The document discusses data structures and algorithms related to graphs, focusing on undirected and unweighted graphs. It introduces graph notation, various graph representations (like adjacency matrices and lists), and basic operations performed on these graphs. Depth-first search and breadth-first search algorithms are also mentioned as methods for exploring graphs.

1. Data Structures and Algorithms – COMS21103
Representing and Exploring Graphs
Depth First Search and Breadth First Search
Benjamin Sach

2. Graph notation recap
G is a Graph,
Vertex
Edge
V is the set of vertices
E is the set of edges
|V | is the number of vertices
|E| is the number of edges
This lecture focuses on exploring undirected and unweighted graphs
1
3
5
2
4
6
8
7
9
v ∈ V is a vertex
(u, v) ∈ E is an edge from u to v
though everything discussed today will work for directed, unweighted graphs too

3. Graph notation recap
an undirected and unweighted graph an directed and unweighted graph
a directed and weighted graphan undirected and weighted graph
374
9
374
2
83

4. Graph notation recap
an undirected and unweighted graph an directed and unweighted graph
a directed and weighted graphan undirected and weighted graph
374
9
374
2
83
Today we’re
focused on
these graphs

5. Graph notation recap
an undirected and unweighted graph an directed and unweighted graph
a directed and weighted graphan undirected and weighted graph
374
9
374
2
83
Today we’re
focused on
these graphs
But it all works for
these graphs too

6. Graph representations
7
1
3
5
2
4
6
9
0 1 1 0 0 0 0 0 0
1 0 1 1 1 0 0 0 0
1 1 0 1 1 0 0 0 0
0 1 1 0 1 1 0 0 0
0 1 1 1 0 1 0 0 0
0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 1 0
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
FROM
TO
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency Matrix
|V |
|V |
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |

7. Graph representations
7
1
3
5
2
4
6
9
0 1 1 0 0 0 0 0 0
1 0 1 1 1 0 0 0 0
1 1 0 1 1 0 0 0 0
0 1 1 0 1 1 0 0 0
0 1 1 1 0 1 0 0 0
0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 1 0
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
FROM
TO
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency Matrix
|V |
|V |
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |

8. Graph representations
7
1
3
5
2
4
6
9
0 1 1 0 0 0 0 0 0
1 0 1 1 1 0 0 0 0
1 1 0 1 1 0 0 0 0
0 1 1 0 1 1 0 0 0
0 1 1 1 0 1 0 0 0
0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 1 0
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
FROM
TO
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency Matrix
|V |
|V |
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |

9. Graph representations
7
1
3
5
2
4
6
9
0 1 1 0 0 0 0 0 0
1 0 1 1 1 0 0 0 0
1 1 0 1 1 0 0 0 0
0 1 1 0 1 1 0 0 0
0 1 1 1 0 1 0 0 0
0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 1 0
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
FROM
TO
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency Matrix
|V |
|V |
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |

10. Graph representations
7
1
3
5
2
4
6
9
0 1 1 0 0 0 0 0 0
1 0 1 1 1 0 0 0 0
1 1 0 1 1 0 0 0 0
0 1 1 0 1 1 0 0 0
0 1 1 1 0 1 0 0 0
0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 1 0
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9
FROM
TO
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency Matrix
|V |
|V |
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |

11. Graph representations
7
1
3
5
2
4
6
9
1
3
6
7
8
9
1 3 54
1
2
2 54
3 62
3 62
54
8 9
7 9
7 8
3
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency List
(using linked lists)
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |
|V | 5 4
4 5
2

12. Graph representations
7
1
3
5
2
4
6
9
1
3
6
7
8
9
1 3 54
1
2
2 54
3 62
3 62
54
8 9
7 9
7 8
3
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency List
(using linked lists)
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |
|V | 5 4
4 5
2

13. Graph representations
7
1
3
5
2
4
6
9
1
3
6
7
8
9
1 3 54
1
2
2 54
3 62
3 62
54
8 9
7 9
7 8
3
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency List
(using linked lists)
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |
|V | 5 4
4 5
2

