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2. Breadth first search Structures for BFS Implementation (Δ, D) – graph Breadth first search

3. BFS by practice

4. We start from a source node. We want to see if we can reach a target node. We proceed by waves. target source Breadth first search

5. We start from a source node. We want to see if we can reach a target node. We proceed by waves. target source 3 waves total. Breadth first search

6. • a Queue, in which we add neighbours and from which we select the next one to visit • a Dictionary: to each node that we visited we bind as value its predecessor. 2 data structures pred: Danish pred: Awin pred: Shivam target pred: Pranil pred: Pranil pred: Pranil source pred: Robin Breadth first search

7. Implementation

8. (Δ, D) – graph

9. Distance The number of hops it takes to go from A to B is called the distance from A to B and denoted d(A,B). 2 hops 1 hop 3 hops d(Pranil,Som) = 3 Breadth first search

10. Distance The number of hops it takes to go from A to B is called the distance from A to B and denoted d(A,B). 2 1 We use the shortest path. d(Danish,Aram) = 2 Breadth first search

11. Diameter The diameter of a graph is the maximum distance between two vertices, and is denoted D. D = 5 Breadth first search

12. (Δ, D) – graph Let Δ be the maximum degree in a graph, i.e. the largest number of neighbours that any node can have. Let D be the diameter. For a given Δ and D, design the graph having as many nodes as possible. Example: we want to create a good network of workstations. Each workstation can be connected to at most 3, and the diameter must be 1. The best (3, 1) – graph has 4 nodes. Breadth first search

13. (Δ, D) – graph Let Δ be the maximum degree in a graph, i.e. the largest number of neighbours that any node can have. Let D be the diameter. For a given Δ and D, design the graph having as many nodes as possible. Practice: draw the best (3, 2) – graph possible. The best (3, 1) – graph has 4 nodes. Breadth first search

14. (Δ, D) – graph Practice: draw the best (3, 2) – graph possible. 5 6 7 8 10 Breadth first search