1. © 2004 Goodrich, Tamassia Trees1 Lecture 5 Graphs Topics Graphs ( 圖 ) Depth First Search ( 深先搜尋 ) Breast First Search ( 寬先搜尋 )

2. © 2004 Goodrich, Tamassia Trees2 Graphs A graph is a pair ( 對 ) (V, E), where V is a set of nodes, called vertices ( 節點 ) E is a collection ( 集合 ) of pairs of vertices, called edges ( 邊 ) Vertices and edges are positions ( 位置 ) and store elements Example: A vertex represents ( 代表 ) an airport ( 機場 ) and stores the three-letter airport code ( 碼 ) An edge represents a flight route ( 飛航路線 ) between two airports and stores the mileage ( 哩數 ) of the route ORD PVD MIA DFW SFO LAX LGA HNL 849 802 1387 1743 1843 1099 1120 1233 337 2555 142

3. © 2004 Goodrich, Tamassia Trees3 Graphs Edge types Directed edge ( 有向邊 ) ordered pair ( 有序對 ) of vertices (u,v) first vertex u is the origin ( 起點 ) second vertex v is the destination( 目的 地 ) e.g., a flight Undirected edge ( 有向邊 ) unordered pair of vertices (u,v) e.g., a flight route Directed graph( 有向圖 ) all the edges are directed e.g., route network Undirected graph( 有向圖 ) all the edges are undirected e.g., flight network ORDPVD flight AA 1206 ORDPVD 849 miles

4. © 2004 Goodrich, Tamassia Trees4 Graphs Applications Electronic circuits ( 電路 ) Printed circuit board ( 電路板 ) Integrated circuit (IC) Transportation ( 交通 ) networks Highway ( 高速公路 ) network Flight ( 飛航 ) network Computer networks Local area network Internet Web Databases Entity-relationship diagram ( 實 體 - 關聯圖 )

5. © 2004 Goodrich, Tamassia Trees5 Graphs Terminology ( 術語 ) End ( 終點 ) vertices (or endpoints) of an edge U and V are the endpoints of a Edges incident ( 接上 ) on a vertex a, d, and b are incident on V Adjacent ( 接鄰 ) vertices U and V are adjacent Degree ( 級 ) of a vertex X has degree 5 Parallel ( 並行 ) edges h and i are parallel edges Self-loop ( 自我迴圈 ) j is a self-loop XU V W Z Y a c b e d f g h i j

6. © 2004 Goodrich, Tamassia Trees6 P1P1 Graphs Terminology (cont.) Path ( 路徑 ) sequence ( 數列 ) of alternating ( 交替 ) vertices and edges begins with a vertex ends with a vertex each edge is preceded ( 之前 ) and followed ( 之後 ) by its endpoints Simple path ( 單純路徑 ) path such that all its vertices and edges are distinct ( 不同 ) Examples P 1 =(V,b,X,h,Z) is a simple path P 2 =(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple XU V W Z Y a c b e d f g hP2P2

7. © 2004 Goodrich, Tamassia Trees7 Graphs Terminology (cont.) Cycle circular sequence of alternating vertices and edges each edge is preceded and followed by its endpoints Simple cycle cycle such that all its vertices and edges are distinct Examples C 1 =(V,b,X,g,Y,f,W,c,U,a,  ) is a simple cycle C 2 =(U,c,W,e,X,g,Y,f,W,d,V,a,  ) is a cycle that is not simple C1C1 XU V W Z Y a c b e d f g hC2C2

8. © 2004 Goodrich, Tamassia Trees8 Graphs Notation n number of vertices m number of edges deg(v) degree of vertex v Property ( 性質 ) 1  v deg(v)  2m Proof: each edge is counted ( 計算 ) twice Property 2 In an undirected graph with no self-loops and no multiple ( 多數 ) edges m  n (n  1)  2 Proof: each vertex has degree at most (n  1) What is the bound ( 極限 ) for a directed graph? Example n  4 m  6 deg(v)  3