14. Graph representations
7
1
3
5
2
4
6
9
1
3
6
7
8
9
1 3 54
1
2
2 54
3 62
3 62
54
8 9
7 9
7 8
3
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency List
(using linked lists)
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |
|V | 5 4
4 5
2

15. Graph representations
7
1
3
5
2
4
6
9
1
3
6
7
8
9
1 3 54
1
2
2 54
3 62
3 62
54
8 9
7 9
7 8
3
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency List
(using linked lists)
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |
|V | 5 4
4 5
2

16. Graph representations
7
1
3
5
2
4
6
9
1
3
6
7
8
9
1 3 54
1
2
2 54
3 62
3 62
54
8 9
7 9
7 8
3
8
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Adjacency List
(using linked lists)
both representations are symmetric because the graphs are undirected
We will in general assume that the vertices are numbered 1, 2, 3 . . . |V |
|V | 5 4
4 5
2

17. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

18. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

19. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

20. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time O(V ) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

21. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time O(deg(u)) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

22. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

23. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2
some vertex u

24. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2
some vertex u
deg(u) = 8

25. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

26. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time
O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

27. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2

28. Basic operations
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2
Adjacency List
(using hash tables)
1
2
3
4
5

29. Basic operations
all three representations work for directed and/or weighted graphs too
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
(with the same complexities)
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2
Adjacency List
(using hash tables)
1
2
3
4
5

30. Basic operations
all three representations work for directed and/or weighted graphs too
Adjacency Matrix
Adjacency Matrix
Adjacency List
(using linked lists)
Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
deg(u) (the degree of u),
is the number of edges leaving u
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
(with the same complexities)
1
2
3
4
5
1
1
2
2
3
2
3
3
4
2
Adjacency List
(using hash tables)
1
2
3
4
5

31. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

32. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Basic hash tables give expected time complexities (constant time ‘on average’)
- no worst case guarantees against collisions

33. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Basic hash tables give expected time complexities (constant time ‘on average’)
- no worst case guarantees against collisions
If you’re interested in hashing with provable guarantees,
come to COMS31900 (Advanced Algorithms)

34. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
Basic hash tables give expected time complexities (constant time ‘on average’)
- no worst case guarantees against collisions
If you’re interested in hashing with provable guarantees,
come to COMS31900 (Advanced Algorithms)
Fortunately, few algorithms need to ask “is there an edge?”
Normally, “tell me all the edges” is fine

35. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

36. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
For this course we’ll stick to Adjacency Matrices and Lists (using linked lists)
(for today just Adjacency Lists)

37. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
For this course we’ll stick to Adjacency Matrices and Lists (using linked lists)
(for today just Adjacency Lists)
One practical reason to stick to standard representations is that
you may not have control over how your graph is stored.

38. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
For this course we’ll stick to Adjacency Matrices and Lists (using linked lists)
(for today just Adjacency Lists)
One practical reason to stick to standard representations is that
you may not have control over how your graph is stored.
it may not even be stored as a graph...

39. Basic operations
Adjacency Matrix Adjacency List
(using linked lists)
Is there an edge from vertex u to v?
List all the edges leaving u ∈ V
Space
Adjacency List
(using hash tables)
Θ(|V |2) Θ(|V | + |E|)
O(1) time
O(V ) time O(deg(u)) time
O(deg(u)) time
Θ(|V | + |E|)
O(deg(u)) time
O(1) time
why don’t we always use these?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges
For this course we’ll stick to Adjacency Matrices and Lists (using linked lists)
(for today just Adjacency Lists)
One practical reason to stick to standard representations is that
you may not have control over how your graph is stored.
it may not even be stored as a graph...
Fortunately in many cases, you can pretend your graph is stored as one of the above.

