1 / 117

Chapter 6 : Graph – Part 1

Chapter 6 : Graph – Part 1. 교수 : 이상환 강의실 : 113,118 호 , 324 호 연구실 : 과학관 204 호 E-mail : sanghwan@kookmin.ac.kr Home : http://netapp.cs.kookmin.ac.kr 면담시간 : 수업시간 전후 1 시간. u. v. Graphs. G = (V,E) V is the vertex set. Vertices are also called nodes and points .

Download Presentation

Chapter 6 : Graph – Part 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 6 : Graph– Part 1 교수 : 이상환 강의실 : 113,118호, 324호 연구실 : 과학관 204호 E-mail : sanghwan@kookmin.ac.kr Home : http://netapp.cs.kookmin.ac.kr 면담시간 : 수업시간 전후 1시간

  2. u • v Graphs • G = (V,E) • Vis thevertexset. • Vertices are also callednodesandpoints. • Eis theedgeset. • Each edge connects two different vertices. • Edges are also calledarcsand lines. • Directed edge has an orientation(u,v).

  3. u v Graphs • Undirected edge has no orientation (u,v). • Undirected graph => no oriented edge. • Directed graph => every edge has an orientation.

  4. 2 3 8 1 10 4 5 9 11 6 7 Undirected Graph

  5. 2 3 8 1 10 4 5 9 11 6 7 Directed Graph (Digraph)

  6. 2 3 8 1 10 4 5 9 11 6 7 Applications—Communication Network • Vertex = city, edge = communication link.

  7. 2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 11 5 6 7 6 7 Driving Distance/Time Map • Vertex = city, edge weight = driving distance/time.

  8. 2 3 8 1 10 4 5 9 11 6 7 Street Map • Some streets are one way.

  9. n = 4 n = 1 n = 2 n = 3 Complete Undirected Graph Has all possible edges.

  10. Number Of Edges—Undirected Graph • Each edge is of the form (u,v), u != v. • Number of such pairs in an nvertex graph isn(n-1). • Since edge(u,v)is the same as edge(v,u),the number of edges in a completeundirected graph isn(n-1)/2. • Number of edges in an undirected graph is<= n(n-1)/2.

  11. Number Of Edges—Directed Graph • Each edge is of the form (u,v), u != v. • Number of such pairs in an nvertex graph isn(n-1). • Since edge(u,v)isnotthe same as edge(v,u),the number of edges in a complete directed graph isn(n-1). • Number of edges in a directed graph is<= n(n-1).

  12. 2 3 8 1 10 4 5 9 11 6 7 Vertex Degree Number of edges incident to vertex. degree(2) = 2, degree(5) = 3, degree(3) = 1

  13. 8 10 9 11 Sum Of Vertex Degrees Sum of degrees =2e(eis number of edges)

  14. 2 3 8 1 10 4 5 9 11 6 7 In-Degree Of A Vertex in-degree is number of incoming edges indegree(2) = 1, indegree(8) = 0

  15. 2 3 8 1 10 4 5 9 11 6 7 Out-Degree Of A Vertex out-degree is number of outbound edges outdegree(2) = 1, outdegree(8) = 2

  16. Sum Of In- And Out-Degrees each edge contributes 1 to the in-degree of some vertex and 1 to the out-degree of some other vertex sum of in-degrees = sum of out-degrees = e, where e is the number of edges in the digraph

  17. Sample Graph Problems • Path problems. • Connectedness problems. • Spanning tree problems.

  18. 2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 11 5 6 7 6 7 Path Finding Path between 1 and 8. Path length is20.

  19. 2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 11 5 6 7 6 7 Another Path Between 1 and 8 Path length is28.

  20. 2 3 8 1 10 4 5 9 11 6 7 Example Of No Path No path between 2 and9.

  21. Connected Graph • Undirected graph. • There is a path between every pair of vertices.

  22. 2 3 8 1 10 4 5 9 11 6 7 Example Of Not Connected

  23. 2 3 8 1 10 4 5 9 11 6 7 Connected Graph Example

  24. 2 3 8 1 10 4 5 9 11 6 7 Connected Components

  25. Connected Component • A maximal subgraph that is connected. • Cannot add vertices and edges from original graph and retain connectedness. • A connected graph has exactly 1component.

  26. 2 3 8 1 10 4 5 9 11 6 7 Not A Component

  27. 2 3 8 1 10 4 5 9 11 6 7 Communication Network Each edge is a link that can be constructed (i.e., a feasible link).

  28. Communication Network Problems • Is the network connected? • Can we communicate between every pair of cities? • Find the components. • Want to construct smallest number of feasible links so that resulting network is connected.

