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資料結構實習 - 十四

資料結構實習 - 十四. Graph. Graph. 0. 1. V ( G 1 )={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} E ( G 1 )={(0, 2), (0, 3), (1, 4), (2, 3), (2, 5), (2, 6), (3, 6), (3, 7), (4, 7), (5, 6), (5, 8), (6, 9), (7, 9),

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資料結構實習 - 十四

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  1. 資料結構實習-十四 Graph

  2. Graph 0 1 V(G1)={0, 1, 2, 3, 4, 5, 6, 7, 8, 9}E(G1)={(0, 2), (0, 3), (1, 4), (2, 3), (2, 5), (2, 6), (3, 6), (3, 7), (4, 7), (5, 6), (5, 8), (6, 9), (7, 9), } 2 3 4 5 6 7 8 9

  3. Adjacency matrix 0 • 以右圖為例: • Adjacency matrices表示 • 輸出結果: 1 2 3 0 1 2 3 0 1 2 3 • (0) (1) (2) (3) • (0) 0 0 0 1 • (1) 0 0 1 1 • 0 1 0 1 • 1 1 1 0

  4. Adjacency matrix – 範例 0 1 2 3 0 0 0 1 2 3 1 2 1 3 G1 The adjacency matrix of G1 2 0 1 2 0 1 2 G3 The adjacency matrix of G3

  5. Adjacency list 0 3 1 2 3 2 1 3 3 0 1 2 0 1 2 3 • (0) -> 3 • (1) -> 2 -> 3 • -> 1 -> 3 • -> 0 -> 1 -> 2 • 以右圖為例: • Adjacency List表示 • 輸出結果:

  6. Adjacency list – 範例 adj_list data link 0 3 1 2 2 3 0 1 2 1 3 0 3 G1 0 1 2 Adjacent list ofG1 0 Adj_list data link 1 1 2 0 G3 Adjacent list ofG3 2

  7. Depth First Search 0 0 • 類似樹的先序尋訪。 • 必須紀錄之前的路徑。 • Stack • Recursive 1 2 1 2 4 3 4 5 6 3 5 6 7 7 0 -> 1 -> 3 -> 7 -> 4 -> 5 -> 2 -> 6

  8. Biconnected Graph 8 • Articulation point • 如果將v 以及所有附著於v 的邊刪除,會產生兩個以上的連通元件(connected graph),我們就稱v 為Articulation point(連接點)。 • biconnected graph • 二連通圖是一個沒有連接點的連通圖形

  9. Biconnected Graph – 範例 8 0 9 7 1 7 1 2 3 5 3 5 4 6 9

  10. A Connected Graph and Its Biconnected Components 0 8 9 8 0 9 1 7 7 1 7 1 7 2 3 5 2 3 3 5 5 4 6 4 6 (a) A connected graph (b) Biconnected components 10

  11. Biconnected Graph – 範例 0 2 1 3 3 0 1 2 3 Biconnected Graph(List) 1：<0,3> 2：<1,2><1,3><2,3> • 以右圖為例: • Biconnected Graph表示 • 程式預期輸出結果:

  12. Biconnected Graph – 演算法 • depth first numbers (dfn) • 在頂點在深度優先搜尋時,頂點被拜訪的順序。 • low(u)是利用 u 的後代以及至多一個後退邊(back edge)可達到的頂點之最小深度優先序號 • low(u) = min {dfn(u), min {low(w) | w is a child of u}, min {dfn(w) | (u, w) is a back edge}} • back edge：在DFS時不被經過的邊。 • u 是一個連接點若且唯若 • u 是一個根節點且至少有兩個子節點 • 或 u 不是根節點但有一個子節點 w 使得 low(w)  >=dfn(u))

  13. Biconnected Graph – 演算法 dfn:0 dfn:9 dfn:8 0 0 0 8 8 low:9 0 9 8 9 9 9 low:8 low:0 dfn:1 7 , 9 7 , 8 1 1 1 1 7 7 7 7 7 7 low:1 7 , 5 dfn:7 low:7 low:5 6 , 7 5 , 6 <7,8> dfn:2 3 , 5 <7,9> low:1 low:2 2 3 2 3 3 5 2 2 3 5 5 3 , 1 5 <7,5> <6,7> <5,6> 4 , 3 <3,5> dfn:4 dfn:5 <2,4> <1,2> <3,1> <4,3> 2 , 4 low:4 low:1 low:5 <0,1> 1 , 2 4 6 6 6 4 4 6 6 4 0 , 1 dfn:3 dfn:6 low:1 low:3 low:6 low:5 Stack 13

  14. A Connected Graph and Its Biconnected Components 0 8 9 8 0 9 1 7 7 1 7 1 7 2 3 5 2 3 3 5 5 4 6 4 6 (a) A connected graph (b) Biconnected components 14

  15. 0 1 2 3 4 5 6 7 8 9 練習 • 右圖為本次練習的題目 • 請自教學網站下載graph.txt。 • 共有十個vertex，以0-9表示。 • 每一行代表一條edge • vertex 之間以tab(\t)隔開

  16. 練習-1 將該圖形資料轉換成Adjacency matrices的格式，並輸出矩陣格式。 針對練習一中完成的矩陣格式，做Depth First Search的動作，並將結果輸出。 針對練習一中完成的矩陣格式，找出該圖形中的所有 Biconnected Componentsgraph，並將結果輸出。

  17. 練習-2 將該圖形資料轉換成Adjacency list的格式，並輸出矩陣格式。 針對練習一中完成的串列格式，做Depth First Search的動作，並將結果輸出。 針對練習一中完成的串列格式，找出該圖形中的所有 Biconnected Componentsgraph，並將結果輸出。

  18. 練習 • DFS: • 0 2 3 6 5 8 9 7 4 1 • Biconnected Components graph • <5,8> • <4,1> • <7,4> • <7,3> <9,7> <6,9> <5,2> <6,5> <6,2> <3,6> <3,0> <2,3><0,2>

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