0、什么是环?
在图论中,环(英语:cycle)是一条只有第一个和最后一个顶点重复的非空路径。
无向图
在有向图中,一个结点经过两种路线到达另一个结点,未必形成环。
有向图
1、拓扑排序
1.1、无向图
使用拓扑排序可以判断一个无向图中是否存在环,具体步骤如下:
- 求出图中所有结点的度。
- 将所有度 <= 1 的结点入队。(独立结点的度为 0)
- 当队列不空时,弹出队首元素,把与队首元素相邻节点的度减一。如果相邻节点的度变为一,则将相邻结点入队。
- 循环结束时判断已经访问的结点数是否等于 n。等于 n 说明全部结点都被访问过,无环;反之,则有环。
1.2、有向图
使用拓扑排序判断无向图和有向图中是否存在环的区别在于:
- 在判断无向图中是否存在环时,是将所有度 <= 1 的结点入队;
- 在判断有向图中是否存在环时,是将所有入度 = 0 的结点入队。
2、DFS
使用 DFS 可以判断一个无向图和有向中是否存在环。深度优先遍历图,如果在遍历的过程中,发现某个结点有一条边指向已访问过的结点,并且这个已访问过的结点不是上一步访问的结点,则表示存在环。
我们不能仅仅使用一个 bool 数组来表示结点是否访问过。规定每个结点都拥有三种状态,白、灰、黑。开始时所有结点都是白色,当访问过某个结点后,该结点变为灰色,当该结点的所有邻接点都访问完,该节点变为黑色。
那么我们的算法可以表示为:如果在遍历的过程中,发现某个结点有一条边指向灰色节点,并且这个灰色结点不是上一步访问的结点,那么存在环。
DFS
#include
#include
#include
using namespace std;
vector<vector<int>> g;
vector<int> color;
int last;
bool hasCycle;
bool topo_sort() {
int n = g.size();
vector<int> degree(n, 0);
queue<int> q;
for (int i = 0; i degree[i] = g[i].size();
if (degree[i] <= 1) {
q.push(i);
}
}
int cnt = 0;
while (!q.empty()) {
cnt++;
int root = q.front();
q.pop();
for (auto child : g[root]) {
degree[child]--;
if (degree[child] == 1) {
q.push(child);
}
}
}
return (cnt != n);
}
void dfs(int root) {
color[root] = 1;
for (auto child : g[root]) {
if (color[child] == 1 && child != last) {
hasCycle = true;
break;
}
else if (color[child] == 0) {
last = root;
dfs(child);
}
}
color[root] = 2;
}
int main() {
int n = 4;
g = vector<vector<int>>(n, vector<int>());
g[0].push_back(1);
g[1].push_back(0);
g[1].push_back(2);
g[2].push_back(1);
g[2].push_back(3);
g[3].push_back(2);
cout <endl; //0,无环
color = vector<int>(n, 0);
last = -1;
hasCycle = false;
dfs(0);
cout <endl; //0,无环
g[0].push_back(3);
g[3].push_back(0);
cout <endl; //1,有环
color = vector<int>(n, 0);
last = -1;
hasCycle = false;
dfs(0);
cout <endl; //1,有环
return 0;
}
#include
#include
#include
using namespace std;
vector<vector<int>> g;
vector<int> color;
int last;
bool hasCycle;
bool topo_sort() {
int n = g.size();
vector<int> degree(n, 0);
queue<int> q;
for (int i = 0; i degree[i] = g[i].size();
if (degree[i] <= 1) {
q.push(i);
}
}
int cnt = 0;
while (!q.empty()) {
cnt++;
int root = q.front();
q.pop();
for (auto child : g[root]) {
degree[child]--;
if (degree[child] == 1) {
q.push(child);
}
}
}
return (cnt != n);
}
void dfs(int root) {
color[root] = 1;
for (auto child : g[root]) {
if (color[child] == 1 && child != last) {
hasCycle = true;
break;
}
else if (color[child] == 0) {
last = root;
dfs(child);
}
}
color[root] = 2;
}
int main() {
int n = 4;
g = vector<vector<int>>(n, vector<int>());
g[0].push_back(1);
g[1].push_back(0);
g[1].push_back(2);
g[2].push_back(1);
g[2].push_back(3);
g[3].push_back(2);
cout <endl; //0,无环
color = vector<int>(n, 0);
last = -1;
hasCycle = false;
dfs(0);
cout <endl; //0,无环
g[0].push_back(3);
g[3].push_back(0);
cout <endl; //1,有环
color = vector<int>(n, 0);
last = -1;
hasCycle = false;
dfs(0);
cout <endl; //1,有环
return 0;
}
3、Union-Find Set
我们可以使用并查集来判断一个图中是否存在环:
对于无向图来说,在遍历边(u-v)时,如果结点 u 和结点 v 的“父亲”相同,那么结点 u 和结点 v 在同一个环中。
