Prof. Qing Wang

1. Lecture-10~11 Lecture Notes: Data Structures and Algorithms Chapter 9 Graph Prof. Qing Wang

2. Table of Contents • Graph Definition and Concepts • Graph ADT and Storage • Graph Traversal and Connectivity • Mini Spanning Tree • Shortest Path • Topological Sorting and Critical Path • Summary Prof. Q.Wang

3. 9.1 Graph Definition and Concepts • Graph • Consists of a set, whose members are called the vertices of G, together with a set E of pairs of distinct vertices from G. • Vertices set • Edge (arc) set Graph＝( V, E ) V = { x | x  Data Object} E = {(x, y) | x, y V } E = {<x, y> | x, y V && Path (x, y)} Prof. Q.Wang

4. Examples Prof. Q.Wang

5. Concepts • Vertex • Arc • Tail or Initial node • Head or Terminal node • Edge v1 v2 G1 v3 v4 v1 v2 G1=(V1,{A1}) V1={v1,v2,v3,v4} A1={<v1,v2>,<v1,v3>,<v3,v4>,<v4,v1>} G2 v3 v4 v5 G2=(V2,{E2}) V2={v1,v2,v3,v4,v5} E2={(v1,v2),(v1,v4),(v2,v3),(v2,v5),(v3,v4),(v3,v5)} Prof. Q.Wang

6. Directed graph • Ordered pair <x,y> • Undirected graph • Unordered pair (x,y) • Network • Weighted pair <x,y> or (x,y) Prof. Q.Wang

7. Completed graph • Completed directed graph • Completed undirected graph • Sparse graph • Dense graph • Sub-graph Prof. Q.Wang

8. Adjacent • Incident • Degree TD(v) • In Degree ID(v) • Out Degree OD(v) TD(v)=ID(v)+OD(v) Prof. Q.Wang

9. Path • Cycle • Simple path • Simple Cycle • Path Length • Generic length • Weighted length • Connected graph • Connected component • Max connected sub-graph • Strongly connected component • Weakly connected component Prof. Q.Wang

10. Prof. Q.Wang

11. Spanning tree • Connected graph • Mini connected sub-graph • Have all the vertices of graph (n) • Only n-1 edges • Without cycle • Spanning forest • Unconnected graph or directed graph • Directed spanning trees A B C A D F B C F E D E Prof. Q.Wang

12. 9.2 ADT and Storage of Graph class Graph { public: Graph ( ); voidInsertVertex ( Type & vertex ); voidInsertEdge( intv1, intv2, intweight ); voidRemoveVertex ( intv ); voidRemoveEdge ( intv1, intv2 ); intIsEmpty ( ); TypeGetWeight ( int v1, int v2 ); intGetFirstNeighbor ( int v ); intGetNextNeighbor ( intv1, intv2 ); } Prof. Q.Wang

13. Multi-linked list v1 v2 v1 v2 G1 G2 v3 v3 v4 v4 v5 v1 v1 v2 v4 v2 v3 v3 v4 v5 Prof. Q.Wang

14. Adjacency Matrix • Graph A = (V, E) • n vertices A.vertex[n] • A.edge[n][n] • Symmetric for undirected graph Prof. Q.Wang

15. Degree and adjacency matrix Prof. Q.Wang

16. Network Prof. Q.Wang

17. // Graph class by adjacency matrix const int MaxEdges = 50; const int MaxVertices = 10; template <class NameType, class DistType> classGraph { private: SeqList<NameType>VerticesList (MaxVertices); DistType Edge[MaxVertices][MaxVertices]; int CurrentEdges; intFindVertex ( SeqList<NameType> & L; const NameType &vertex ) { return L.Find (vertex); } Prof. Q.Wang

18. intGetVertexPos ( Const NameType &vertex ) { returnFindVertex (VerticesList, vertex );} public: Graph ( intsz = MaxNumEdges ); intGraphEmpty ( ) const { return VerticesList.IsEmpty ( );} int GraphFull( ) const{ returnVerticesList.IsFull( ) || CurrentEdges ==MaxEdges;} intNumberOfVertices ( ) { returnVerticesList.last +1;} intNumberOfEdges ( ) { returnCurrentEdges;} Prof. Q.Wang

19. NameType GetValue ( inti ) { returni >= 0 &&i <= VerticesList.last ?VerticesList.data[i] :NULL;} DistType GetWeight ( intv1, intv2 ); intGetFirstNeighbor ( intv ); intGetNextNeighbor ( intv1, int v2 ); void InsertVertex ( NameType &vertex ); voidInsertEdge ( int v1, int v2, DistTypeweight ); voidRemoveVertex ( int v ); voidRemoveEdge ( intv1, intv2 ); } Prof. Q.Wang

20. //Implementation of functions //constructor template <class NameType, class DistType> Graph<NameType, DistType> :: Graph( int sz) { for ( int i = 0;i < sz;i++ ) for ( intj = 0;j < sz;j++ ) Edge[i][j] = 0; CurrentEdges = 0; } //get the weight of arc (v1, v2) template <class NameType, class DistType> DistType Graph<NameType, DistType> :: GetWeight( int v1, int v2 ) { Prof. Q.Wang

