<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>

class Solution:
    def numBusesToDestination(self, routes: List[List[int]], source: int, target: int) -> int:
        if source == target: return 0
        INF = 2**20
        station2bus, E = collections.defaultdict(list), collections.defaultdict(set)
        for i, stations in enumerate(routes):
            for s in stations:
                station2bus[s].append(i)
        # print(station2bus)
        for _, buses in station2bus.items():
            if len(buses)>1:
                N = len(buses)
                # print("buses=", buses)
                for i in range(N):
                    _buses = buses.copy()
                    _buses[i], _buses[N-1] = _buses[N-1], _buses[i]
                    _buses.pop()
                    # print("e=",buses[i], _buses)
                    E[buses[i]]= E[buses[i]].union(set(_buses))
        # print("edges=",E)

        # lazy djikstra on weight of 1
        def dijkstra(src_bus):
            n_buses = len(routes)            
            dist = [INF]*n_buses
            dist[src_bus] = 0
            H = [(dist[src_bus], src_bus)]
            while H:
                d, u = heapq.heappop(H)
                if dist[u]<d: continue 
                for v in E[u]:
                    if dist[u] + 1 < dist[v]:
                        dist[v] = dist[u] + 1
                        heapq.heappush(H, (dist[v], v))
            return dist    

        out= INF    
        for src_bus in station2bus[source]:
            dist = dijkstra(src_bus)
            # print("src_bus=",src_bus, dist)
            for dst_bus in station2bus[target]:
                # print("dst_bus=", dst_bus, dist[dst_bus])
                out = min(out, dist[dst_bus])                
        
        return  out +1 if out<INF else -1 # out is actually dist
        
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