStr = retStr.insert(i, "#");
}
retStr.insert(0, "$");
// Process retStr
vector<int> p(retStr.size(), 0);
int mx = 0, id = 0, resLen = 0, resCenter = 0;
for (int i = 1; i < retStr.size(); ++i) {
p[i] = mx > i ? min(p[2 * id - i], mx - i) : 1;
while (retStr[i + p[i]] == retStr[i - p[i]]) ++p[i];
if (mx < i + p[i]) {
mx = i + p[i];
id = i;
}
if (resLen < p[i]) {
resLen = p[i];
resCenter = i;
}
}
return s.substr((resCenter - resLen) / 2, resLen - 1);
}
};
```