  29. 2 3 8 1 10 4 5 9 11 6 7 Cycles And Connectedness Removal of an edge that is on a cycle does not affect connectedness.

  30. Cycles And Connectedness 2 Connected subgraph with all vertices and minimum number of edges has no cycles. 3 8 1 10 4 5 9 11 6 7

  31. Tree • Connected graph that has no cycles. • nvertex connected graph withn-1edges.

  32. Spanning Tree • Subgraph that includes all vertices of the original graph. • Subgraph is a tree. • If original graph has n vertices, the spanning tree has n vertices and n-1 edges.

  33. 2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 8 11 5 6 2 7 6 7 Minimum Cost Spanning Tree • Tree cost is sum of edge weights/costs.

  34. A Spanning Tree Spanning tree cost = 51. 2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 8 11 5 6 2 7 6 7

  35. Minimum Cost Spanning Tree Spanning tree cost = 41. 2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 8 11 5 6 2 7 6 7

  36. A Wireless Broadcast Tree 2 Source = 1, weights = needed power. Cost = 4+2+5+7+8+6+4+4+8+3 = 51. 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 8 11 5 6 2 7 6 7

  37. Graph Representation • Adjacency Matrix • Adjacency Lists • Linked Adjacency Lists • Array Adjacency Lists

  38. 1 2 3 4 5 2 1 3 2 1 3 4 4 5 5 Adjacency Matrix • 0/1 n x nmatrix, wheren =# of vertices • A(i,j) = 1 iff (i,j) is an edge 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0

  39. 1 2 3 4 5 2 1 0 1 0 1 0 3 2 1 0 0 0 1 1 3 0 0 0 0 1 4 4 1 0 0 0 1 5 5 0 1 1 1 0 Adjacency Matrix Properties • Diagonal entries are zero. • Adjacency matrix of an undirected graph is symmetric. • A(i,j) = A(j,i)for alli andj.

  40. 1 2 3 4 5 2 1 0 0 0 1 0 3 2 1 0 0 0 1 1 3 0 0 0 0 0 4 4 0 0 0 0 1 5 5 0 1 1 0 0 Adjacency Matrix (Digraph) • Diagonal entries are zero. • Adjacency matrix of a digraph need not be symmetric.

  41. Adjacency Matrix • n2bits of space • For an undirected graph, may store only lower or upper triangle (exclude diagonal). • (n-1)n/2 bits • O(n)time to find vertex degree and/or vertices adjacent to a given vertex.

  42. 2 3 1 4 5 Adjacency Lists • Adjacency list for vertexiis a linear list ofvertices adjacent from vertexi. • An array ofnadjacency lists. aList[1] = (2,4) aList[2] = (1,5) aList[3] = (5) aList[4] = (5,1) aList[5] = (2,4,3)

  43. 2 aList[1] 2 4 3 [2] 1 5 [3] 5 1 [4] 5 1 4 aList[5] 2 4 3 5 Linked Adjacency Lists • Each adjacency list is a chain. Array Length = n # of chain nodes = 2e (undirected graph) # of chain nodes = e (digraph)

  44. 2 aList[1] 2 4 3 [2] 1 5 [3] 5 1 [4] 5 1 4 aList[5] 2 4 3 5 Array Adjacency Lists • Each adjacency list is an array list. Array Length = n # of list elements = 2e (undirected graph) # of list elements = e (digraph)

  45. Weighted Graphs • Cost adjacency matrix. • C(i,j) = cost of edge (i,j) • Adjacency lists =>each list element is a pair(adjacent vertex, edge weight)

  46. Number Of C++ Classes Needed • Graph representations • Adjacency Matrix • Adjacency Lists • Linked Adjacency Lists • Array Adjacency Lists • 3 representations • Graph types • Directed and undirected. • Weighted and unweighted. • 2 x 2 = 4graph types • 3 x 4 = 12 C++ classes

  47. 2 3 8 1 4 5 9 10 6 7 11 Graph Search Methods A vertex u is reachable from vertex v iff there is a path from v to u.

  48. 2 3 8 1 4 5 9 10 6 7 11 Graph Search Methods A search method starts at a given vertex v and visits/labels/marks every vertex that is reachable from v.

  49. Graph Search Methods • Many graph problems solved using a search method. • Path from one vertex to another. • Is the graph connected? • Find a spanning tree. • Etc. • Commonly used search methods: • Depth-first search. • Breadth-first search.

  50. Depth-First Search DFS(v) { Label vertex v as reached. for (each unreached vertex u adjacent from v) DFS(u); }

More Related