6.1.3 Graph representation

1. 6.1.3 Graph representation

2. 0 1 2 3 G1 0 1 2 0 4 G3 2 1 5 6 3 7 G4 常見的表示法 • Adjacency matrices • Adjacency lists • Adjacency multilists

3. 0 1 2 3 G1 0 1 2 G3 Adjacency matrix • 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)

4. 0 1 2 3 1 G1 2 0 0 1 2 3 1 2 2 G3 3 0 3 1 0 0 1 2 Adjacency lists • 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 */

5. 1 2 0 0 1 2 G3 Adjacency lists, by array

6. 0 1 2 3 G1 Adjacency multilists list1 m vertex1 vertex2 list2 0 1 N1 N3 N0 0 2 N2 N3 N1 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]; 0 3 NIL N4 N2 1 2 N4 N5 N3 1 3 NIL N5 N4 2 3 NIL NIL N5

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. 例如:地毯式搜索 Breadth-First Search 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);

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. 0 1 2 3 1 2 3 2 3 0 G1 3 1 0 0 1 2 Spanning Tree • A spanning tree is a minimal subgraph G’, such that V(G’)=V(G) and G’ is connected • Weight and MST 0 0 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. 9 0 8 9 0 7 8 1 7 1 7 2 3 5 1 7 4 6 2 3 3 5 5 4 6 A connected graph and its biconnected components • 為何沒有任一個 點/邊 可能存在於兩個或多個biconnected component中?

16. 1 3 2 6 4 5 5 9 10 0 8 9 3 7 2 6 4 1 7 4 8 8 1 7 3 1 6 2 3 5 10 5 9 7 0 8 9 4 6 2 DFS spanning tree of the graph in 6.19(a) • Root at 3 • Back edge and cross edge

17. dfn() and low() • Observation • 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. 1 3 2 6 4 5 5 9 10 0 8 9 3 7 2 6 4 1 7 4 8 8 1 7 3 1 6 2 3 5 10 5 9 7 0 8 9 4 6 2 A example: dfs() and low() • 請自行Trace Biconnected() • Hint: 將Unvisited edge跟back edge送入Stack, 到Articulation Point 再一次輸出