# Python O(N) concise solution with detailed explanation, passed updated testcases

• In this problem, you need to deal with 3 cases:

1. There is a loop in the graph, and no vertex has more than 1 parent.
Example:
Input: [[1,2], [2,3], [3,4], [4,1], [1,5]]
Output: [4,1]

In this case, you can simply output the edge in the loop that occurs last.
Union-find can be used to check whether an directed graph contains a cycle or not. At first, every vertex is an independent subset. For each edge, join the subsets that are on both sides of the edge. If both the vertices are in the same subset, a cycle is found.

2. A vertex has more than 1 parent, but there isn't a loop in the graph.
Example:
Input: [[1,2], [1,3], [2,3]]
Output: [2,3]

This case is also easy. You can just return the last edge that changes the tree into a graph. You can use an array of booleans to indicate whether a vertex has already got a parent.

3. A vertex has more than 1 parent, and is part of a loop.
Example:
Input: [[2,1], [3,1], [4,2], [1,4]]
Output: [2,1]

Case 3 is a mixture of case 1 and case 2. If you detect both cases, do the following:
a. Find the vertex that has multiple parents. It is obvious that this vertex is also in the loop. In the example above, node 1 is what we are looking for.
b. Starting from this vertex, use DFS to find the last edge that forms the cycle.
c. Return this edge. In the example above, it is (2, 1).

``````class Solution(object):
def union(self, a, b):
self.uf[self.find(b)] = self.find(a)

def find(self, a):
while self.uf[a] != a:
a = self.uf[a]
return a

def detectCycle(self, V):
self.visited[V] = True
if self.visited[nextV]:
return (V, nextV)
ret = self.detectCycle(nextV)
if ret[0]:
return ret
return (None, None)

def findRedundantDirectedConnection(self, edges):
"""
:type edges: List[List[int]]
:rtype: List[int]
"""
self.uf = [0] + [i + 1 for i in range(len(edges))]
self.adjList = [[] for i in range(len(edges) + 1)]      # Adjancency List
hasFather = [False] * (len(edges) + 1)                  # Whether a vertex has already got a parent
criticalEdge = None

for i, edge in enumerate(edges):
w, v = edge[0], edge[1]
if hasFather[v]:
criticalEdge = (w, v)                           # If a vertex has more than one parent, record the last edge
hasFather[v] = True
if self.find(w) == self.find(v):                    # If a loop is found, record the edge that occurs last
cycleEdge = (w, v)
self.union(w, v)

if not criticalEdge:                                    # Case 1
return cycleEdge
self.visited = [False] * (len(edges) + 1)
(w, v) = self.detectCycle(criticalEdge[1])
return (w, v) if w else criticalEdge                    # Case 2 and 3

``````

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