Trail: a walk that does not repeat any edges.
Closed Trail(Circuit): a trail that begins and ends at the same vertex.
Eulerian Circuit: a circuit that includes every edge of the graph.
Eulerian Trail: a trail that includes every edge of the graph.
在无向图中，图 $G$ 有欧拉回路当且仅当每个点的度数为偶数。
在有向图中，图 $G$ 有欧拉回路当且仅当每个点的入度和出度相同。
在无向图中，图 $G$ 有欧拉路径(非欧拉回路)当且仅当只有两个点的度数为奇数，其余都是偶数。
在有向图中，图 $G$ 有欧拉路径(非欧拉回路)当且仅当只有两个点的入度和出度不同，且其中一个点的“入度 $-$ 出度=1”，另一个点的“出度 $-$ 入度=1”。
输出欧拉回路和欧拉路径的常用算法是 Hierholzer's Algorithm 和 Fleury's Algorithm，复杂度分别为 $O(E)$ 和 $O(E^2)$。
Hierholzer's Algorithm的正确性：因为一个具有欧拉回路的图G能够拆分成多个边不相交的环，因此这个算法正确。
1 Find a circuit called $R_1$ 2 mark all edges of $R_1$ 3 i=1; 4 while(true) 5 { 6 if $R_1$ contains all edges of G, then return; 7 else 8 { 9 let $v_i$ be a vertex on $R_i$ that is incident with unmarked edge; 10 build a circuit $Q_i$ from $v_i$; 11 mark all edges of $Q_i$; 12 $R_{i+1} = R_i \cup Q_i$; 13 i++; 14 } 15 }
1 import java.util.LinkedList; 2 import java.util.List; 3 import java.util.Scanner; 4 5 public class EularTrail { 6 int n; 7 int m; 8 int head[]; 9 int edge[][]; 10 int next[]; 11 int visited[]; 12 int ee = 0; 13 int indegree[]; 14 int outdegree[]; 15 LinkedList<Pair> queue = new LinkedList<Pair>(); 16 List<Integer> tmpList = new LinkedList<Integer>(); 17 List<Integer> resultList = new LinkedList<Integer>(); 18 19 int index[][]; 20 int start = 1; 21 int end = 0; 22 public boolean eularTrail(){ 23 calculateDegree(); 24 boolean isEularCircuit = hasEularCircuit(); 25 boolean isEularTrail = hasEularTrail(); 26 if(isEularTrail || isEularCircuit){ 27 tmpList.add(start); 28 findCycle(start, 0); 29 while(!queue.isEmpty()){ 30 Pair u = queue.removeLast(); //注意这里只能用removeLast而不能用removeFirst 31 findCycle(u.vertex,u.pos); 32 } 33 printEular(); 34 return true; 35 } 36 else return false; 37 } 38 39 /** 40 * 判定输入的路径是否是欧拉路径 41 * @return 42 */ 43 public boolean checkEularTrail(){ 44 System.out.println("###################"); 45 System.out.println("#Check Eular Trail#"); 46 System.out.println("###################"); 47 System.out.println("Input path:"); 48 Scanner in = new Scanner(System.in); 49 boolean[] pathVisited = new boolean[m]; 50 for(int i=0;i<m;i++){ 51 pathVisited[i] = false; 52 } 53 int[] sequence = new int[m+1]; 54 for(int i=0;i<m+1;i++){ 55 sequence[i] = in.nextInt(); 56 } 57 for(int i=0;i<m;i++){ 58 int begin = sequence[i]; 59 int end = sequence[i+1]; 60 pathVisited[index[begin][end]]=true; 61 } 62 for(int i=0;i<m;i++){ 63 if(pathVisited[i]==false){ 64 System.out.println("\nNot a Eular Trail!"); 65 return false; 66 } 67 } 68 System.out.println("\nIs a Eular Trail!"); 69 return true; 70 } 71 72 private void calculateDegree() { 73 for(int i=1;i<=n;i++){ 74 for(int j=head[i];j!=-1;j=next[j]){ 75 outdegree[i]++; 76 indegree[edge[j][1]]++; 77 } 78 } 79 } 80 private void printEular(){ 81 for(int i=0;i<resultList.size();i++){ 82 System.out.println(resultList.get(i)); 83 } 84 } 85 private boolean hasEularTrail() { 86 int start_count = 0; 87 int end_count = 0; 88 for(int i=1;i<=n;i++){ 89 if(indegree[i]!=outdegree[i]){ 90 if((indegree[i]-outdegree[i])!=1 && (indegree[i]-outdegree[i])!