# 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 :^)

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'def longest_palindrome(s):\n    n = len(s)\n    if n <= 1:\n        return s\n    max, index = 0,  -1\n    max2, index2 = 0, -1\n    d1 = [0]*n\n    i, l, r = 0, 0, -1\n    while i < n:\n        if i > r:\n            k = 1\n        else:\n            k = min(d1[l + r - i], r - i + 1)\n        while 0 <= i - k and i + k < n and s[i - k] == s[i + k]:\n            k += 1\n        if k > max:\n            max, index = k, i  # s[i-k+1:i+k], 2k-1\n        d1[i] = k\n        k -= 1\n        if i + k > r:\n            l, r = i - k, i + k\n        i += 1\n\n    d2 = [0]*n\n    i, l, r = 0, 0, -1\n    while i < n:\n        if i > r:\n            k = 0\n        else:\n            k = min(d2[l + r - i + 1], r - i + 1)\n        while 0 <= i - k - 1 and i + k < n and s[i - k - 1] == s[i + k]:\n            k += 1\n        if k > max2:\n            max2, index2 = k, i  # s[i-k:i+k], 2k\n        d2[i] = k\n        k -= 1\n        if i + k > r:\n            l, r = i - k - 1, i + k\n        i += 1\n\n    index = index - max + 1  # start\n    max = 2 * max - 1        # len\n    index2 -= max2           # start\n    max2 *= 2                # len\n    start, ln = 0, 0\n    if max == max2:\n        if index <= index2:\n            start, ln = index, max\n        else:\n            start, ln = index2, max2\n    elif max > max2:\n        start, ln = index, max\n    else:\n        start, ln = index2, max2\n    return s[start: start + ln]\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'longest_palindrome'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 6490