1. Algorithms and Data Structures
Lecture 9

2. Agenda: Graph traverse Breadth-first search algorithm
Depth-first search algorithm
Topological sort

3. Graphs: traverse
Sometimes we need to perform some procedure on all vertices of a graph
Task can be accomplished visiting each vertex of a graph (from vertex from vertex) and performing required procedure
Such task is known as graph traverse
Let’s examine two well-known graph traverse algorithms: breadth-first search and depth-first search algorithms

4. Graphs: traverse
The ideas behind graph traverse algorithms are the following:
For given graph G and some vertex s, algorithm builds paths from vertex s to all vertices, attainable from s
Paths built from a given vertex constitute a tree
Some vertices of a graph (unattainable from vertex s) may remain out of a tree; procedure is repeated until all the vertices of a graph will be covered by some tree
Trees (of a given graph) constitute a forest

5. Graphs: traverse

6. Graphs: traverse

7. Graphs: traverse

8. Graphs: breadth-first search
Breadth-first search is one of graph traverse algorithms
Algorithm is often used by other algorithms
Algorithm is applicable for both directed and undirected graphs

9. Graphs: breadth-first search
Input: graph G
Output: breadth-first forest
Any vertex of a G may be colored in white, gray or black color
White vertices are unvisited, gray – discovered vertices, black – already processed vertices (finished)
Q – auxiliary storage (queue) of FIFO type; holds discovered vertices (gray)
Initially, G consists of white vertices only; finally, all the vertices of a G are black

10. Graphs: breadth-first search
Step 1: all the vertices are marked as white
Step 2: start with any white vertex s; it is marked as gray and added to the Q
Step 3: get first vertex x from the Q
Step 4: each adjacent to x white vertex is added to the Q and marked as gray
Step 5: vertex x is removed from the Q, marked as black and “some” vertex processing procedure is performed (e.g. printing)
Step 6: if Q is non-empty go to step 3
Step 7: continue from step 2 until G has no more white vertices

11. Graphs: breadth-first search, sample

12. Graphs: breadth-first search, sample

13. Graphs: breadth-first search, sample

14. Graphs: breadth-first search, sample

15. Graphs: breadth-first search, sample

16. Graphs: breadth-first search, sample

17. Graphs: breadth-first search, sample on usage

18. Graphs: breadth-first search, sample on usage

19. Graphs: breadth-first search, sample on usage

20. Graphs: breadth-first search, sample on usage

21. Graphs: breadth-first search
For given graph G and some white vertex s, algorithm builds paths from vertex s to all vertices, attainable from s
Paths are shortest
Paths built from a given vertex constitute a breadth-first tree
Breadth-first trees (constructed while considering distinct white vertices) constitute a breadth-first forest
Changing order of considered white vertices may produce another breadth-first forest

22. Graphs: breadth-first search

23. Graphs: breadth-first search

24. Graphs: depth-first search
Depth-first search is one of graph traverse algorithms
Algorithm is often used by other algorithms
Algorithm is applicable for both directed and undirected graphs

25. Graphs: depth-first search
Input: graph G
Output: depth-first forest
Any vertex of a G may be colored in white, gray or black color
White vertices are unvisited, gray – discovered vertices, black – already processed vertices (finished)
Initially, G consists of white vertices only; finally, all the vertices of G are black

26. Graphs: depth-first search
Step 1: all the vertices are marked as white
Step 2: start with any white vertex s
Step 3: vertex s is marked as gray
Step 4: for each adjacent to s white vertex algorithm performs recursive reentrance (steps 3-5)
Step 5: if there are no more adjacent to s white vertices in G, algorithm marks vertex s as black and performs some procedure under the vertex (e.g. prints data)
Step 7: continue from step 2 until G has no more white vertices

27. Graphs: depth-first search, sample

28. Graphs: depth-first search, sample

29. Graphs: depth-first search, sample on usage

30. Graphs: depth-first search, sample on usage

31. Graphs: depth-first search, sample on usage

32. Graphs: depth-first search
For given graph G and some white vertex s, algorithm builds paths from vertex s to all vertices, attainable from s
Paths built from a given vertex constitute a depth-first tree
Breadth-first trees (constructed while considering distinct white vertices) constitute a depth-first forest
Changing order of considered white vertices may produce another depth-first forest

33. Graphs: depth-first search

34. Graphs: depth-first search

35. Graphs: topological sort
Let G=<V,E> is a directed acyclic graph
Let’s define comparison operation on V
Vertices u and v are comparable and u<v if exists directed edge (u,v)
Vertices u and w are comparable and u<w if exist some vertex v and edges (u,v) and (v,w); u<v & v<w => u<w
Otherwise two vertices are incomparable
Graph may not have cycles as relation u<u is invalid

36. Graphs: topological sort
The task of topological sort assumes rearrangement of all graph vertices to a linear sequence
Input: set of vertices V={v1,v2,…,vn}
Output: sequence of vertices S=(v1,v2,…,vn), where vi<vk while i<k, or vi and vk are incomparable at all

37. Graphs: topological sort

38. Graphs: topological sort

39. Graphs: topological sort
Algorithm:
Step 1: Allocate storage S for sorted vertices, |S|=|V|
Step 2: Perform Depth-First Search under the graph G; each finished (black) vertex is added to the end of the sequence S
Step 3: Sequence S is topologically sorted set of vertices

40. Graphs: topological sort, sample

41. Graphs: topological sort, sample

42. Graphs: topological sort, sample

43. Graphs: topological sort, sample

44. Graphs: topological sort, sample

45. Graphs: topological sort, sample

46. Graphs: topological sort, sample on usage

47. Graphs: topological sort, sample on usage

48. Graphs: topological sort, sample on usage

49. Q & A