21. if ( v1 != -1 &&v2 != -1 ) returnEdge[v1][v2]; else return 0; } //Get the first adjacent node template <class NameType, class DistType> intGraph<NameType, DistType>:: GetFirstNeighbor ( const intv ) { if ( v != -1 ) { for ( intcol = 0;col < VerticesList.last;col++ ) if ( Edge[v][col] > 0 && Edge[v][col] < max ) returncol; } Prof. Q.Wang

22. return-1; } //Get the next adjacent of v1 after v2 template <class NameType, class DistType> int Graph<NameType, DistType> :: GetNextNeighbor ( intv1, intv2 ) { intcol; if ( v1 != -1 &&v2 != -1 ) { for ( col = v2+1;col < VerticesList.last;col++ ) if ( Edge[v1][col] > 0 && Edge[v1][col] < max ) return col; } return-1; } Prof. Q.Wang

23. Adjacency List • Undirected graph Prof. Q.Wang

24. Out arcs In arcs Adjacency list Reverse adjacency list • Directed graph • Adjacency list • Reverse adjacency list Prof. Q.Wang

25. Network • Adjacency list • Reverse adjacency list Adjacency list Prof. Q.Wang

26. //Graph class by adjacency list const intDefaultSize = 10; template <class DistType> classGraph; template <class DistType> struct Edge { friend classGraph<NameType, DistType>; intdest; DistTypecost; Edge<DistType> *link; Edge ( ) { } Edge ( int D, DistTypeC ) : dest (D), cost (C), link (NULL) { } int operator != ( Edge<DistType>& E ) const { returndest != E.dest;} } Prof. Q.Wang

27. template <class NameType, class DistType> structVertex{ friend classGraph<NameType, DistType>; NameType data; Edge<DistType> *adj; } template <class NameType, class DistType> class Graph { private: Vertex<NameType, DistType> *NodeTable; intNumVertices; intMaxVertices; int NumEdges; Prof. Q.Wang

28. int GetVertexPos ( NameType & vertex ); public: Graph ( intsz ); ~Graph ( ); intGraphEmpty ( ) const { return NumVertices == 0;} int GraphFull ( ) const { returnNumVertices == MaxVertices;} NameType GetValue ( int i ) { returni >= 0 &&i < NumVertices? NodeTable[i].data: NULL;} intNumberOfVertices ( ) { return NumVertices;} intNumberOfEdges ( ) { returnNumEdges;} Prof. Q.Wang

29. void InsertVertex ( NameType &vertex ); voidRemoveVertex ( intv ); voidInsertEdge ( int v1, int v2, DistTypeweight ); voidRemoveEdge ( int v1, int v2 ); DistType GetWeight ( int v1, int v2 ); intGetFirstNeighbor ( int v ); intGetNextNeighbor ( int v1, intv2 ); } Prof. Q.Wang

30. template <class NameType, class DistType> //Constructor Graph<NameType, DistType> :: Graph ( intsz = DefaultSize ) : NumVertices (0), MaxVertices (sz), NumEdges (0){ intn, e, k, j;NameTypename, tail, head; DistTypeweight; NodeTable = newVertex<Nametype>[MaxVertices]; cin >> n; for ( inti = 0; i < n;i++) {cin >> name;InsertVertex ( name );} cin >> e; for ( i = 0;i < e; i++) { cin >> tail >> head >> weight; k = GetVertexPos ( tail );j = GetVertexPos ( head ); InsertEdge ( k, j, weight ); } } Prof. Q.Wang

31. //Deconstructor template <class NameType, class DistType> Graph<NameType, DistType> :: ~Graph ( ) { for ( inti = 0;i < NumVertices;i++ ) { Edge<DistType> *p = NodeTable[i].adj; while ( p != NULL ) { NodeTable[i].adj = p→link;deletep; p = NodeTable[i].adj; } } delete [ ] NodeTable; } Prof. Q.Wang

32. //Implementation of functions //get the serial number ofvertex template <class NameType, class DistType> intGraph<NameType, DistType> :: GetVertexPos ( const NameType &vertex ) { for ( inti =0; i < NumVertices;i++ ) if ( NodeTable[i].data == vertex ) returni; return-1; } Prof. Q.Wang

33. //Get the first adjacent ofv template <Class NameType, class DistType> intGraph<NameType, DistType> :: GetFirstNeighbor ( int v ) { if ( v != -1 ) { Edge<DistType> *p = NodeTable[v].adj; if ( p != NULL ) returnp→dest; } return-1; } Prof. Q.Wang

34. //Get the next adjacent ofv1 afterv2 template <Class NameType, class DistTypeType> intGraph<NameType, DistType> :: GetNextNeighbor ( intv1, int v2 ) { if ( v1 != -1 ) { Edge<DistType> *p = NodeTable[v1].adj; while ( p != NULL ) { if ( p→dest == v2 &&p→link != NULL ) returnp→link→dest; elsep = p→link; } } return-1; } Prof. Q.Wang

