1. Chapter 13 Graph Algorithms
Graph Theory
Chapter 13 Graph Algorithms
大葉大學 資訊工程系 黃鈴玲

2. Contents 13.4 Breadth-First Search 13.5 Depth-First Search
13.6 Connected Components
13.7 Dijkstra’s Shortest Path Algorithm

3. 13.4 Breadth-First Search
先進先出
Three operations: (1) Enqueue(Q, u): adds u to the tail of Q. (2) Dequeue(Q): removes one item from the head of the nonempty queue Q and returns that element. (3) Empty(Q): returns True if Q is empty, False otherwise.

4. // 每次mark一點就把 i + 1
若G不是connected，要執行到 所有點都mark為止。

5. Example 13.5 T1 T2 T3 T4 T5 s  
y=6    Q: 1 6 8 5 9
x=1

6. source     2 7 4 3

8. Ex: Find a BFS forest for the following graph G with source v1

9. 13.5 Depth-First Search

11. Ex: Find a DFS forest for the following graph G with source v1

12. 13.6 Connected Components 用DFS拜訪所有點 Component number 1 2 3 4 1 1 3 2 2

14. 13.7 Dijkstra’s Shortest Path algorithm
加Father[.] of size n: 記錄shortest path的上一點
Father[i] =1 if W1i
Father[u] =v if pathcost[u] = pathcost[v] +Wuv

15. Father[3] =1
第1個for迴圈
source
Father[4] =1
Umin={3}

16. 第2個for迴圈 Father[3] =1 Father[2] =3 i=1 Father[5] =3 Father[4] =3
source
Father[5] =3
Father[4] =3
Father[6] =3
Umin={4}

17. Father[3] =1 Father[2] =3 i=2 Father[5] =4 Father[4] =3 Father[6] =3
source
Father[5] =4
Father[4] =3
Father[6] =3
Umin={5}

18. Father[3] =1 Father[2] =3 i=3 Father[5] =4 Father[6] =5 Father[4] =3
source
Father[5] =4
Father[4] =3
Father[6] =5
Umin={2}

19. Father[3] =1 Father[2] =3 i=4 Father[5] =4 Father[4] =3 Father[6] =2
source
Father[5] =4
Father[4] =3
Father[6] =2
end

20. Ex v1 v2 v3 v4 v5 v6 v7 U Shortest u0-v3 path: u0, v5, v4 v5 v7 v6 v2
Source u0 Ex 8 v1 16 v5 10 13 v7 v6 9 11 7 17 v2 6 v4
pathcost
father
minimum 14 5 v3 v1 v2 v3 v4 v5 v6 v7 U
(, -)
{v1,…,v7}
(, -)
(13, u0)
(16, u0)
(8, u0)
{v1,…,v7}-v5
{v1,v3,v4,v6,v7}
(18, v5)
(13, u0)
(25, v5)
(15, v5)
(, -)
{v1,v3,v6,v7}
(18, v5)
(, -)
(25, v5)
(15, v5)
{v3,v6,v7}
(18, v5)
(20, v4)
(, -)
{v6,v7}
(20, v4)
(, -)
(, -)
Shortest u0-v3 path:
u0,
v5, v4

21. Ex: Find the distances from 1 to x for each x=2, 3,…,8, and determine their shortest paths.5 4 1 2 2 3 1 3 1 5 2 4 5 3 5