1.  Last lesson  Graphs  Today  Graphs (Implementation, Traversal)

2. Implementation  There are different approaches  Adjacency list  Each node has a list of outgoing edges  Perhaps also list of incoming edges  nodes stored as array, or list, or symbol table (with label as key)  Adjacency matrix  Two-dimensional array A  Entry A[i][j] is 1 (or cost) if there is an edge from v i to v j  Often also keep symbol table to map label to index of node  Incidence matrix  Two-dimensional array I  Entry I[i][j] is 1 (or cost) if node v i is connected with edge e j

3. Adjacency List NodesAdjacencies ab, e ba, c, e cb, d dc, e ea, b, d b c d e a

4. Adjacency List Adjacency list representation of the graph shown above. The elements in list i represent nodes adjacent to i and the cost of the connecting edge.

5. Adjacency and Incidence Matrix  Adjacency Matrix  Incidence Matrix abcde a01001 b10101 c01010 d00101 e11010 123456 a110000 b011010 c000011 d000101 e101100 b c d e a 1 2 3 4 5 6

6. Graph Data Structure: Basic Operations  Insert node  Insert edge from one node to another  Delete edge  Delete node  Must also delete adjacent edges  Given edge, find start and finish node  Go though all nodes  Given node, go through edges out of it  Given node, go through edges into it

7. Complex Operations  Find whether there is a path from one node to another  If edges have costs, find least cost path  Find all nodes reachable from one node  Find whether there is a cycle  Find a sequence of nodes that is consistent with all directed edges  No edge goes backwards in the sequence

8. Graph Traversal  Do something with each node  “visit” the node  Different algorithms visit in different orders  Can be used as basis for many code cliches  E.g. count the nodes  E.g. find maximum/minimum/sum of a quantity  Also the basis for important graph calculations

9. Overview  Keep a collection of nodes that are waiting to be visited  Start at one node s  When you visit node v  Consider all nodes that can be reached in one step from v  They are the finish for each edge starting at v  Any such node can be added to the waiting collection  Unless its already there  Vital choice: how to choose next node from the waiting collection?  Breadth-first Search (BFS)  Depth-first Search (DFS)  Topological Sort

10. Breadth-first Search  Waiting nodes kept in queue  Initially, insert one node  Dequeue a node  Visit it  Enqueue any adjacent node that is not already visited or waiting  Repeat till queue is empty  This will visit all nodes which can be reached from initial one  Repeat with a new (as yet unvisited) initial node  Do BFS from each  Until all nodes have been visited

11. BFS Example         rstu vwxy

12.   0      rstu vwxy s Q:

13. BFS Example 1  0 1     rstu vwxy w Q: r

14. BFS Example 1  0 1 2 2   rstu vwxy r Q: tx

15. BFS Example 1 2 0 1 2 2   rstu vwxy Q: txv

16. BFS Example 1 2 0 1 2 2 3  rstu vwxy Q: xvu

17. BFS Example 1 2 0 1 2 2 3 3 rstu vwxy Q: vuy

18. BFS Example 1 2 0 1 2 2 3 3 rstu vwxy Q: uy

19. BFS Example 1 2 0 1 2 2 3 3 rstu vwxy Q: y

20. BFS Example 1 2 0 1 2 2 3 3 rstu vwxy Q: Ø

21. Implementation  Local variables for the traversal routine  Queue of waiting nodes  Currently visited node  Need to find all nodes adjacent to it  Call routine of Graph class  Efficient with Adjacency List representation  Way to test if a node has been visited or is waiting  Symbol table as Set  Or, keep extra boolean in each node object  Run-time cost  O(E+V) with adjacency list representation