<p>There is a country of <code>n</code> cities numbered from <code>0</code> to <code>n - 1</code>. In this country, there is a road connecting <b>every pair</b> of cities.</p>

<p>There are <code>m</code> friends numbered from <code>0</code> to <code>m - 1</code> who are traveling through the country. Each one of them will take a path consisting of some cities. Each path is represented by an integer array that contains the visited cities in order. The path may contain a city <strong>more than once</strong>, but the same city will not be listed consecutively.</p>

<p>Given an integer <code>n</code> and a 2D integer array <code>paths</code> where <code>paths[i]</code> is an integer array representing the path of the <code>i<sup>th</sup></code> friend, return <em>the length of the <strong>longest common subpath</strong> that is shared by <strong>every</strong> friend&#39;s path, or </em><code>0</code><em> if there is no common subpath at all</em>.</p>

<p>A <strong>subpath</strong> of a path is a contiguous sequence of cities within that path.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 5, paths = [[0,1,<u>2,3</u>,4],
                       [<u>2,3</u>,4],
                       [4,0,1,<u>2,3</u>]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> The longest common subpath is [2,3].
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 3, paths = [[0],[1],[2]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There is no common subpath shared by the three paths.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 5, paths = [[<u>0</u>,1,2,3,4],
                       [4,3,2,1,<u>0</u>]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> The possible longest common subpaths are [0], [1], [2], [3], and [4]. All have a length of 1.</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>m == paths.length</code></li>
	<li><code>2 &lt;= m &lt;= 10<sup>5</sup></code></li>
	<li><code>sum(paths[i].length) &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= paths[i][j] &lt; n</code></li>
	<li>The same city is not listed multiple times consecutively in <code>paths[i]</code>.</li>
</ul>

class RabinKarp: 

    def __init__(self, s): 
        """Calculate rolling hash of s"""
        self.m = 1_111_111_111_111_111_111
        self.pow = [1]
        self.roll = [0] # rolling hash 

        p = 1_000_000_007
        for x in s: 
            self.pow.append(self.pow[-1] * p % self.m)
            self.roll.append((self.roll[-1] * p + x) % self.m)

    def query(self, i, j): 
        """Return rolling hash of s[i:j]"""
        return (self.roll[j] - self.roll[i] * self.pow[j-i]) % self.m


class Solution:
    def longestCommonSubpath(self, n: int, paths: List[List[int]]) -> int:
        rks = [RabinKarp(x) for x in paths]
        
        def fn(x):
            """Return True if a common subpath of length x is found."""
            seen = set()
            for rk, path in zip(rks, paths): 
                vals = {rk.query(i, i+x) for i in range(len(path)-x+1)}
                if not seen: seen = vals
                else: seen &= vals
                if not seen: return False 
            return True 
            
        lo, hi = 0, len(paths[0])
        while lo < hi: 
            mid = lo + hi + 1 >> 1
            if fn(mid): lo = mid 
            else: hi = mid - 1
        return lo
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