# Longest Palindromic Substring (Linear)

A palindrome is a word, phrase, or sequence that reads the same backward as forward, e.g.,
'madam' or 'racecar'. Even the letter 'x' is considered a palindrome. 

For this Kata, you are given a string ```s```. Write a function that returns the longest _contiguous_ palindromic substring in ```s``` (it could be the entire string). In the event that there are multiple longest palindromic substrings, return the first to occur.

I'm not trying to trick you here:
- You can assume that all inputs are valid strings.
- Only the letters a-z will be used, all lowercase (your solution should, in theory, extend to more than just the letters a-z though). 

**NOTE:** Quadratic asymptotic complexity _(O(N^2))_ or above will __NOT__ work here.

-----

## Examples

### Basic Tests

```
Input: "babad"
Output: "bab"
(Note: "bab" occurs before "aba")
```
```
Input: "abababa"
Output: "abababa"
```
```
Input: "cbbd"
Output: "bb"
```

### Edge Cases

```
Input: "ab"
Output: "a"
```
```
Input: ""
Output: ""
```
-----

## Testing

Along with the example tests given:

 - There are **500** tests using strings of length in range [1 - 1,000]
 - There are **50** tests using strings of length in range [1,000 - 10,000]
 - There are **5** tests using strings of length in range [10,000 - 100,000]
 
All test cases can be passed within 10 seconds, but non-linear solutions will time out every time. _Linear performance is essential_. 

## Good Luck!

-----

This problem was inspired by [this](https://leetcode.com/problems/longest-palindromic-substring/) challenge on LeetCode. Except this is the performance version :^)

Difficulty: MEDIUM
Starter code: def longest_palindrome(s):
	

Solutions:
def longest_palindrome(s, sep=' '):
	t = sep + sep.join(s) + sep
	r = 0
	c = 0
	spans = []
	for (i, _) in enumerate(t):
		span = min(spans[2 * c - i], r - i - 1) if i < r else 0
		while span <= i < len(t) - span and t[i - span] == t[i + span]:
			span += 1
		(r, c) = max((r, c), (i + span, i))
		spans.append(span)
	span = max(spans)
	middle = spans.index(span)
	return t[middle - span + 1:middle + span].replace(sep, '')


def longest_palindrome(s):
	if s == '':
		t = '^#$'
	else:
		t = '^#' + '#'.join(s) + '#$'
	c = 0
	r = 0
	p = [0] * len(t)
	for i in range(1, len(t) - 1):
		mirror = 2 * c - i
		p[i] = max(0, min(r - i, p[mirror]))
		while t[i + 1 + p[i]] == t[i - 1 - p[i]]:
			p[i] += 1
		if i + p[i] > r:
			c = i
			r = i + p[i]
	(k, i) = (0, 0)
	for (j, e) in enumerate(p):
		if e > k:
			(k, i) = (e, j)
	(deb, fin) = ((i - k) // 2, (i + k) // 2)
	return s[deb:fin]


def longest_palindrome(s):
	if is_palindrome(s):
		return s
	max_pal = ''
	for i in range(len(s)):
		temp = check(i, i, s)
		if len(max_pal) < len(temp):
			max_pal = temp
	return max_pal

def check(li, ri, s):
	if li > 0 and ri < len(s):
		if is_palindrome(s[li - 1:ri + 2]):
			return check(li - 1, ri + 1, s)
		elif is_palindrome(s[li:ri + 2]):
			return check(li, ri + 1, s)
		elif is_palindrome(s[li - 1:ri + 1]):
			return check(li - 1, ri, s)
	return s[li:ri + 1]

def is_palindrome(s):
	return s == s[::-1]


import re

def longest_palindrome(string):
	if len(string) < 2:
		return string
	string = '|' + '|'.join(string) + '|'
	LPS = [0 for _ in range(len(string))]
	C = 0
	R = 0
	for i in range(len(string)):
		iMirror = 2 * C - i
		if R > i:
			LPS[i] = min(R - i, LPS[iMirror])
		else:
			LPS[i] = 0
		try:
			while string[i + 1 + LPS[i]] == string[i - 1 - LPS[i]]:
				LPS[i] += 1
		except:
			pass
		if i + LPS[i] > R:
			C = i
			R = i + LPS[i]
	(r, c) = (max(LPS), LPS.index(max(LPS)))
	return string[c - r:c + r].replace('|', '')


def modi(s):
	return ''.join(['#' + i for i in s]) + '#'

def demodi(s):
	return ''.join([i for i in s if i != '#'])

def longest_palindrome(s):
	s = modi(s)
	(lenth, c, r, maxl, p) = (len(s), 0, 0, 0, [0] * len(s))
	for i in range(lenth):
		mir = 2 * c - i
		if i < r:
			p[i] = min(r - i, p[mir])
		(a, b) = (i + (1 + p[i]), i - (1 + p[i]))
		while a < lenth and b >= 0 and (s[a] == s[b]):
			p[i] += 1
			a += 1
			b -= 1
		if i + p[i] > r:
			c = i
			r = i + p[i]
			if p[i] > maxl:
				maxl = p[i]
	mai = p.index(maxl)
	return demodi(s[mai - maxl:mai + maxl + 1])


