G64ADS Advanced Data Structures

1. G64ADSAdvanced Data Structures Graph Algorithms 1

2. Going from A to B … • What data structure to use? • How to think about the problem? C A B F D E H G 2

3. Going from A to B … Strip away the irrelevant details … Vertices C A B F D E H G 3

4. Going from A to B … Strip away the irrelevant details … Edges C A B F D E H G 4

5. Going from A to B … Strip away the irrelevant details … Weights 4 C 3 A B 3 F 4 4 D E 4 5 6 H G 5

6. Going from A to B … Strip away the irrelevant details … Now, much simpler … 4 C 3 A B 3 F 4 4 D E 4 5 6 H G 6

7. Graphs • Are a tool for modeling real world problems. • Allow us to abstract details and focus on the problem. • We can represent our domain as graphs apply algorithms to solve our problem.

8. Simple Graphs

9. Directed Graphs

10. 4 3 3 4 4 4 5 6 Weighted Graphs

11. Path and Cycle

12. Representation of Graphs • Adjacency matrix • Adjacency list

13. Representation of Graphs • Adjacency list

14. Topological Sort • Ordering of vertices in a directed acyclic graph, such that if there is a path from vi to vj, the vj appears after vi in the ordering

15. Topological Sort • Application • Given a number of tasks, there are often a number of constraints between the tasks: • task A must be completed before task B can start • These tasks together with the constraints form a directed acyclic graph • A topological sort of the graph gives an order in which the tasks can be scheduled while still satisfying the constraints

16. Topological Sort • Application • Consider someone getting ready to go out for dinner • He must wear the following: • jacket, shirt, briefs, socks, tie, Blackberry, etc. • There are certain constraints: • the pants really should go on after the briefs, • the Blackberry must be clipped on the belt • socks are put on before shoes

17. Topological Sort • Application • Getting dress graph

18. Topological Sort • Application • Getting dress graph • One topological sort is: • briefs, pants, wallet, keys, belt, Blackberry, socks, shoes, shirt, tie, jacket, iPod, watch

19. Topological Sort • Application • Getting dress graph another possible solution: briefs, socks, pants, shirt, belt, tie, jacket, wallet, keys, Blackberry, iPod, watch, shoes • One topological sort is: • briefs, pants, wallet, keys, belt, Blackberry, socks, shoes, shirt, tie, jacket, iPod, watch Not unique,

20. Topological Sort • Course prerequisite structure represented in an acyclic graph

21. Topological Sort • Topological Sort (Relabel): Given a directed acyclic graph (DAG), relabel the vertices such that every directed edge points from a lower-numbered vertex to a higher numbered one 6 1 5 8 3 2 7 4 3 5 2 7 8 4 1 6

22. Topological Sort • Topological Sort (rearrange): Given a directed acyclic graph (DAG), rearrange its vertices on a horizontal line such that all the directed edges points from left to right. 6 1 5 8 3 2 7 4 7 5 6 2 1 4 8 3

23. web.mit.edu/jorlin/www/15.082/Animations/Topological_Sorting.pptweb.mit.edu/jorlin/www/15.082/Animations/Topological_Sorting.ppt Following slides are taken from

24. 1 2 3 4 5 6 7 8 Node Indegree 2 2 3 2 1 1 0 2 7 0 next LIST Initialization 6 1 “Next” will be the label of nodes in the topological order. Determine the indegree of each node LIST is the set of nodes with indegree of 0. 5 8 3 2 7 4

25. LIST Select a node from LIST 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 5 8 3 2 7 7 4 1 0 1 next 1 2 3 4 5 6 7 8 Node Indegree 2 2 3 1 2 0 1 1 0 2 7 5

26. LIST Select a node from LIST 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 8 3 2 7 7 4 1 2 0 1 next 1 2 3 4 5 6 7 8 Node Indegree 2 2 1 3 0 1 2 0 1 1 0 0 2 4 5 7 6