对于有向图来说,在遍历边(u->v)时,如果结点 u 的“父亲”是结点 v,那么结点 u 和结点 v 在同一个环中。
#include
#include
#include
using namespace std;
vectorint, int>> g;vector<int> father;int findFather(int x) {int a = x;while (x != father[x]) {
x = father[x];
}while (a != father[a]) {int z = a;
a = father[a];
father[z] = x;
}return x;
}void Union(int a, int b) {int fa = findFather(a);int fb = findFather(b);
father[a] = father[b] = min(fa, fb);
}bool isCyclicUnirectedGraph() {for (int i = 0; i int u = g[i].first;int v = g[i].second;if (father[u] == father[v]) {return true;
}
Union(u, v);
}return false;
}bool isCyclicDirectedGraph() {for (int i = 0; i int u = g[i].first;int v = g[i].second;if (father[u] == v) {return true;
}
father[v] = findFather(u);
}return false;
}int main() {// Undirected acyclic graph// 0// / \// 1 2
g.push_back(make_pair(0, 1));
g.push_back(make_pair(0, 2));for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //0,无环// Undirected cyclic graph// 0// / \// 1———2
g.push_back(make_pair(1, 2));vector<int>().swap(father);for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //1,有环// Directed acyclic graph// 0// / \// v v// 1——>2vectorint, int>>().swap(g);
g.push_back(make_pair(0, 1));
g.push_back(make_pair(1, 2));
g.push_back(make_pair(0, 2));vector<int>().swap(father);for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //0,无环// Directed cyclic graph// 0// / ^// v \// 1——>2
g.pop_back();
g.push_back(make_pair(2, 0));vector<int>().swap(father);for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //1,有环return 0;
}
#include
#include
#include
using namespace std;
vectorint, int>> g;vector<int> father;int findFather(int x) {int a = x;while (x != father[x]) {
x = father[x];
}while (a != father[a]) {int z = a;
a = father[a];
father[z] = x;
}return x;
}void Union(int a, int b) {int fa = findFather(a);int fb = findFather(b);
father[a] = father[b] = min(fa, fb);
}bool isCyclicUnirectedGraph() {for (int i = 0; i int u = g[i].first;int v = g[i].second;if (father[u] == father[v]) {return true;
}
Union(u, v);
}return false;
}bool isCyclicDirectedGraph() {for (int i = 0; i int u = g[i].first;int v = g[i].second;if (father[u] == v) {return true;
}
father[v] = findFather(u);
}return false;
}int main() {// Undirected acyclic graph// 0// / \// 1 2
g.push_back(make_pair(0, 1));
g.push_back(make_pair(0, 2));for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //0,无环// Undirected cyclic graph// 0// / \// 1———2
g.push_back(make_pair(1, 2));vector<int>().swap(father);for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //1,有环// Directed acyclic graph// 0// / \// v v// 1——>2vectorint, int>>().swap(g);
g.push_back(make_pair(0, 1));
g.push_back(make_pair(1, 2));
g.push_back(make_pair(0, 2));vector<int>().swap(father);for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //0,无环// Directed cyclic graph// 0// / ^// v \// 1——>2
g.pop_back();
g.push_back(make_pair(2, 0));vector<int>().swap(father);for (int i = 0; i 3; i++) {
father.push_back(i);
}cout <endl; //1,有环return 0;
}