[Leetcode] 684. Redundant Connection

In this problem, a tree is an undirected graph that is connected and has no cycles.

The given input is a graph that started as a tree with N nodes (with distinct values 1, 2, ..., N), with one additional edge added. The added edge has two different vertices chosen from 1 to N, and was not an edge that already existed.

The resulting graph is given as a 2D-array of edges. Each element of edges is a pair [u, v] with u < v, that represents an undirected edge connecting nodes u and v.

Return an edge that can be removed so that the resulting graph is a tree of N nodes. If there are multiple answers, return the answer that occurs last in the given 2D-array. The answer edge [u, v] should be in the same format, with u < v.

Example 1:

Input: [[1,2], [1,3], [2,3]]
Output: [2,3]
Explanation: The given undirected graph will be like this:
  1
 / 
2 - 3

Example 2:

Input: [[1,2], [2,3], [3,4], [1,4], [1,5]]
Output: [1,4]
Explanation: The given undirected graph will be like this:
5 - 1 - 2
    |   |
    4 - 3


1. 1st use union-find method. find the group of each connected node. and check new connected edge to be in the previous group

  #initialize tree
        def constructUnion(N):            #union-size
            for i in range(1, N+1):
                idLst[i] = i
                sizeLst[i] = 1
        
        
        #find group id
        def find(p):  
            while(p != idLst[p]):
                idLst[p] = idLst[idLst[p]]
                p = idLst[p]
            return p
        
        
        # check whether they are connected
        def isConnected(p, q):
            return find(p) == find(q)
        
        #Union two element into same group
        def union(p, q):
            idp = find (p)
            idq = find(q)
            
            if idp !=idq:
                if sizeLst[p] > sizeLst[q]:
                    idLst[idq] = idp
                    sizeLst[idp] += sizeLst[idq]
                    
                else:
                    idLst[idp] = idq
                    sizeLst[idq] += sizeLst[idp]
            
                    
        N = 1000
        idLst = [0] * (N+1)
        sizeLst = [0] * (N+1)
        
        constructUnion(N)
        for eg in edges:
            if isConnected(eg[0], eg[1]): return eg
            #print ("pair for cycle : ", eg)
            union(eg[0], eg[1])
                
        return []

--reference:  https://www.cs.princeton.edu/~rs/AlgsDS07/01UnionFind.pdf

　　2. use naive way to detect the cycle exist or not in the graph

1) get the edge from reversed order, and delete from the graph gTmp

2)use DFS to judge whether circle exists in the gTmp.  the starting node is selected from the input list backwards

 

def cycleIterative(v, visited, parent, gTmp):
            visited[v] = True
            
            for i in gTmp[v]:
                if not visited[i]:
                    if cycleIterative(i, visited, v, gTmp):
                        return True
                elif parent != i:
                    return True
            return False

        g = defaultdict(list)                   # default dictionary to store graph
        
        for edge in edges:
            g[edge[0]].append(edge[1])
            g[edge[1]].append(edge[0])
        
        def cycle(gTmp):
            visited = defaultdict()
            for v in gTmp:
                visited[v] = False
            for v in gTmp:
                if not visited[v]:
                    if cycleIterative(v, visited, -1, gTmp):
                        return True
            return False    
        
        for pair in edges[::-1]:
            gTmp = deepcopy(g)
            #remove the pair edge from g
            gTmp[pair[0]].remove(pair[1])
            gTmp[pair[1]].remove(pair[0])
            print ("edge: ", gTmp)
            
            if not cycle(gTmp):
                print ("pair for cycle : ", pair)
        return []