40. Intersection Graphs

41. Intersection Graphs
the intersections of these circles
correspond to the graph below

42. Intersection Graphs
each circle is a vertex
the intersections of these circles
correspond to the graph below

43. Intersection Graphs
the intersections of these circles
correspond to the graph below

44. Intersection Graphs
each intersection
is an edge
the intersections of these circles
correspond to the graph below

45. Intersection Graphs
each intersection
is an edge
the intersections of these circles
correspond to the graph below

46. Intersection Graphs
each intersection
is an edge
we can pretend we have
an Adjacency Matrix
without building one
even if we only store the circles,
the intersections of these circles
correspond to the graph below

47. Other intersection graphs. . .
(these are 1D intervals on a line spread out for visual clarity)
both of these ‘collections’ also
correspond to the example graph

48. Other intersection graphs. . .
= an edge
= a vertex
(these are 1D intervals on a line spread out for visual clarity)
both of these ‘collections’ also
correspond to the example graph

49. Other intersection graphs. . .
= an edge
= a vertex
there is an edge if two intervals intersect
each interval is a vertex
(these are 1D intervals on a line spread out for visual clarity)
both of these ‘collections’ also
correspond to the example graph

50. Other intersection graphs. . .
= an edge
= a vertex
there is an edge if two intervals intersect
each interval is a vertex
(these are 1D intervals on a line spread out for visual clarity)
we can pretend we have
an Adjacency Matrix
without building one
Again,
both of these ‘collections’ also
correspond to the example graph

51. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

52. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

53. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

54. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

55. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

56. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

57. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

58. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

59. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move

60. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move
storing this uses much less space than
storing this this graph

61. Configuration Graphs
This is the configuration graph for the
(4 disk) towers of Hanoi
a node is a state that the
game can be in
there is an edge if you can get from
one state to another in a single move
storing this uses much less space than
storing this this graph
we can pretend we have
an Adjacency List
without building one

62. Exploring a graph
you are here
BAG

63. Exploring a graph
you are here
BAG Goal: visit every vertex in a
connected graph efficiently

64. Exploring a graph
you are here
BAG Goal: visit every vertex in a
connected graph efficiently
The bag (secretly a data structure)
We can put a vertex in the bag
We can take a vertex from the bag
We don’t care how it works (for now)
(the bag doesn’t forget or change things)
but we don’t know which one we’ll get

65. Exploring a graph
BAG Goal: visit every vertex in a
connected graph efficiently
We going to use an adjacency list so the
graph is really stored like this:
The bag (secretly a data structure)
We can put a vertex in the bag
We can take a vertex from the bag
We don’t care how it works (for now)
(the bag doesn’t forget or change things)
but we don’t know which one we’ll get

66. Exploring a graph
you are here
BAG Goal: visit every vertex in a
connected graph efficiently

67. Exploring a graph
vertex s
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

68. Exploring a graph
vertex s
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

69. Exploring a graph
vertex s
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

70. Exploring a graph
vertex s
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

71. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

72. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

73. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

74. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

75. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently
back in!
in twice!

76. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

77. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

78. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

79. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently
already marked
(do nothing)

80. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

81. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

82. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

83. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

84. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

85. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

86. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently

87. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently
Some time later. . .

88. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently
Some time later. . . ?

89. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Goal: visit every vertex in a
connected graph efficiently
Some time later. . . ?
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
3. How fast is it? (in the worst case)

90. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
3. How fast is it? (in the worst case)

91. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
1. Does TRAVERSE always stop?

92. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
1. Does TRAVERSE always stop?

93. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
back in!
in twice!
1. Does TRAVERSE always stop?

94. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
back in!
in twice!
1. Does TRAVERSE always stop?
We may put a vertex in the bag many times but each edge is only
processed twice (once in each direction)

95. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
back in!
in twice!
1. Does TRAVERSE always stop?
We may put a vertex in the bag many times but each edge is only
processed twice (once in each direction)
never seen
again!

96. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
back in!
in twice!
1. Does TRAVERSE always stop?
We may put a vertex in the bag many times but each edge is only
processed twice (once in each direction)
never seen
again!
Important later: number of ‘puts’ 2|E|

97. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
back in!
in twice!
1. Does TRAVERSE always stop?
We may put a vertex in the bag many times but each edge is only
processed twice (once in each direction)
never seen
again!
Important later: number of ‘puts’ 2|E|
TRAVERSE(s)
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag

98. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
back in!
in twice!
1. Does TRAVERSE always stop?
We may put a vertex in the bag many times but each edge is only
processed twice (once in each direction)
never seen
again!
Important later: number of ‘puts’ 2|E|

99. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
back in!
in twice!
1. Does TRAVERSE always stop? Yes
We may put a vertex in the bag many times but each edge is only
processed twice (once in each direction)
never seen
again!
Important later: number of ‘puts’ 2|E|

100. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
3. How fast is it? (in the worst case)

101. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?

102. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
Proof (by induction)
(unless it was already marked)

103. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

104. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

105. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

106. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

107. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

108. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

109. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

110. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

111. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

112. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
there could be a ‘shortcut’
(unless it was already marked)

113. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
there could be a ‘shortcut’
(unless it was already marked)

114. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
there could be a ‘shortcut’
(unless it was already marked)

115. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
there could be a ‘shortcut’
(unless it was already marked)

116. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
there could be a ‘shortcut’
(unless it was already marked)

117. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
there could be a ‘shortcut’
(unless it was already marked)

118. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

119. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

120. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

121. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

122. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

123. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

124. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
(unless it was already marked)

125. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
Yes
(unless it was already marked)

126. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
Yes
The correctness doesn’t depend
on how the bag works!
(unless it was already marked)

127. Exploring a graph
BAG
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
2. Does it always visit every vertex?
When a vertex on this path is marked, the next vertex is put in the bag
Eventually, that vertex is removed from the bag and is marked.
Consider any path from s to some v:
s
v
Proof (by induction)
Yes
The correctness doesn’t depend
on how the bag works!
(but this doesn’t tell you anything about
which order vertices are marked in)
(unless it was already marked)

128. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)

129. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Assumption
(we’ll come back to this)
bag operations
take O(1) time

130. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Assumption
(we’ll come back to this)
bag operations
take O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

131. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Assumption
(we’ll come back to this)
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

132. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

133. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

134. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
(store an array where mark[u] = 1 iff u is marked)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

135. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
(store an array where mark[u] = 1 iff u is marked)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(1) time every time we put into the bag
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

136. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
(store an array where mark[u] = 1 iff u is marked)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(1) time every time we put into the bag
What is the total number of bag operations?
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

137. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
(store an array where mark[u] = 1 iff u is marked)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(1) time every time we put into the bag
What is the total number of bag operations?
we do at most 2|E| put operations
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

138. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
(store an array where mark[u] = 1 iff u is marked)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(1) time every time we put into the bag
What is the total number of bag operations?
we do at most 2|E| put operations
so we do at most 2|E| take operations
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

139. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
(store an array where mark[u] = 1 iff u is marked)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(1) time every time we put into the bag
What is the total number of bag operations?
we do at most 2|E| put operations
so we do at most 2|E| take operations
The overall time complexity is O(|E|)
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

140. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Time complexity:
Assumption
(we’ll come back to this)
(store an array where mark[u] = 1 iff u is marked)
O(1) time every time we take from the bag
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(1) time every time we put into the bag
What is the total number of bag operations?
we do at most 2|E| put operations
so we do at most 2|E| take operations
The overall time complexity is O(|E|)
O(|E|)
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

141. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Assumption
(we’ll come back to this)
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(|E|)
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

142. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Assumption
(we’ll come back to this)
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(|E|)
What if we had used an
adjacency matrix?
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

143. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Assumption
(we’ll come back to this)
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(|E|)
What if we had used an
adjacency matrix?
finding all edges leaving u would have taken O(|V |) time
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

144. Exploring a graph
Questions:
1. Does TRAVERSE always stop?
2. Does it always visit every vertex?
Yes
Yes
3. How fast is it? (in the worst case)
Assumption
(we’ll come back to this)
put s into the bag
while the bag is not empty
take u from the bag
if u unmarked
mark u
for every edge (u, v)
put v into the bag
TRAVERSE(s)
bag operations
take O(1) time
O(|E|)
What if we had used an
adjacency matrix?
finding all edges leaving u would have taken O(|V |) time
0 1 1 0 0
1 0 1 1 1
1 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1
2
3
4
5
1 2 3 4 5
overall, the time complexity would have been O(|V |2) instead of O(|E|).
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges

