LeetCode 261. Graph Valid Tree（判断图是否为树）

原题网址：https://leetcode.com/problems/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.

For 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.

Hint:

1. Given n = 5 and edges = [[0, 1], [1, 2], [3, 4]], what should your return? Is this case a valid tree?
2. According to the definition of tree on Wikipedia: “a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any connected graph without simple cycles is a tree.”

注意，这里的树是普通的树，不是二叉树！

思路：要判断一个图是否为树，首先要知道树的定义。

一棵树必须具备如下特性：

（1）是一个全连通图（所有节点相通）

（2）无回路

其中（2）等价于：（3）图的边数=节点数-1

因此我们可以利用特性（1）（2）或者（1）（3）来判断。

方法一：广度优先搜索。要判断连通性，广度优先搜索法是一个天然的选择，时间复杂度O(n)，空间复杂度O(n)。

public class Solution {    public boolean validTree(int n, int[][] edges) {        Map<Integer, Set<Integer>> graph = new HashMap<>();        for(int i=0; i<edges.length; i++) {            for(int j=0; j<2; j++) {                Set<Integer> pairs = graph.get(edges[i][j]);                if (pairs == null) {                    pairs = new HashSet<>();                    graph.put(edges[i][j], pairs);                }                pairs.add(edges[i][1-j]);            }        }        Set<Integer> visited = new HashSet<>();        Set<Integer> current = new HashSet<>();        visited.add(0);        current.add(0);        while (!current.isEmpty()) {            Set<Integer> next = new HashSet<>();            for(Integer node: current) {                Set<Integer> pairs = graph.get(node);                if (pairs == null) continue;                for(Integer pair: pairs) {                    if (visited.contains(pair)) return false;                    next.add(pair);                    visited.add(pair);                    graph.get(pair).remove(node);                }            }            current = next;        }        return visited.size() == n;    }}

方法二：深度优先搜索，搜索目标是遍历全部节点。参考文章：http://buttercola.blogspot.com/2015/08/leetcode-graph-valid-tree.html

public class Solution {    private boolean[] visited;    private int visits = 0;    private boolean isTree = true;    private void check(int prev, int curr, List<Integer>[] graph) {        if (!isTree) return;        if (visited[curr]) {            isTree = false;            return;        }        visited[curr] = true;        visits ++;        for(int next: graph[curr]) {            if (next == prev) continue;            check(curr, next, graph);            if (!isTree) return;        }            }    public boolean validTree(int n, int[][] edges) {        visited = new boolean[n];        List<Integer>[] graph = new List[n];        for(int i=0; i<n; i++) graph[i] = new ArrayList<>();        for(int[] edge: edges) {            graph[edge[0]].add(edge[1]);            graph[edge[1]].add(edge[0]);        }        check(-1, 0, graph);        return isTree && visits == n;    }}

方法三：按节点大小对边进行排序，原理类似并查集。

public class Solution {    public boolean validTree(int n, int[][] edges) {        if (edges.length != n-1) return false;        Arrays.sort(edges, new Comparator<int[]>() {           @Override           public int compare(int[] e1, int[] e2) {               return e1[0] - e2[0];           }        });        int[] sets = new int[n];        for(int i=0; i<n; i++) sets[i] = i;        for(int i=0; i<edges.length; i++) {            if (sets[edges[i][0]] == sets[edges[i][1]]) return false;            if (sets[edges[i][0]] == 0) {                sets[edges[i][1]] = 0;            } else if (sets[edges[i][1]] == 0) {                sets[edges[i][0]] = 0;            } else {                sets[edges[i][1]] = sets[edges[i][0]];            }        }        return true;    }}

方法四：Union-Find

public class Solution {    public boolean validTree(int n, int[][] edges) {        if (edges.length != n-1) return false;        int[] roots = new int[n];        for(int i=0; i<n; i++) roots[i] = i;        for(int i=0; i<edges.length; i++) {            int root1 = root(roots, edges[i][0]);            int root2 = root(roots, edges[i][1]);            if (root1 == root2) return false;            roots[root2] = root1;        }        return true;    }    private int root(int[] roots, int id) {        if (id == roots[id]) return id;        return root(roots, roots[id]);    }}

参考文章：

http://blog.csdn.net/dm_vincent/article/details/7655764

http://www.elvisyu.com/graph-valid-tree-union-and-find/

http://blog.csdn.net/pointbreak1/article/details/48796691

http://buttercola.blogspot.com/2015/08/leetcode-graph-valid-tree.html