[LeetCode] 261. Graph Valid Tree 图是否是树

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.”

Note: 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.

给一个无向图，判断其是否为一棵树。如果是树的话，所有的节点必须是连接的，也就是说必须是连通图，而且不能有环，所以就变成了验证是否是连通图和是否含有环。

解法1: DFS

解法2: BFS

解法3: Union Find

Java: DFS

public boolean validTree(int n, int[][] edges) {
    HashMap<Integer, ArrayList<Integer>> map = new HashMap<Integer, ArrayList<Integer>>();
    for(int i=0; i<n; i++){
        ArrayList<Integer> list = new ArrayList<Integer>();
        map.put(i, list);
    }

    for(int[] edge: edges){
        map.get(edge[0]).add(edge[1]);
        map.get(edge[1]).add(edge[0]);
    }

    boolean[] visited = new boolean[n];

    if(!helper(0, -1, map, visited))
        return false;

    for(boolean b: visited){
        if(!b)
            return false;
    }

    return true;
}

public boolean helper(int curr, int parent, HashMap<Integer, ArrayList<Integer>> map, boolean[] visited){
    if(visited[curr])
        return false;

    visited[curr] = true;

    for(int i: map.get(curr)){
        if(i!=parent && !helper(i, curr, map, visited)){
            return false;
        }
    }   

    return true;
}　

Java: BFS

public boolean validTree(int n, int[][] edges) {
    HashMap<Integer, ArrayList<Integer>> map = new HashMap<Integer, ArrayList<Integer>>();
    for(int i=0; i<n; i++){
        ArrayList<Integer> list = new ArrayList<Integer>();
        map.put(i, list);
    }

    for(int[] edge: edges){
        map.get(edge[0]).add(edge[1]);
        map.get(edge[1]).add(edge[0]);
    }

    boolean[] visited = new boolean[n];

    LinkedList<Integer> queue = new LinkedList<Integer>();
    queue.offer(0);
    while(!queue.isEmpty()){
        int top = queue.poll();
        if(visited[top])
            return false;

        visited[top]=true;

        for(int i: map.get(top)){
            if(!visited[i])
                queue.offer(i);
        }
    }

    for(boolean b: visited){
        if(!b)
            return false;
    }

    return true;　

Java：BFS

public class Solution {
    /**
     * @param n an integer
     * @param edges a list of undirected edges
     * @return true if it's a valid tree, or false
     */
    public boolean validTree(int n, int[][] edges) {
        if (n == 0) {
            return false;
        }

        if (edges.length != n - 1) {
            return false;
        }

        Map<Integer, Set<Integer>> graph = initializeGraph(n, edges);

        // bfs
        Queue<Integer> queue = new LinkedList<>();
        Set<Integer> hash = new HashSet<>();

        queue.offer(0);
        hash.add(0);
        while (!queue.isEmpty()) {
            int node = queue.poll();
            for (Integer neighbor : graph.get(node)) {
                if (hash.contains(neighbor)) {
                    continue;
                }
                hash.add(neighbor);
                queue.offer(neighbor);
            }
        }

        return (hash.size() == n);
    }

    private Map<Integer, Set<Integer>> initializeGraph(int n, int[][] edges) {
        Map<Integer, Set<Integer>> graph = new HashMap<>();
        for (int i = 0; i < n; i++) {
            graph.put(i, new HashSet<Integer>());
        }

        for (int i = 0; i < edges.length; i++) {
            int u = edges[i][0];
            int v = edges[i][1];
            graph.get(u).add(v);
            graph.get(v).add(u);
        }

        return graph;
    }
}

Java: Union Find　　

public class Solution {
      class UnionFind{
        HashMap<Integer, Integer> father = new HashMap<Integer, Integer>();
        UnionFind(int n){
            for(int i = 0 ; i < n; i++) {
                father.put(i, i);
            }
        }
        int compressed_find(int x){
            int parent =  father.get(x);
            while(parent!=father.get(parent)) {
                parent = father.get(parent);
            }
            int temp = -1;
            int fa = father.get(x);
            while(fa!=father.get(fa)) {
                temp = father.get(fa);
                father.put(fa, parent) ;
                fa = temp;
            }
            return parent;

        }

        void union(int x, int y){
            int fa_x = compressed_find(x);
            int fa_y = compressed_find(y);
            if(fa_x != fa_y)
                father.put(fa_x, fa_y);
        }
    }
    /**
     * @param n an integer
     * @param edges a list of undirected edges
     * @return true if it's a valid tree, or false
     */
    public boolean validTree(int n, int[][] edges) {
        // tree should have n nodes with n-1 edges
        if (n - 1 != edges.length) {
            return false;
        }