=-1){ 91 return false; 92 } 93 else if((indegree[i]-outdegree[i])==1){ 94 end = i; 95 end_count++; 96 } 97 else if((indegree[i]-outdegree[i])==-1){ 98 start = i; 99 start_count++; 100 } 101 } 102 } 103 if(start_count!=1 || end_count!=1) return false; 104 return true; 105 } 106 107 private void findCycle(int i,int begin) { 108 DFS(i,i,begin); 109 resultList.addAll(begin,tmpList); 110 tmpList.clear(); 111 } 112 private boolean DFS(int source,int i,int begin){ 113 if(outdegree[i]==0){ 114 return true; 115 } 116 for(int j=head[i];j!=-1;j=next[j]){ 117 int end = edge[j][1]; 118 119 if(visited[j]==0){ 120 visited[j] = 1; 121 outdegree[i]--; 122 if(outdegree[i]>0&&!queue.contains(i)){ 123 queue.add(new Pair(i,begin+tmpList.size())); 124 } 125 tmpList.add(end); 126 if(end==source){ 127 if(outdegree[end]>0&&!queue.contains(end)){ 128 queue.add(new Pair(end,begin+tmpList.size())); 129 } 130 return false; 131 } 132 boolean flag = DFS(source,edge[j][1],begin); 133 if(flag==false){ 134 return false; 135 } 136 } 137 } 138 return true; 139 } 140 141 private boolean hasEularCircuit() { 142 for(int i=1;i<=n;i++){ 143 if(indegree[i]!=outdegree[i]){ 144 return false; 145 } 146 } 147 return true; 148 } 149 public void input(){ 150 System.out.println("Input Graph:"); 151 Scanner in = new Scanner(System.in); 152 n = in.nextInt(); 153 m = in.nextInt(); 154 head = new int[n+1]; 155 next = new int[m]; 156 edge = new int[m][2]; 157 visited = new int[m]; 158 indegree = new int[n+1]; 159 outdegree = new int[n+1]; 160 index = new int[n+1][n+1]; 161 for(int i=1;i<=n;i++){ 162 head[i] = -1; 163 outdegree[i] = 0; 164 indegree[i] = 0; 165 } 166 for(int i=0;i<m;i++){ 167 next[i] = -1; 168 visited[i] = 0; 169 } 170 for(int i=1;i<=m;i++){ 171 int begin = in.nextInt(); 172 int end = in.nextInt(); 173 edge[ee][0] = begin; 174 edge[ee][1] = end; 175 next[ee] = head[begin]; 176 head[begin] = ee; 177 index[begin][end] = ee; 178 ee++; 179 } 180 } 181 public static void main(String[] args) { 182 EularTrail hierholzer = new EularTrail(); 183 hierholzer.input(); 184 hierholzer.eularTrail(); 185 //boolean flag = hierholzer.checkEularTrail(); 186 //System.out.println(flag); 187 } 188 }
1 import java.util.LinkedList; 2 import java.util.List; 3 import java.util.Scanner; 4 5 class Pair{ 6 int vertex; 7 int pos; 8 public Pair(int vertex,int pos){ 9 this.vertex = vertex; 10 this.pos = pos; 11 } 12 @Override 13 public String toString() { 14 return "Pair [vertex=" + vertex + ", pos=" + pos + "]"; 15 } 16 } 17 public class EularCircuit { 18 int n; 19 int m; 20 int head[]; 21 int edge[][]; 22 int next[]; 23 int visited[]; 24 int ee = 0; 25 int indegree[]; 26 int outdegree[]; 27 LinkedList<Pair> queue = new LinkedList<Pair>(); 28 List<Integer> tmpList = new LinkedList<Integer>(); 29 List<Integer> resultList = new LinkedList<Integer>(); 30 31 int index[][]; 32 public boolean eularCircuit(){ 33 calculateDegree(); 34 boolean isEular = hasEularCircuit(); 35 if(!isEular) return false; 36 tmpList.add(1); 37 findCycle(1, 0); 38 while(!queue.isEmpty()){ 39 Pair u = queue.removeLast(); //注意这里只能用removeLast而不能用removeFirst 40 findCycle(u.vertex,u.pos); 41 } 42 printEular(); 43 return true; 44 } 45 /** 46 * 判定输入的路径是否是欧拉回路 47 * @return 48 */ 49 public boolean checkEularCircuit(){ 50 System.out.println("#####################"); 51 System.out.println("#Check Eular Circuit#"); 52 System.out.println("#####################"); 53 System.out.println("Input path:"); 54 Scanner in = new Scanner(System.in); 55 boolean[] pathVisited = new boolean[m]; 56 for(int i=0;i<m;i++){ 57 pathVisited[i] = false; 58 } 59 int[] sequence = new int[m+1]; 60 for(int i=0;i<m+1;i++){ 61 sequence[i] = in.nextInt(); 62 } 63 if(sequence[0]!