Graph Valid Tree

Given n nodes labeled from 0 to n - 1 and a list of undirected edges (each edge is a pair of nodes), write a function to check whether these edges make up a valid tree.

Notice

You can assume that no duplicate edges will appear in edges. Since all edges are undirected, [0, 1] is the same as [1, 0] and thus will not appear together in edges.

Example

Given n = 5 and edges = [[0, 1], [0, 2], [0, 3], [1, 4]], return true.

Given n = 5 and edges = [[0, 1], [1, 2], [2, 3], [1, 3], [1, 4]], return false.

首先看下树的定义：详见维基百科,只要没有回路的连通图就是树.

如果无向简单图G有有限个顶点（设为n个顶点），那么G 是一棵树还等价于：G是连通的，有n − 1条边，并且G没有简单回路.

后面这条定义更加关键,　也就是说我们先判断是不是有n-1边，不是则肯定没法构成树．

之后我们判断是否有简单回路．有回路，则肯定所有节点之间不是全部联通的．可以中途退出．

如何判断是否有回路呢，可以使用并查集，在并查集中，所有联通的块都有同一个祖先节点．如果新加入的这条边，两个节点的祖先节点一样，说明已经联通，会构成环，则说明不合格．

并查集大小是n,合并和查找操作O(n)级别的，所以最终代码时间复杂度为O(nlog*n)级别的，也就是Ｏ(n)级别．另外这题顶点已经预先给定，所以用数组是一个比hashmap更节省空间的选择．

代码如下：

class Solution:
    # @param {int} n an integer
    # @param {int[][]} edges a list of undirected edges
    # @return {boolean} true if it's a valid tree, or false
    def validTree(self, n, edges):
        if not n:
            return True
        if n - 1 != len(edges):
            return False
        UF = UnionFind(n)
        for edge in edges:
            if UF.find(edge[0]) == UF.find(edge[1]):
                return False
            UF.union(edge[0], edge[1])
        return True 
class UnionFind(object):
    def __init__(self, n):
        self.id = range(n)
        self.sz = [1] * n
        
    def union(self, x, y):  # union is carried out between fathers
        i = self.find(x)
        j = self.find(y)
        if self.sz[i] > self.sz[j]:
            i, j = j, i
        self.id[j] = i
        self.sz[i] += self.sz[j]
        
    def find(self, i):
        while i != self.id[i]:
            self.id[i] = self.id[self.id[i]]
            i = self.id[i]
        return i