<p>You are given an array <code>routes</code> representing bus routes where <code>routes[i]</code> is a bus route that the <code>i<sup>th</sup></code> bus repeats forever.</p>

<ul>
	<li>For example, if <code>routes[0] = [1, 5, 7]</code>, this means that the <code>0<sup>th</sup></code> bus travels in the sequence <code>1 -&gt; 5 -&gt; 7 -&gt; 1 -&gt; 5 -&gt; 7 -&gt; 1 -&gt; ...</code> forever.</li>
</ul>

<p>You will start at the bus stop <code>source</code> (You are not on any bus initially), and you want to go to the bus stop <code>target</code>. You can travel between bus stops by buses only.</p>

<p>Return <em>the least number of buses you must take to travel from </em><code>source</code><em> to </em><code>target</code>. Return <code>-1</code> if it is not possible.</p>

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

<pre>
<strong>Input:</strong> routes = [[1,2,7],[3,6,7]], source = 1, target = 6
<strong>Output:</strong> 2
<strong>Explanation:</strong> The best strategy is take the first bus to the bus stop 7, then take the second bus to the bus stop 6.
</pre>

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

<pre>
<strong>Input:</strong> routes = [[7,12],[4,5,15],[6],[15,19],[9,12,13]], source = 15, target = 12
<strong>Output:</strong> -1
</pre>

<p>&nbsp;</p>

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

<ul>
	<li><code>1 &lt;= routes.length &lt;= 500</code>.</li>
	<li><code>1 &lt;= routes[i].length &lt;= 10<sup>5</sup></code></li>
	<li>All the values of <code>routes[i]</code> are <strong>unique</strong>.</li>
	<li><code>sum(routes[i].length) &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= routes[i][j] &lt; 10<sup>6</sup></code></li>
	<li><code>0 &lt;= source, target &lt; 10<sup>6</sup></code></li>
</ul>

from collections import defaultdict

class Solution:
    def numBusesToDestination(self, routes: List[List[int]], source: int, target: int) -> int:
        # bus route -> node
        # bus routes with same stop -> strongly connected
        
        # param gives us bus -> stop
        # we maintain stop -> bus; bus routes at same stop mean they are connected

        # start at any bus that stops at `source`
        # end at any bus that stops at `target`

        if source == target:
            return 0

        n = len(routes)
        stop_to_routes = defaultdict(list)
        for i in range(n):
            for stop in routes[i]:
                stop_to_routes[stop].append(i)

        graph = [set() for _ in range(n)] # route -> set of routes
        for routes in stop_to_routes.values():
            for i in routes:
                for j in routes:
                    if i == j:
                        continue
                    graph[i].add(j)

        start_candidates = set(stop_to_routes[source])
        end_candidates = set(stop_to_routes[target])
        min_num_buses = None
        for start in start_candidates:
            queue = deque([(start, 1)])
            visited = [False] * n
            while queue:
                route, num_buses = queue.popleft()
                if route in end_candidates:
                    min_num_buses = min(min_num_buses, num_buses) if min_num_buses is not None else num_buses
                    break
                for neighbor in graph[route]:
                    if visited[neighbor]:
                        continue
                    visited[neighbor] = True
                    queue.append((neighbor, num_buses+1))
        return min_num_buses if min_num_buses is not None else -1
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