35. //Get the cost betweenv1 andv2 template <Class NameType, class DistType> DistType Graph<NameType, DistType> :: GetWeight ( int v1, int v2) { if ( v1 != -1 &&v2 != -1 ) { Edge<DistType> *p = NodeTable[v1].adj; while ( p != NULL ) { if ( p→dest == v2) returnp→cost; elsep = p→link; } } return 0; } Prof. Q.Wang

36. Adjacency Multi-list • Undirected graph • Edge • Mark • Vertex1, Vertex2 • Path1, Path2 (pointers to next edge incident with vertex1 or vertex2) • Vertex mark vertex1 vertex2 path1 path2 data Firstout Prof. Q.Wang

37. Example1 of AML Prof. Q.Wang

38. Directed graph • Edge • Mark • Vertex1 (initial node) • Vertex2 (terminal node) • Path1 (link to next arc beginning from vertex1) • Path2 (link to next arc ending at vertex2) • Vertex • Data • Firstin (point to the first arc starting from this vertex) • Firstout (point to the first arc ending at this vertex) mark vertex1 vertex2 path1 path2 data Firstin Firstout Prof. Q.Wang

39. Example2 of AML Prof. Q.Wang

40. Cross linked list mark tailvex headvex hlink tlink 0 1 v1 v2 data Firstin Firstout 2 3 v3 v4 0 v1 m 0 1 m 0 2 1 v2 2 v3 m 2 0 m 2 3 3 v4 m 3 0 m 3 1 m 3 2 Prof. Q.Wang

41. 9.3 Traversal and Connectivity • Traversal • Requirement • Visit each vertex one time and only one time • Approaches • Depth-first traversal • Breadth-first traversal • Auxiliary Boolean array • Visited[] Prof. Q.Wang

42. Depth First Search (DFS) • Steps • Searching • Backtracking (recursion involved) Searching Backtracking Prof. Q.Wang

43. // DFS algorithm template<class NameType, class DistType> voidGraph <NameType, DistType> :: DFS ( ) { int * visited = newint [NumVertices]; for ( inti = 0;i < NumVertices;i++ ) visited [i] = 0; DFS (0, visited ); delete [ ] visited; } template<class NameType, class DistType> voidGraph<NameType, DistType> :: DFS ( const intv, int visited [ ] ) { Prof. Q.Wang

44. cout << GetValue (v) << ‘ ’; //visit v visited[v] = 1; //markingv intw = GetFirstNeighbor (v); //Get the first adjacent of v : w while ( w != -1 ) { if ( !visited[w] ) DFS ( w, visited ); //if w has not been visited, DFS(w) w = GetNextNeighbor ( v, w ); //Get the next adjacent of v afterw } } • Time Analysis • O(n+e) for adjacency list • O(n2) for adjacency matrix Prof. Q.Wang

45. Breadth First Search (BFS) • Steps • Search • Search in next level (queue involved) Searching Prof. Q.Wang

46. //BFS algorithm template<class NameType, class DistType> voidGraph <NameType, DistType> :: BFS ( intv ) { int * visited = new int[NumVertices]; for ( inti = 0;i < NumVertices;i++ ) visited[i] = 0; cout << GetValue (v) << ' '; visited[v] = 1; Queue<int> q;q.EnQueue (v); while ( !q.IsEmpty ( ) ) { v = q.DeQueue ( ); intw = GetFirstNeighbor (v); Prof. Q.Wang

47. while ( w != -1 ) { if ( !visited[w] ) { cout << GetValue (w) << ‘ ’; visited[w] = 1;q.EnQueue (w); } w = GetNextNeighbor (v, w); } } delete [ ] visited; } • Time Analysis • d0 + d1 + … + dn-1 = O(e) for adjacency list, di is the degree of node i • O(n2) for adjacency matrix Prof. Q.Wang

48. Connected component • Connected graph • Visit all the nodes within one searching • Spanning tree • Depth first spanning tree • Breadth first spanning tree • Unconnected graph • Call DFS or BFS several times until all the nodes have been visited • Spanning forest Prof. Q.Wang

49. Unconnected graph Example 0 A 4 3 2 1 1 B 5 0 2 C 0 3 D 0 4 E 6 5 0 5 F 6 4 1 6 G 5 4 7 H 9 8 8 I 9 7 9 J 8 7 10 K 13 12 11 11 L 14 10 12 M 14 10 13 N 10 Connected components of the unconnected graph 14 O 12 11 Prof. Q.Wang

50. //Connected components template<class NameType, class DistType> voidGraph<NameType, DistType> :: Components ( ) { int *visited = new int[NumVertices]; for ( int i = 0; i < NumVertices;i++ ) visited[i] = 0; for ( i = 0;i < NumVertices;i++ ) if ( !visited[i] ) { DFS ( i, visited ); OutputNewComponent ( ); //output a connected component } delete [ ] visited; } Prof. Q.Wang