1. 6.1.3 Graph representation

2. 常見的表示法 Adjacency matrices Adjacency lists Adjacency multilists 1 2 31 2 3 G1
常見的表示法
Adjacency matrices
Adjacency lists
Adjacency multilists 2 1 G3 1 2 3 4 6 5 7 G4

3. Adjacency matrix n*n 陣列 n*(n-1),即 O(n2) If the matrix is sparse ?G3
n*n 陣列
n*(n-1),即 O(n2)
If the matrix is sparse ?
大部分元素是0
e << (n2/2) 1 2 3 G1

4. Adjacency lists n個linked list 2 1 G3 1 2 #define MAX_VERTICES 50G3 1 2
n個linked list
#define MAX_VERTICES 50
typedef struct node *node_ptr;
typedef struct node {
int vertex;
node_ptr link;
} node;
node_ptr graph[MAX_VERTICES];
int n = 0; /* number of nodes */ 3 1 2 1 2 3 G1

5. Adjacency lists, by array2 1 G3 1 2

6. Adjacency multilists 1 2 3 G1 m vertex1 vertex2 list1 list2 N01 2 3 G1
Adjacency multilists m
vertex1
vertex2
list1
list2 N0
N1 N3 N1
N2 N3
typedef struct edge *edge_ptr;
Typedef struct edge {
int marked;
int vertex1;
int vertex2;
edge_ptr path1;
edge_ptr path2;
} edge;
edge_ptr graph[MAX_VERTICES]; N2
NIL N4 N3
N4 N5 N4
NIL N5 N5
NIL NIL

7. Weighted edges
Cost
Weight field
Network

8. 6.2 Elementary graph operations

9. Outlines Operations similar to tree traversals
Depth-First Search (DFS)
Breadth-First Search (BFS)
Is it a connected graph?
Spanning trees
Biconnected components

10. Depth-First Search Adjacency list: O(e) Adjacency Mtx: O(n2) 例如:老鼠走迷宮
int visited[MAX_VERTICES];
void dfs(int v) {
node_ptr w;
visited[v] = TRUE;
printf(“%5d”, v);
for (w = graph[v]; w; w = w->link)
if(!visited[w->vertex])
dfs(w->vertex); }

11. Breadth-First Search 例如:地毯式搜索 void bfs(int v) { node_ptr w;
queue_ptr front, rear;
front=rear=NULL;
printf(“%5d”,v);
visited[v]=TRUE;
addq(&front, &rear, v);
while(front) {
v = deleteq(&front);
for(w=graph[v]; w; w=w->link)
if(!visited[w->vertex]) {
printf(“%5d”, w->vertex);
addq(&front, &rear, w->vertex);
visited[w->vertex] = TRUE; }
typedef struct queue *queue_ptr;
typedef struct queue {
int vertex;
queue_ptr link; };
void addq(queue_ptr *, queue_ptr *, int);
Int deleteq(queue_ptr);
例如:地毯式搜索

12. Connected component Is it a connected graph?
BFS(v) or DFS(v)
Find out connected component
void connected(void){
int i;
for (i=0;i<n;i++){
if(!visited[i]){
dfs(i);
printf(“\n”); }

13. Spanning Tree
A spanning tree is a minimal subgraph G’, such that V(G’)=V(G) and G’ is connected
Weight and MST 3 1 2 1 2 3 G1 1 2 1 2 3 3
DFS(0)
BFS(0)

14. Biconnected components
Definition: Articulation point (關節點)
原文請參閱課本
如果將vertex v以及連接的所有edge去除,產生 graph G’, G’至少有兩個connected component, 則v稱為 articulation point
Definition: Biconnected graph
定義為無Articulation point的connected graph
Definition: Biconnected component
Graph G中的Biconnected component H, 為G中最大的biconnected subgraph; 最大是指G中沒有其他subgraph是biconnected且包含入H

15. A connected graph and its biconnected components1 2 3 4 5 7 6 8 9 1 7 8 7 9 1 2 3 4 5 7 6 3 5
為何沒有任一個邊可能存在於兩個或多個biconnected component中?

16. DFS spanning tree of the graph in 6.19(a)
Root at 3
Back edge and cross edge 1 2 3 4 5 7 6 8 9 10 1 2 3 4 5 7 6 8 9 10

17. dfn() and low() Observation low(u): u及後代,其back edge可達vertex之最小dfn()
若root有兩個以上child, 則為articulation point
若vertex u有任一child w, 使得w及w後代無法透過back edge到u的祖先, 則為 articulation point
low(u): u及後代,其back edge可達vertex之最小dfn()
low(u) = min{ dfn(u), min{low(w)|w是u的child}, min{dfn(w)|(u,w)是back edge}}

18. A example: dfs() and low()1 2 3 4 5 7 6 8 9 10 1 2 3 4 5 7 6 8 9 10
請自行Trace Biconnected()
Hint: 將Unvisited edge跟back edge送入Stack, 到Articulation Point 再一次輸出