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Graph

Graph. Data Structures. Linear data structures: Array, linked list, stack, queue Non linear data structures: Tree, binary tree, graph and digraph. Graph. A graph is a non-linear structure (also called a network ) Unlike a tree or binary tree, a graph does not have a root

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Graph

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  1. Graph

  2. Data Structures • Linear data structures: • Array, linked list, stack, queue • Non linear data structures: • Tree, binary tree, graph and digraph

  3. Graph • A graph is a non-linear structure (also called a network) • Unlike a tree or binary tree, a graph does not have a root • Any node (vertex, v) can be connected to any other node by an edge (e) • Can have any number of edges (E) and nodes (set of Vertex V) • Application: the highway system connecting cities on a map a graph data structure

  4. u • v Graphs • G = (V,E) • V is the vertex set (nodes set). • Vertices are also called nodes and points. • E is the edge set. • Each edge connects two different vertices. • Edges are also called arcs and lines. • Directed edge has an orientation (u,v).

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

  6. 2 3 8 1 10 4 5 9 11 6 7 Undirected Graph G = (V, E) V = {1,2,…11} E = { (2,1), (4,1), …}

  7. 2 3 8 1 10 4 5 9 11 6 7 Directed Graph (Digraph) G = (V, E) V = {1,2,…11} E = { (1,2), (1,4), …}

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

  9. 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. • Find a shortest path from city 1 to city 5 • 125 (12) • 14675 (14)

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

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

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

  13. n = 4 n = 1 n = 2 n = 3 Complete directed Graph Has all possible edges.

  14. Number Of Edges—Directed Graph • Each edge is of the form (u,v), u != v. • Number of such pairs in an n vertex graph is n(n-1). • Since edge (u,v) is not the same as edge (v,u), the number of edges in a complete directed graph is n(n-1). • Number of edges in a directed graph is <= n(n-1).

  15. 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

  16. 8 10 9 11 Sum Of Vertex Degrees Sum of degrees = 2e (e is number of edges)

  17. 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

  18. 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

  19. 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

  20. Graph Operations And Representation

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

  22. 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 is 20.

  23. 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 is 28.

  24. 2 3 8 1 10 4 5 9 11 6 7 Example Of No Path No path between 2 and 9.

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

  26. 2 3 8 1 10 4 5 9 11 6 7 Example Of Not Connected

  27. 2 3 8 1 10 4 5 9 11 6 7 Connected Graph Example

  28. 2 3 8 1 10 4 5 9 11 6 7 Connected Components

  29. Connected Component • A maximal subgraph that is connected. • A connected graph has exactly 1 component.

  30. 2 3 8 1 10 4 5 9 11 6 7 Not A Component

  31. 2 3 8 1 10 4 5 9 11 6 7 Communication Network Each edge is a link that can be constructed

  32. Communication Network Problems • Is the network connected? • Can we communicate between every pair of cities? • Find the components. • Want to construct smallest number of links so that resulting network is connected (finding a minimum spanning tree).

  33. 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.

  34. 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

  35. Tree • Connected graph that has no cycles. • n vertex connected graph with n-1 edges.

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

  37. 1 2 3 4 5 2 1 3 2 1 3 4 4 5 5 Adjacency Matrix • 0/1 n x n matrix, where n = # 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

  38. 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 alliand j.

  39. 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.

  40. Adjacency Matrix • n2bits of space • For an undirected graph, may store only lower or upper triangle (exclude diagonal). • (n-1)n/2 bits

  41. 2 3 1 4 5 Adjacency Lists • Adjacency list for vertex i is a linear list ofvertices adjacent from vertex i. • An array of n adjacency lists. aList[1] = (2,4) aList[2] = (1,5) aList[3] = (5) aList[4] = (5,1) aList[5] = (2,4,3)

  42. 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)

  43. 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)

  44. 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 = 4 graph types • 3 x 4 = 12 C++ classes

  45. Abstract Class Graph template<class T> class graph { public: // ADT methods come here // implementation independent methods come here }; http://www.boost.org/doc/libs/1_42_0/libs/graph/doc/index.html

  46. Abstract Methods Of Graph // ADT methods virtual ~graph() {} virtual int numberOfVertices() const = 0; virtual int numberOfEdges() const = 0; virtual bool existsEdge(int, int) const = 0; virtual void insertEdge(edge<T>*) = 0; virtual void eraseEdge(int, int) = 0; virtual int degree(int) const = 0; virtual int inDegree(int) const = 0; virtual int outDegree(int) const = 0;

  47. Abstract Methods Of Graph // ADT methods (continued) virtual bool directed() const = 0; virtual bool weighted() const = 0; virtual vertexIterator<T>* iterator(int) = 0; virtual void output(ostream&) const = 0;

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