<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 __init__(self):
    self.kBase = 165_131
    self.kHash = 8_417_508_174_513

  def longestCommonSubpath(self, n: int, paths: list[list[int]]) -> int:
    l = 0
    r = len(paths[0])

    while l < r:
      m = l + (r - l + 1) // 2
      if self._checkCommonSubpath(paths, m):
        l = m
      else:
        r = m - 1

    return l

  def _checkCommonSubpath(self, paths: list[list[int]], m: int) -> bool:
    """
    Returns True if there's a common subpath of length m for all the paths.
    """
    # Calculate the hash values for subpaths of length m for every path.
    hashSets = [self._rabinKarp(path, m) for path in paths]

    # Check if there is a common subpath of length m.
    for subpathHash in hashSets[0]:
      if all(subpathHash in hashSet for hashSet in hashSets):
        return True

    return False

  def _rabinKarp(self, path: list[int], m: int) -> set[int]:
    """Returns the hash values for subpaths of length m in the path."""
    hashes = set()
    maxPower = 1
    hash = 0

    for i, num in enumerate(path):
      hash = (hash * self.kBase + num) % self.kHash
      if i >= m:
        hash = (hash - path[i - m] * maxPower %
                self.kHash + self.kHash) % self.kHash
      else:
        maxPower = maxPower * self.kBase % self.kHash
      if i >= m - 1:
        hashes.add(hash)

    return hashes
        
3100