1. 1 Chapter 15 Graphs, Trees, and Networks

2. 2 Chap.15 15.1 Undirected Graphs 15.2 Directed Graphs 15.3 Trees 15.4 Networks 15.5 Graph Algorithms 15.5.1 Iterators 15.5.3 Generating a Minimum Spanning Tree 15.5.4 Finding the Shortest Path

3. 3 Chap.15 (Cont.) 15.6 Network Class ( 軟體開發三道工序 ): 15.6.1 Method Specifications 15.6.2 Data Structure Design 15.6.3 Method Definitions

4. 4 Graph theory 圖論 [Wiki] In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pair-wise relations between objects from a certain collection. ( 研究 某 collection 內 objects 的兩兩關係 )

5. 5 Applications of Graphs 許多實際問題可用 graph 來表示 例如：網址連結的結構，可用有向圖 (directed graph) 頂點 (vertex) 是網頁網址，如果有一個 B 網址的網頁連 結到 A 網址的網頁 ，便有個連結 (edge) 在網址 A,B 之間 類似的 在 travel 問題，生物學，電腦晶片設計，以及 其他許多領域。 Graph 可增加 weight ( 權重 ) 延伸為 weighted graph ( 加權圖 ) 例如 weight 能代表連結 (edge) 的長度

6. 6 15.1 Undirected Graphs

7. 7 Example: X Y Z T ANS: 5 vertices 6 edges S

8. 8 Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet

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12. 12 X Y Z T X, Y, T, Z, S has length 4. Shortest path from X to S = ? Ans: 2 S In general, how can the smallest path between 2 vertices be determined? We will cover that.

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14. 14 X Y Z T X, Y, T, Z, X is a cycle. Y, T, S, Z, T, Y is not a cycle. (repeated edges: YT and TY) S Is Y, Z, T, S, Z, X, Y a cycle? Yes!

15. 15 Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet This undirected graph is acyclic.

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18. 18 15.2 Directed Graphs

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24. 24 15.3 Trees

25. 25 Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet

26. 26 15.4 Networks

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32. 32 Semantic Network A vertex is related to another by some semantics, but not vise versa. For example, Taipei is a city of Taiwan. And, both Taiwan and Japan are countries of Asia. Taipei Taiwan cityOf AsiaJapan countryOf

33. 33 15.5 Graph Algorithms

34. 34 15.5.1 Iterators

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36. 36 x v Y Z T S Assume the vertices are entered in alphabetical order. Perform a breadth-first iteration from S. Ans: S, T, Z, Y, V, X

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41. 41 X V Y Z T S

42. 42 Assume the edges are inserted into the graph as follows: S, T (Same as T, S) S, Z T, Y T, Z V, X V, Z X, Y X, Z Perform a breadth-first iteration from S.

43. 43 queueVertex returned by next( ) S T, ZS Z, YT Y, V, XZ V, XY XV X enqueue dequeue

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47. 47 For a breadth-first iteration of a directed graph, the algorithm is the same, but “neighbor” takes into account the direction of the arrow. A B C D Starting at A, the neighbors of A are B and C, but not D. D NEVER gets enqueued, and never gets returned by next( ).

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49. 49 Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet

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51. 51 A Stack!

52. 52 X V Y Z T S

53. 53 Assume the edges are inserted into the graph as follows: S, T // Same as T, S S, Z T, Y T, Z V, X V, Z X, Y X, Z Perform a depth-first iteration from S.

54. 54 StackVertex returned by next( ) S Z, T S X, V, TZ Y, V, TX V, TY T V T Top Bottom

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58. 58 15.5.3 Generating a Minimum Spanning Tree

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77. 77 15.5.4 Finding the Shortest Path

78. 78 Shortest Path from A to B C 7 2 A 10 B 1. start with A. check A’s neighbors. save in priority queue (pq): ; 2. remove. check C’s neighbors. we found is shorter than 1. replace B’s weight sum with 9 2. insert to pq 3. make C the predecessor of B now, pq: 3. remove we got B. DONE!

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92. 92 15.6 A Network Class

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95. 95 15.6.1 Method Specifications of the Network Class

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111. A TreeMap This is called “neighborMap”.

114. keyvalue keyvalue

116. 116 For example, in the outer treeMap key “karen” has the neighborMap below as its value: (mark, 10.0) (w1, weight1) (don, 7.4) (w2, weight2) (courtney, 14.2) (w3, weight3) This neighborMap is by itself another inner treeMap as below: key value Don (7.4) Courtney (14.2) Mark (10.0)