def get_func(s, i):
	(p2, p3) = ('', '')
	(s2, s3) = (s[i:i + 2], s[i:i + 3])
	if s2[0] == s2[-1]:
		p2 = s2
		(u, v) = (i - 1, i + 2)
		while u > -1 and v < len(s):
			if s[u] != s[v]:
				break
			p2 = s[u] + p2 + s[v]
			(u, v) = (u - 1, v + 1)
	if s3[0] == s3[-1]:
		p3 = s3
		(u, v) = (i - 1, i + 3)
		while u > -1 and v < len(s):
			if s[u] != s[v]:
				break
			p3 = s[u] + p3 + s[v]
			(u, v) = (u - 1, v + 1)
	return p3 if len(p3) > len(p2) else p2

def set_pal(p, idxp, pal, idx):
	if len(p) > len(pal) or (len(p) == len(pal) and idxp < idx):
		(pal, idx) = (p, idxp)
	return (pal, idx)

def longest_palindrome(s):
	if len(s) < 2:
		return s
	pal = s[0]
	idx = 1
	j = len(s) // 2 - 1
	k = j + 1
	if len(s) > 3:
		for i in range(j):
			(pal, idx) = set_pal(get_func(s, j - i - 1), j - i, pal, idx)
			(pal, idx) = set_pal(get_func(s, k + i - 1), k + i, pal, idx)
			if len(s) < len(pal) + 2 * i:
				break
	if len(pal) < 3 and len(s) % 2 != 0:
		(pal, idx) = set_pal(get_func(s, len(s) - 3), len(s) - 2, pal, idx)
	if len(pal) < 2:
		sub = s[len(s) - 2:len(s)]
		if sub[0] == sub[-1]:
			pal = sub
	return pal


def longest_palindrome(s):
	s = '_' + '_'.join(s) + '_'
	n = len(s)
	a = [0] * n
	rb = ml = mctr = ctr = c = 0
	for i in range(1, n):
		a[i] = min(rb - i, a[2 * ctr - i]) if rb > i else 1
		c = a[i]
		while i + c < n and i >= c and (s[i - c] == s[i + c]):
			c += 1
		a[i] = c
		if i + c > rb:
			rb = i + c
			ctr = i
		if c > ml:
			ml = c
			mctr = i
	return s[mctr - ml + 1:mctr + ml].replace('_', '')


def longest_palindrome(S):
	T = [0] * (2 * len(S) + 3)
	sen_char = '@'
	start_sen = '!'
	end_sen = '#'
	for i in range(len(T)):
		if i == 0:
			T[i] = start_sen
		elif i % 2 == 0 and i < len(T) - 1:
			s_index = (i - 1) // 2
			T[i] = S[s_index]
		elif i % 2 == 1 and i < len(T) - 1:
			T[i] = sen_char
		else:
			T[i] = end_sen
	P = [0] * len(T)
	center = right = 0
	max_len = index = 0
	for i in range(1, len(T) - 1):
		mirror = 2 * center - i
		if i < right:
			P[i] = min(right - i, P[mirror])
		while T[i + P[i] + 1] == T[i - (P[i] + 1)]:
			P[i] += 1
		if i + P[i] > right:
			right = i + P[i]
			center = i
		if P[i] > max_len:
			max_len = P[i]
			index = i
	t_arr = T[index - max_len:index + max_len + 1]
	word_arr = [c for c in t_arr if c != sen_char and c != start_sen and (c != end_sen)]
	word = ''.join(word_arr)
	return word


def longest_palindrome(s):
	n = len(s)
	if n <= 1:
		return s
	(max, index) = (0, -1)
	(max2, index2) = (0, -1)
	d1 = [0] * n
	(i, l, r) = (0, 0, -1)
	while i < n:
		if i > r:
			k = 1
		else:
			k = min(d1[l + r - i], r - i + 1)
		while 0 <= i - k and i + k < n and (s[i - k] == s[i + k]):
			k += 1
		if k > max:
			(max, index) = (k, i)
		d1[i] = k
		k -= 1
		if i + k > r:
			(l, r) = (i - k, i + k)
		i += 1
	d2 = [0] * n
	(i, l, r) = (0, 0, -1)
	while i < n:
		if i > r:
			k = 0
		else:
			k = min(d2[l + r - i + 1], r - i + 1)
		while 0 <= i - k - 1 and i + k < n and (s[i - k - 1] == s[i + k]):
			k += 1
		if k > max2:
			(max2, index2) = (k, i)
		d2[i] = k
		k -= 1
		if i + k > r:
			(l, r) = (i - k - 1, i + k)
		i += 1
	index = index - max + 1
	max = 2 * max - 1
	index2 -= max2
	max2 *= 2
	(start, ln) = (0, 0)
	if max == max2:
		if index <= index2:
			(start, ln) = (index, max)
		else:
			(start, ln) = (index2, max2)
	elif max > max2:
		(start, ln) = (index, max)
	else:
		(start, ln) = (index2, max2)
	return s[start:start + ln]


def longest_palindrome(s):
	tst = '#'.join('^{}$'.format(s))
	n = len(tst)
	p = [0] * n
	c = 0
	r = 0
	for i in range(1, n - 1):
		p[i] = r > i and min(r - i, p[2 * c - i])
		while tst[i + 1 + p[i]] == tst[i - 1 - p[i]]:
			p[i] += 1
		if i + p[i] > r:
			(c, r) = (i, i + p[i])
	(mx, ctri) = max(((n, -i) for (i, n) in enumerate(p)))
	return s[(-ctri - mx) // 2:(-ctri + mx) // 2]