1 Chapter 15 Graphs, Trees, and Networks

2 Chap Undirected Graphs 15.2 Directed Graphs 15.3 Trees 15.4 Networks 15.5 Graph Algorithms Iterators Generating a Minimum Spanning Tree Finding the Shortest Path

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

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 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

Undirected Graphs

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

8 Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet

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

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Directed Graphs

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Trees

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Networks

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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

Graph Algorithms

Iterators

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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 X V Y Z T S

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

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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 Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet

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51 A Stack!

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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 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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Generating a Minimum Spanning Tree

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Finding the Shortest Path

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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A Network Class

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Method Specifications of the Network Class

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A TreeMap This is called “neighborMap”.

keyvalue keyvalue

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)