<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:
        paths = [[x + 1 for x in path] for path in paths]

        def check(x):
            if x > len(paths[mi]):
                return False
            mod = (1 << 128) - 159
            mul = 1 << 17
            prv_mul = (1 << (17 * (x - 1))) % mod
            h = set()

            cur = 0
            for j in range(x):
                cur = (mul * cur + paths[mi][j]) % mod
            h.add(cur)
            for j in range(x, len(paths[mi])):
                cur = (mul * (cur - paths[mi][j - x] * prv_mul) + paths[mi][j]) % mod
                h.add(cur)

            for i in range(m):
                if i == mi: continue
                nh = set()
                cur = 0
                for j in range(x):
                    cur = (mul * cur + paths[i][j]) % mod
                if cur in h:
                    nh.add(cur)
                for j in range(x, len(paths[i])):
                    cur = (mul * (cur - paths[i][j - x] * prv_mul) + paths[i][j]) % mod
                    if cur in h:
                        nh.add(cur)
                h = nh
                if not h:
                    break
                        
            return bool(h)

        mi = 0
        m = len(paths)
        for i in range(1, m):
            if len(paths[i]) < len(paths[mi]):
                mi = i
        l, r = 0, len(paths[mi])
        while l < r:
            mid = (l + r) // 2 + 1
            if check(mid):
                l = mid
            else:
                r = mid - 1
        return l
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