<p>Given a weighted undirected connected graph with <code>n</code>&nbsp;vertices numbered from <code>0</code> to <code>n - 1</code>,&nbsp;and an array <code>edges</code>&nbsp;where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>, weight<sub>i</sub>]</code> represents a bidirectional and weighted edge between nodes&nbsp;<code>a<sub>i</sub></code>&nbsp;and <code>b<sub>i</sub></code>. A minimum spanning tree (MST) is a subset of the graph&#39;s edges that connects all vertices without cycles&nbsp;and with the minimum possible total edge weight.</p>

<p>Find <em>all the critical and pseudo-critical edges in the given graph&#39;s minimum spanning tree (MST)</em>. An MST edge whose deletion from the graph would cause the MST weight to increase is called a&nbsp;<em>critical edge</em>. On&nbsp;the other hand, a pseudo-critical edge is that which can appear in some MSTs but not all.</p>

<p>Note that you can return the indices of the edges in any order.</p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/ex1.png" style="width: 259px; height: 262px;" /></p>

<pre>
<strong>Input:</strong> n = 5, edges = [[0,1,1],[1,2,1],[2,3,2],[0,3,2],[0,4,3],[3,4,3],[1,4,6]]
<strong>Output:</strong> [[0,1],[2,3,4,5]]
<strong>Explanation:</strong> The figure above describes the graph.
The following figure shows all the possible MSTs:
<img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/msts.png" style="width: 540px; height: 553px;" />
Notice that the two edges 0 and 1 appear in all MSTs, therefore they are critical edges, so we return them in the first list of the output.
The edges 2, 3, 4, and 5 are only part of some MSTs, therefore they are considered pseudo-critical edges. We add them to the second list of the output.
</pre>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/ex2.png" style="width: 247px; height: 253px;" /></p>

<pre>
<strong>Input:</strong> n = 4, edges = [[0,1,1],[1,2,1],[2,3,1],[0,3,1]]
<strong>Output:</strong> [[],[0,1,2,3]]
<strong>Explanation:</strong> We can observe that since all 4 edges have equal weight, choosing any 3 edges from the given 4 will yield an MST. Therefore all 4 edges are pseudo-critical.
</pre>

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

<ul>
	<li><code>2 &lt;= n &lt;= 100</code></li>
	<li><code>1 &lt;= edges.length &lt;= min(200, n * (n - 1) / 2)</code></li>
	<li><code>edges[i].length == 3</code></li>
	<li><code>0 &lt;= a<sub>i</sub> &lt; b<sub>i</sub> &lt; n</code></li>
	<li><code>1 &lt;= weight<sub>i</sub>&nbsp;&lt;= 1000</code></li>
	<li>All pairs <code>(a<sub>i</sub>, b<sub>i</sub>)</code> are <strong>distinct</strong>.</li>
</ul>

import heapq

class Solution:
    def findCriticalAndPseudoCriticalEdges(self, n: int, edges: List[List[int]]) -> List[List[int]]:
        # first run prim's to get mst edge sum + initial mst edge set
        # for each edge, remove from graph and run prim's 
        #       - if edge in initial mst edge set + edge sum > mst edge sum --> critical
        #       - if edge sum == mst edge sum --> 
        #         - keep running union of mst edge sets + intersection ==> used for pseudo critical calc

        def prim(graph, removed_edge=None, include_edge=None):
            if include_edge:
                heap = [(0, include_edge[0], -1)]
            else:
                heap = [(0, 0, -1)]
            seen = set()
            total = 0
            
            while heap:
                weight, node, prev = heapq.heappop(heap)
                if node in seen:
                    continue
                seen.add(node)
                if prev != -1:
                    total += weight

                if len(seen) == n:
                    return total

                for neighbor, next_weight in graph[node]:
                    next_edge = (neighbor, node) if neighbor < node else (node, neighbor)
                    if removed_edge and removed_edge == next_edge:
                        continue 
                    if include_edge and include_edge == next_edge:
                        next_weight = 0
                    if neighbor not in seen:
                        heapq.heappush(heap, (next_weight, neighbor, node))
            
            return -1
        
        graph = collections.defaultdict(list)
        for a, b, w in edges:
            graph[a].append((b, w))
            graph[b].append((a, w))
            
            
        mst_total_weight = prim(graph)
        if mst_total_weight < 0:
            return [[], []]

        critical_list = []
        pseudo_list = []

        for ind, (a, b, w) in enumerate(edges):
            curr_edge = (a, b) if a < b else (b, a)
            curr_total_weight = prim(graph, curr_edge, None)
            if curr_total_weight > mst_total_weight or curr_total_weight < 0:
                critical_list.append(ind) 
                continue 
                
            curr_total_weight2 = prim(graph, None, (a, b))
            if mst_total_weight == (curr_total_weight2 + w):
                pseudo_list.append(ind)
                continue 
            curr_total_weight3 = prim(graph, None, (b, a)) 
            if mst_total_weight == (curr_total_weight3 + w):
                pseudo_list.append(ind)
            
        return [critical_list, pseudo_list]


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