=sequence[m]) { 64 System.out.println("\nNot a Eular Circuit!"); 65 return false; 66 } 67 for(int i=0;i<m;i++){ 68 int begin = sequence[i]; 69 int end = sequence[i+1]; 70 pathVisited[index[begin][end]]=true; 71 } 72 for(int i=0;i<m;i++){ 73 if(pathVisited[i]==false){ 74 System.out.println("\nNot a Eular Circuit!"); 75 return false; 76 } 77 } 78 System.out.println("\nIs a Eular Circuit!"); 79 return true; 80 } 81 public void input(){ 82 System.out.println("Input Graph:"); 83 Scanner in = new Scanner(System.in); 84 n = in.nextInt(); 85 m = in.nextInt(); 86 head = new int[n+1]; 87 next = new int[m]; 88 edge = new int[m][2]; 89 visited = new int[m]; 90 indegree = new int[n+1]; 91 outdegree = new int[n+1]; 92 index = new int[n+1][n+1]; 93 for(int i=1;i<=n;i++){ 94 head[i] = -1; 95 outdegree[i] = 0; 96 indegree[i] = 0; 97 } 98 for(int i=0;i<m;i++){ 99 next[i] = -1; 100 visited[i] = 0; 101 } 102 for(int i=1;i<=m;i++){ 103 int begin = in.nextInt(); 104 int end = in.nextInt(); 105 edge[ee][0] = begin; 106 edge[ee][1] = end; 107 next[ee] = head[begin]; 108 head[begin] = ee; 109 index[begin][end] = ee; 110 ee++; 111 } 112 } 113 114 private void calculateDegree() { 115 for(int i=1;i<=n;i++){ 116 for(int j=head[i];j!=-1;j=next[j]){ 117 outdegree[i]++; 118 indegree[edge[j][1]]++; 119 } 120 } 121 } 122 private boolean hasEularCircuit() { 123 for(int i=1;i<=n;i++){ 124 if(indegree[i]!=outdegree[i]){ 125 return false; 126 } 127 } 128 return true; 129 } 130 /*从i开始寻找一个环，即 i->....->i */ 131 private void findCycle(int i,int begin) { 132 DFS(i,i,begin); 133 resultList.addAll(begin,tmpList); 134 tmpList.clear(); 135 } 136 137 private boolean DFS(int source,int i,int begin){ 138 if(outdegree[i]==0){ 139 return true; 140 } 141 for(int j=head[i];j!=-1;j=next[j]){ 142 int end = edge[j][1]; 143 144 if(visited[j]==0){ 145 visited[j] = 1; 146 outdegree[i]--; 147 if(outdegree[i]>0&&!queue.contains(i)){ 148 queue.add(new Pair(i,begin+tmpList.size())); 149 } 150 tmpList.add(end); 151 if(end==source){ 152 if(outdegree[end]>0&&!queue.contains(end)){ 153 queue.add(new Pair(end,begin+tmpList.size())); 154 } 155 return false; 156 } 157 boolean flag = DFS(source,edge[j][1],begin); 158 if(flag==false){ 159 return false; 160 } 161 } 162 } 163 return true; 164 } 165 /*输出欧拉回路*/ 166 private void printEular(){ 167 for(int i=0;i<resultList.size();i++){ 168 System.out.println(resultList.get(i)); 169 } 170 } 171 172 public static void main(String[] args) { 173 EularCircuit hierholzer = new EularCircuit(); 174 hierholzer.input(); 175 hierholzer.eularCircuit(); 176 boolean flag = hierholzer.checkEularCircuit(); 177 System.out.println(flag); 178 } 179 180 }
其中，EularCircuit 类的功能：
- 输入图
- 检测给定序列是否为欧拉回路
- 输出欧拉回路
EularTrail 类的功能：
- 输入图
- 检测给定序列是否为欧拉路径
- 输出欧拉路径
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