<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 Solution:
    def longestCommonSubpath(self, n: int, paths: List[List[int]]) -> int:

        def isDuplicate(path, k, p, idd, l):
            
            # double hashing w/ rolling hash
            base1, base2 = 256, 257
            mod1, mod2 = (1<<63)-1,  (1<<61)-1
            h1, h2 = 0, 0
            pk1, pk2 = pow(base1, k, mod1), pow(base2, k, mod2)

            for i in range(k):
                h1 = (h1 * base1 + path[i]) % mod1
                h2 = (h2 * base2 + path[i]) % mod2

            comb_hash = (h1, h2)
            p[comb_hash].add(idd)

            if len(p[comb_hash]) == l:
                return 1

            for i in range(k, len(path)):
                h1 = (h1 * base1 - path[i-k] * pk1 + path[i]) % mod1
                h2 = (h2 * base2 - path[i-k] * pk2 + path[i]) % mod2

                comb_hash = (h1, h2)
                p[comb_hash].add(idd)

                if len(p[comb_hash]) == l:
                    return 1

            return 0

        def rabin_karp(k):
            
            p, l = defaultdict(set), len(paths)

            for i in range(l):
                val = isDuplicate(paths[i], k, p, i, l)

                if val == 1:
                    return True

            return False

        l, r = 1, math.inf
        maxx = 0

        for path in paths:
            r = min(r, len(path))

        while l <= r:
            m = l+(r-l)//2

            if rabin_karp(m):
                maxx = m
                l = m+1
            else:
                r = m-1

        return maxx
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