9. © 2004 Goodrich, Tamassia Trees9 Graphs Operations ( 操作 ) Vertices and edges are positions store elements Accessor ( 擷取 ) methods endVertices(e): an array of the two endvertices of e opposite(v, e): the vertex opposite ( 對方 ) of v on e areAdjacent(v, w): true iff v and w are adjacent replace(v, x): replace ( 取代 ) element at vertex v with x replace(e, x): replace element at edge e with x Update ( 更新 ) methods insertVertex(o): insert a vertex storing element o insertEdge(v, w, o): insert ( 插入 ) an edge (v,w) storing element o removeVertex(v): remove ( 刪除 ) vertex v (and its incident edges) removeEdge(e): remove edge e Iterator ( 重複子 ) methods incidentEdges(v): edges incident to v vertices(): all vertices in the graph edges(): all edges in the graph

10. © 2004 Goodrich, Tamassia Trees10 Graphs Represented ( 表現 ) by edge List ( 邊串列 ) Vertex list structure Vertex object Element ( 元素 ) Reference ( 參考 ) to position ( 位置 ) in vertex sequence ( 數列 ) Edge object element origin vertex object destination vertex object reference to position in edge sequence Vertex sequence sequence of vertex objects Edge sequence sequence of edge objects v u w ac b a z d uv w z bcd

11. © 2004 Goodrich, Tamassia Trees11 Graphs Represented by adjacency List ( 接鄰 串列 ) Edge list structure Incidence sequence for each vertex sequence of references to edge objects of incident edges Augmented ( 擴增式 ) edge objects references to associated ( 有關 ) positions in incidence sequences of end vertices u v w ab a uv w b

12. © 2004 Goodrich, Tamassia Trees12 Graphs Represented by adjacency matrix ( 接鄰矩陣 ) Augmented vertex objects Integer key (index, 索引 ) associated ( 關聯 ) with vertex 2D-array adjacency array Reference to edge object for adjacent vertices Null for nonadjacent vertices The “old fashioned ( 樣式 )” version just has 0 for no edge and 1 for edge u v w ab 012 0  1  2  a uv w 01 2 b

13. © 2004 Goodrich, Tamassia Trees13 Graphs n vertices, m edges no parallel edges no self-loops Bounds are “big-Oh” Edge List Adjacency List Adjacency Matrix Space n  m n2n2 incidentEdges( v ) mdeg(v)n areAdjacent ( v, w ) mmin(deg(v), deg(w))1 insertVertex( o ) 11n2n2 insertEdge( v, w, o ) 111 removeVertex( v ) mdeg(v)n2n2 removeEdge( e ) 111 Asymptotic Performance ( 漸進式效能 )

14. © 2004 Goodrich, Tamassia Trees14 Graphs A subgraph ( 子圖 ) S of a graph G is a graph such that The vertices of S are a subset ( 子集 ) of the vertices of G The edges of S are a subset of the edges of G A spanning subgraph ( 展 距子圖 ) of G is a subgraph that contains all the vertices of G Subgraph Spanning subgraph

15. © 2004 Goodrich, Tamassia Trees15 Graphs A graph is connected ( 相連 ) if there is a path between every pair of vertices A connected component ( 部分 ) of a graph G is a maximal connected subgraph ( 最大相連子 圖 ) of G Connected graph Non connected graph with two connected components

16. © 2004 Goodrich, Tamassia Trees16 Graphs A (free) tree is an undirected graph T such that T is connected T has no cycles This definition of tree is different from the one of a rooted tree A forest ( 森林 ) is an undirected graph without cycles The connected components of a forest are trees Tree Forest

17. © 2004 Goodrich, Tamassia Trees17 Graphs A spanning tree of a connected graph is a spanning subgraph that is a tree A spanning tree is not unique unless the graph is a tree Spanning trees have applications to the design of communication networks A spanning forest of a graph is a spanning subgraph that is a forest Graph Spanning tree

18. © 2004 Goodrich, Tamassia Trees18 Depth-First Search Depth-first search (DFS) is a general ( 通用 ) technique for traversing ( 走 ) a graph A DFS traversal of a graph G Visits ( 拜訪 ) all the vertices and edges of G Determines ( 決定 ) whether G is connected Computes ( 計算 ) the connected components of G Computes a spanning forest of G DFS on a graph with n vertices and m edges takes O(n  m ) time DFS can be further extended to solve other graph problems Find and report a path between two given vertices Find a cycle in the graph Depth-first search is to graphs what Euler tour is to binary trees

