<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>

from collections import defaultdict, Counter

class Solution:
    def longestCommonSubpath(self, n: int, paths: List[List[int]]) -> int:
        if not paths or not paths[0]:
            return 0
        
        MOD = (1 << 128) - 159
        BASE = 1 << 17

        def rollingHash(path_idx: int, length: int, cntrs: Dict[int, "Counter"]):
            if length > len(paths[path_idx]):
                return
            
            value = 0
            for i in range(length):
                value = (value * BASE + paths[path_idx][i]) % MOD
            cntrs[value][path_idx] += 1

            # base_to_length = (1 << (17 * length)) % MOD
            base_to_length_1 = (1 << (17 * (length - 1))) % MOD
            for i in range(length, len(paths[path_idx])):
                # curr = value * BASE + paths[path_idx][i]
                # value = (curr - prev) % MOD
                prev = (value - base_to_length_1 * paths[path_idx][i - length]) * BASE
                curr = paths[path_idx][i]
                value = (prev + curr) % MOD
                cntrs[value][path_idx] += 1


        def checkPath(length: int) -> bool:
            cntrs = defaultdict(Counter)
            rollingHash(0, length, cntrs)
            for i in range(1, len(paths)):
                rollingHash(i, length, cntrs)
            
            return any(len(cntr) == len(paths) for cntr in cntrs.values())
        
        lo, hi = 0, max(len(path) for path in paths)
        while lo < hi:
            mid = lo + (hi - lo + 1) // 2
            if checkPath(mid):
                lo = mid
            else:
                hi = mid - 1
        
        return lo
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