27. LIST Select a node from LIST 3 6 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 8 3 2 7 7 4 1 3 1 2 0 next 1 2 3 4 5 6 7 8 Node Indegree 1 2 2 1 0 3 2 1 0 0 1 0 1 0 2 4 7 5 6 2

28. LIST Select a node from LIST 3 6 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 4 1 3 2 0 next 1 2 3 4 5 6 7 8 Node Indegree 0 1 2 1 0 2 3 0 1 2 1 0 1 0 0 2 4 5 7 2 6 1

29. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 1 2 5 0 4 3 next 1 2 3 4 5 6 7 8 Node Indegree 2 1 0 2 0 1 2 3 2 1 0 0 1 0 1 0 2 1 4 7 5 1 2 6

30. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 4 1 6 1 0 2 4 5 3 6 next 1 2 3 4 5 6 7 8 Node Indegree 1 0 2 0 1 2 3 2 1 2 0 1 0 1 1 0 0 1 0 2 6 4 1 2 5 7 8

31. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 7 2 5 5 2 2 8 8 3 4 7 7 4 4 1 6 1 4 5 6 2 3 7 0 next 1 2 3 4 5 6 7 8 Node Indegree 2 1 0 0 1 2 0 3 2 1 0 1 2 0 1 0 1 0 2 1 0 8 3

32. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 7 8 2 5 5 2 2 8 8 3 3 4 7 7 4 4 1 6 4 1 6 7 8 5 3 2 0 next List is empty. The algorithm terminates with a topological order of the nodes 1 2 3 4 5 6 7 8 Node Indegree 2 0 1 0 1 2 0 2 3 1 2 0 1 0 1 0 1 0 2 0 1 3

33. Example • Consider the following DAG with six vertices

34. Example • Let us define the table of in-degrees

35. Example • And a queue into which we can insert vertices 1 and 6 Queue

36. Example • We dequeue the head (1), decrement the in-degree of all adjacent vertices, and enqueue 2 Queue Sort 1

37. Example • We dequeue 6 and decrement the in-degree of all adjacent vertices Queue Sort 1, 6

38. Example • We dequeue 2, decrement, and enqueue vertex 5 Queue Sort 1, 6, 2

39. Example • We dequeue 5, decrement, and enqueue vertex 3 Queue Sort 1, 6, 2, 5

40. Example • We dequeue 3, decrement 4, and add 4 to the queue Queue Sort 1, 6, 2, 5, 3

41. Example • We dequeue 4, there are no adjacent vertices to decrement the in degree Queue Sort 1, 6, 2, 5, 3, 4

42. Example • The queue is now empty, so a topological sort is 1, 6, 2, 5, 3, 4

43. Topological Sort

44. Topological Sort

45. Shortest-Path Algorithm • Find the “shortest” path from point A to point B • 􀂄“Shortest” in time, distance, cost, … • 􀂄Numerous applications • Map navigation • Flight itineraries • Circuit wiring • Network routing

46. Shortest Path Problems • Input is a weighted graph where each edge (vi,vj) has cost ci,j to traverse the edge • 􀂄Cost of a path v1v2…vN • Unweighted path length is N-1, number of edges on path

47. Shortest Path Problems • Single-source shortest path problem • Given a weighted graph G=(V,E), and a start vertex s, find the minimum weighted path from s to every other vertex in G

48. Negative Weights • Graphs can have negative weights • e.g., arbitrage • Shortest positive-weight path is a net gain • Path may include individual losses • Problem: Negative weight cycles • Allow arbitrarily-low path costs • Solution • Detect presence of negative-weight cycles

49. Shortest Path Problems • Unweighted shortest-path problem: O(|E|+|V|) • Weighted shortest-path problem • No negative edges: O(|E| log |V|) • Negative edges: O(|E|·|V|) • Acyclic graphs: O(|E|+|V|) • No asymptotically faster algorithm for single-source/single-destination shortest path problem

50. Unweighted Shortest Paths • No weights on edges • Find shortest length paths • Same as weighted shortest path with all weights equal • Breadth-first search