19. © 2004 Goodrich, Tamassia Trees19 Depth-First Search The algorithm uses a mechanism ( 機制 ) for setting ( 設定 ) and getting “labels” ( 標籤 ) of vertices and edges Algorithm DFS(G, v) Input graph G and a start vertex v of G Output labeling of the edges of G in the connected component of v as discovery edges and back edges setLabel(v, VISITED) for all e  G.incidentEdges(v) if getLabel(e)  UNEXPLORED w  opposite(v,e) if getLabel(w)  UNEXPLORED setLabel(e, DISCOVERY) DFS(G, w) else setLabel(e, BACK) Algorithm DFS(G) Input graph G Output labeling of the edges of G as discovery ( 發現 ) edges and back ( 回頭 ) edges for all u  G.vertices() setLabel(u, UNEXPLORED) for all e  G.edges() setLabel(e, UNEXPLORED) for all v  G.vertices() if getLabel(v)  UNEXPLORED DFS(G, v)

20. © 2004 Goodrich, Tamassia Trees20 Depth-First Search - Example DB A C ED B A C ED B A C E discovery edge back edge A visited vertex A Unexplored ( 未展開 ) vertex unexplored edge

21. © 2004 Goodrich, Tamassia Trees21 Depth-First Search - Example (cont.) DB A C E DB A C E DB A C E D B A C E

22. © 2004 Goodrich, Tamassia Trees22 Depth-First Search and Maze Traversal The DFS algorithm is similar to a classic ( 經典 ) strategy ( 策略 ) for exploring ( 探索 ) a maze ( 迷宮 ) We mark each intersection ( 交口 ), corner ( 角落 ) and dead end (vertex) visited We mark each corridor ( 走 廊 ) (edge ) traversed We keep track of the path back to the entrance ( 入口 ) (start vertex) by means of a rope ( 繩索 ) (recursion stack)

23. © 2004 Goodrich, Tamassia Trees23 Depth-First Search - Properties Property 1 DFS(G, v) visits all the vertices and edges in the connected component of v Property 2 The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v DB A C E

24. © 2004 Goodrich, Tamassia Trees24 Depth-First Search - Analysis Setting/getting a vertex/edge label takes O(1) time Each vertex is labeled twice once as UNEXPLORED once as VISITED Each edge is labeled twice once as UNEXPLORED once as DISCOVERY or BACK Method incidentEdges is called once for each vertex DFS runs in O(n  m) time provided ( 設若 ) the graph is represented by the adjacency list structure Recall that  v deg(v)  2m

25. © 2004 Goodrich, Tamassia Trees25 Depth-First Search - Path Finding We can specialize ( 特例化 ) the DFS algorithm to find a path between two given vertices u and z using the template ( 樣板 ) method pattern We call DFS(G, u) with u as the start vertex We use a stack S to keep track of the path between the start vertex and the current vertex As soon as destination vertex z is encountered ( 碰到 ), we return the path as the contents ( 內容 ) of the stack Algorithm pathDFS(G, v, z) setLabel(v, VISITED) S.push(v) if v  z return S.elements() for all e  G.incidentEdges(v) if getLabel(e)  UNEXPLORED w  opposite(v,e) if getLabel(w)  UNEXPLORED setLabel(e, DISCOVERY) S.push(e) pathDFS(G, w, z) S.pop(e) else setLabel(e, BACK) S.pop(v)

26. © 2004 Goodrich, Tamassia Trees26 Depth-First Search - Cycle Finding We can specialize the DFS algorithm to find a simple cycle using the template method pattern We use a stack S to keep track of the path between the start vertex and the current ( 現在 ) vertex As soon as a back edge (v, w) is encountered, we return the cycle as the portion ( 部分 ) of the stack from the top to vertex w Algorithm cycleDFS(G, v, z) setLabel(v, VISITED) S.push(v) for all e  G.incidentEdges(v) if getLabel(e)  UNEXPLORED w  opposite(v,e) S.push(e) if getLabel(w)  UNEXPLORED setLabel(e, DISCOVERY) pathDFS(G, w, z) S.pop(e) else T  new empty stack repeat o  S.pop() T.push(o) until o  w return T.elements() S.pop(v)