        UnionFind uf = new UnionFind(n);

        for (int i = 0; i < edges.length; i++) {
            if (uf.compressed_find(edges[i][0]) == uf.compressed_find(edges[i][1])) {
                return false;
            }
            uf.union(edges[i][0], edges[i][1]);
        }
        return true;
    }
}　　

Python: DFS

class Solution(object):
    def validTree(self, n, edges):
        lookup = collections.defaultdict(list)
        for edge in edges:
            lookup[edge[0]].append(edge[1])
            lookup[edge[1]].append(edge[0])
        visited = [False] * n

        if not self.helper(0, -1, lookup, visited):
            return False

        for v in visited:
            if not v:
                return False

        return True

    def helper(self, curr, parent, lookup, visited):
        print curr, visited
        if visited[curr]:
            return False
        visited[curr] = True
        for i in lookup[curr]:
            if (i != parent and not self.helper(i, curr, lookup, visited)):
                return False

        return True

if __name__ == '__main__':
    print Solution().validTree(5, [[0, 1], [0, 2], [0, 3], [1, 4]])
    print Solution().validTree(5, [[0, 1], [1, 2], [2, 3], [1, 3], [1, 4]])　

Python: BFS, Time: O(|V| + |E|), Space: O(|V| + |E|)

class Solution(object):
    # @param {integer} n
    # @param {integer[][]} edges
    # @return {boolean}
    def validTree(self, n, edges):
        if len(edges) != n - 1:  # Check number of edges.
            return False

        # init node's neighbors in dict
        neighbors = collections.defaultdict(list)
        for u, v in edges:
            neighbors[u].append(v)
            neighbors[v].append(u)

        # BFS to check whether the graph is valid tree.
        visited = {}
        q = collections.deque([0])
        while q:
            curr = q.popleft()
            visited[curr] = True
            for node in neighbors[curr]:
                if node not in visited:
                    visited[node] = True
                    q.append(node)

        return len(visited) == n

Python: Union Find

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):
        # Write your code here
        root = [i for i in range(n)]
        for i in edges:
            root1 = self.find(root, i[0])
            root2 = self.find(root, i[1])
            if root1 == root2:
                return False
            else:
                root[root1] = root2
        return len(edges) == n - 1

    def find(self, root, e):
        if root[e] == e:
            return e
        else:
            root[e] = self.find(root, root[e])
            return root[e]　　

C++: DFS

class Solution {
public:
    bool validTree(int n, vector<pair<int, int>>& edges) {
        vector<vector<int>> g(n, vector<int>());
        vector<bool> v(n, false);
        for (auto a : edges) {
            g[a.first].push_back(a.second);
            g[a.second].push_back(a.first);
        }
        if (!dfs(g, v, 0, -1)) return false;
        for (auto a : v) {
            if (!a) return false;
        }
        return true;
    }
    bool dfs(vector<vector<int>> &g, vector<bool> &v, int cur, int pre) {
        if (v[cur]) return false;
        v[cur] = true;
        for (auto a : g[cur]) {
            if (a != pre) {
                if (!dfs(g, v, a, cur)) return false;
            }
        }
        return true;
    }
};

C++: BFS　　

class Solution {
public:
    bool validTree(int n, vector<pair<int, int>>& edges) {
        vector<unordered_set<int>> g(n, unordered_set<int>());
        unordered_set<int> s{{0}};
        queue<int> q{{0}};
        for (auto a : edges) {
            g[a.first].insert(a.second);
            g[a.second].insert(a.first);
        }
        while (!q.empty()) {
            int t = q.front(); q.pop();
            for (auto a : g[t]) {
                if (s.count(a)) return false;
                s.insert(a);
                q.push(a);
                g[a].erase(t);
            }
        }
        return s.size() == n;
    }
};

C++: Union Find　　

class Solution {
public:
    bool validTree(int n, vector<pair<int, int>>& edges) {
        vector<int> roots(n, -1);
        for (auto a : edges) {
            int x = find(roots, a.first), y = find(roots, a.second);
            if (x == y) return false;
            roots[x] = y;
        }
        return edges.size() == n - 1;
    }
    int find(vector<int> &roots, int i) {
        while (roots[i] != -1) i = roots[i];
        return i;
    }
};

　　

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