27. © 2004 Goodrich, Tamassia Trees27 Breadth-First Search Breadth-first search (BFS) is a general technique for traversing a graph A BFS traversal of a graph G Visits all the vertices and edges of G Determines whether G is connected Computes the connected components of G Computes a spanning forest of G BFS on a graph with n vertices and m edges takes O(n  m ) time BFS can be further ( 更 進一步 ) extended ( 擴充 ) to solve other graph problems Find and report a path with the minimum ( 最小 ) number of edges between two given vertices Find a simple cycle, if there is one

28. © 2004 Goodrich, Tamassia Trees28 Breadth-First Search The algorithm uses a mechanism for setting and getting “labels” of vertices and edges Algorithm BFS(G, s) L 0  new empty sequence L 0.insertLast(s) setLabel(s, VISITED ) i  0 while  L i.isEmpty() L i  1  new empty sequence for all v  L i.elements() for all e  G.incidentEdges(v) if getLabel(e)  UNEXPLORED w  opposite(v,e) if getLabel(w)  UNEXPLORED setLabel(e, DISCOVERY ) setLabel(w, VISITED ) L i  1.insertLast(w) else setLabel(e, CROSS ) i  i  1 Algorithm BFS(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u  G.vertices() setLabel(u, UNEXPLORED ) for all e  G.edges() setLabel(e, UNEXPLORED ) for all v  G.vertices() if getLabel(v)  UNEXPLORED BFS(G, v)

29. © 2004 Goodrich, Tamassia Trees29 Breadth-First Search - Example C B A E D discovery edge cross ( 反 ) edge A visited vertex A unexplored vertex unexplored edge L0L0 L1L1 F CB A E D L0L0 L1L1 F CB A E D L0L0 L1L1 F

30. © 2004 Goodrich, Tamassia Trees30 Breadth-First Search - Example (cont.) CB A E D L0L0 L1L1 F CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2

31. © 2004 Goodrich, Tamassia Trees31 Breadth-First Search - Example (cont.) CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2 CB A E D L0L0 L1L1 F L2L2

32. © 2004 Goodrich, Tamassia Trees32 Breadth-First Search - Properties Notation ( 符號 ) G s : connected component of s Property 1 BFS(G, s) visits all the vertices and edges of G s Property 2 The discovery edges labeled by BFS(G, s) form a spanning tree T s of G s Property 3 For each vertex v in L i The path of T s from s to v has i edges Every path from s to v in G s has at least i edges CB A E D L0L0 L1L1 F L2L2 CB A E D F

33. © 2004 Goodrich, Tamassia Trees33 Breadth-First Search - Analysis Setting/getting a vertex/edge label takes O(1) time Each vertex is labeled twice once as UNEXPLORED once as VISITED Each edge is labeled twice once as UNEXPLORED once as DISCOVERY or CROSS Each vertex is inserted once into a sequence L i Method incidentEdges is called once for each vertex BFS runs in O(n  m) time provided the graph is represented by the adjacency list structure Recall that  v deg(v)  2m

34. © 2004 Goodrich, Tamassia Trees34 Breadth-First Search - Applications Using the template method pattern, we can specialize the BFS traversal of a graph G to solve ( 解決 ) the following problems in O(n  m) time Compute the connected components of G Compute a spanning forest of G Find a simple cycle in G, or report that G is a forest Given two vertices of G, find a path in G between them with the minimum number of edges, or report that no such path exists ( 存在 )

35. © 2004 Goodrich, Tamassia Trees35 DFS vs. BFS CB A E D L0L0 L1L1 F L2L2 CB A E D F DFSBFS ApplicationsDFSBFS Spanning forest, connected components, paths, cycles  Shortest paths  Biconnected components 

36. © 2004 Goodrich, Tamassia Trees36 DFS vs. BFS (cont.) Back edge (v,w) w is an ancestor ( 祖 先 ) of v in the tree of discovery edges Cross edge (v,w) w is in the same level as v or in the next level in the tree of discovery edges CB A E D L0L0 L1L1 F L2L2 CB A E D F DFSBFS