145. How should we implement the bag?
BAG
Queue Stack
Operations take O(1) time
Depth First SearchBreadth First Search
but in different orders with different types of bag
TRAVERSE visits every vertex in a connected graph in O(|E|) time
with a Queue the algorithm is called with a Stack the algorithm is called
Applications Applications
Shortest paths in unweighted graphs
Max-Flow
Testing whether a graph is bipartite
Finding (strongly) connected components
Topicologically sorting a Directed Acyclic Graph
Testing for planarity

146. How should we implement the bag?
BAG
Queue Stack
Operations take O(1) time
Depth First SearchBreadth First Search
but in different orders with different types of bag
TRAVERSE visits every vertex in a connected graph in O(|E|) time
with a Queue the algorithm is called with a Stack the algorithm is called
Applications Applications
Shortest paths in unweighted graphs
Max-Flow
Testing whether a graph is bipartite
Finding (strongly) connected components
Topicologically sorting a Directed Acyclic Graph
Testing for planarity
(and it works for directed graphs too)

147. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex

148. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
distance = 1

149. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
distance = 3

150. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
distance = 6

151. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex

152. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

153. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

154. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

155. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

156. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

157. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

158. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

159. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

160. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

161. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

162. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
these are all in front of these
in the queue
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

163. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
these are all in front of these
in the queue
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

164. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

165. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

166. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

167. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

168. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

169. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

170. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

171. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

172. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

173. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

174. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

175. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

176. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Inductive Hypothesis: Assume that BFS marks all vertices at distance (i − 1)
before marking any vertex at distance > (i − 1).
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

177. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Inductive Hypothesis: Assume that BFS marks all vertices at distance (i − 1)
before marking any vertex at distance > (i − 1).
When BFS marks a vertex at distance (i − 1),
it puts all its neighbours in the queue.
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

178. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Inductive Hypothesis: Assume that BFS marks all vertices at distance (i − 1)
before marking any vertex at distance > (i − 1).
When BFS marks a vertex at distance (i − 1),
it puts all its neighbours in the queue.
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
v is u’s neighbour iff
(u, v) ∈ E
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

179. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Inductive Hypothesis: Assume that BFS marks all vertices at distance (i − 1)
before marking any vertex at distance > (i − 1).
When BFS marks a vertex at distance (i − 1),
Immediately after marking all vertices at distance (i − 1):
it puts all its neighbours in the queue.
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
v is u’s neighbour iff
(u, v) ∈ E
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

180. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Inductive Hypothesis: Assume that BFS marks all vertices at distance (i − 1)
before marking any vertex at distance > (i − 1).
When BFS marks a vertex at distance (i − 1),
Immediately after marking all vertices at distance (i − 1):
it puts all its neighbours in the queue.
every vertex at distance i is in the queue.
no vertex at distance > i has been put in the queue.
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
v is u’s neighbour iff
(u, v) ∈ E
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

181. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Inductive Hypothesis: Assume that BFS marks all vertices at distance (i − 1)
before marking any vertex at distance > (i − 1).
When BFS marks a vertex at distance (i − 1),
Immediately after marking all vertices at distance (i − 1):
it puts all its neighbours in the queue.
every vertex at distance i is in the queue.
no vertex at distance > i has been put in the queue.
because we haven’t marked any of its neighbours
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
(they are all at distance > (i − 1))
v is u’s neighbour iff
(u, v) ∈ E
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

182. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This is why it’s called
Breadth-First Search
Base Case: BFS marks all vertices at distance 0 from s
before marking any vertex at distance > 0 .
Inductive Hypothesis: Assume that BFS marks all vertices at distance (i − 1)
before marking any vertex at distance > (i − 1).
When BFS marks a vertex at distance (i − 1),
Immediately after marking all vertices at distance (i − 1):
it puts all its neighbours in the queue.
every vertex at distance i is in the queue.
no vertex at distance > i has been put in the queue.
because we haven’t marked any of its neighbours
Therefore, all vertices at distance i will be marked before any vertex at distance > i
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Proof by induction (on the distance from s)
(they are all at distance > (i − 1))
v is u’s neighbour iff
(u, v) ∈ E
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

183. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

184. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
we can use this to find the shortest path
from s to every other vertex in O(|V | + |E|) time
With a little bit of book-keeping,
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

185. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This also works for directed, unweighted graphs
we can use this to find the shortest path
from s to every other vertex in O(|V | + |E|) time
With a little bit of book-keeping,
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

186. Shortest paths in unweighted graphs
s
Goal: find the distance from the
source s to every other vertex
Breadth-First Search (BFS)
is TRAVERSE using a queue as the bag
Lemma In BFS, the vertices
are marked in distance order (smallest first)
This also works for directed, unweighted graphs
we can use this to find the shortest path
from s to every other vertex in O(|V | + |E|) time
With a little bit of book-keeping,
It does not work for weighted graphs
TRAVERSE(s) - with a queue
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue

187. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s

188. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s How does this affect the time complexity?

189. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
Time complexity:
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s How does this affect the time complexity?

190. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
Time complexity:
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s How does this affect the time complexity?
O(|V |) time to initialise an array dist

191. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
Time complexity:
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s How does this affect the time complexity?
O(|V |) time to initialise an array dist
O(1) time to set the distance to s

192. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
Time complexity:
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s How does this affect the time complexity?
O(|V |) time to initialise an array dist
O(1) time to set the distance to s
O(1) time whenever we put into the queue
(so this doesn’t change the complexity)

193. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
Time complexity:
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s How does this affect the time complexity?
O(|V |) time to initialise an array dist
O(1) time to set the distance to s
O(1) time whenever we put into the queue
(so this doesn’t change the complexity)
so the overall complexity is O(|V | + |E|)

194. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s

195. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?

196. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Lemma In BFS, the vertices are marked in
distance order (smallest first)

197. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
Lemma In BFS, the vertices are marked in
distance order (smallest first)

198. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Lemma In BFS, the vertices are marked in
distance order (smallest first)

199. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)

200. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
BFS sets dist(v) = dist(u) + 1

201. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
BFS sets dist(v) = dist(u) + 1

202. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
BFS sets dist(v) = dist(u) + 1
length < dist(u) + 1
s
v

203. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Let x be the previous vertex on this path
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
BFS sets dist(v) = dist(u) + 1
length < dist(u) + 1
s
v

204. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Let x be the previous vertex on this path
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
BFS sets dist(v) = dist(u) + 1
length < dist(u) + 1
s
v
x

205. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Let x be the previous vertex on this path
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
BFS sets dist(v) = dist(u) + 1
length < dist(u) length = 1
s
v
x

206. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Let x be the previous vertex on this path
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
BFS sets dist(v) = dist(u) + 1
length < dist(u) length = 1
s
v
x

207. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Let x be the previous vertex on this path
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
x has distance < dist(u) so was marked before u
(by the Lemma) and ‘discovered’ v first
BFS sets dist(v) = dist(u) + 1
length < dist(u) length = 1
s
v
x

208. Shortest Paths using BFS
for all v, set dist(v) = ∞ (NEW!)
set dist(s) = 0 (NEW!)
put s into the queue
while the queue is not empty
take u from the queue
if u unmarked
mark u
for every edge (u, v)
put v into the queue
if dist(v) = ∞ (NEW!)
dist(v) = dist(u) + 1 (NEW!)
BFS(s)
dist(v) gives the distance
between s and v
We’ve added four (NEW!) lines to TRAVERSE
to track the distances from s
Why does this work?
Correctness sketch: (using the Lemma)
For each v, dist(v) is set when it is first ‘discovered’
and inserted into the queue. (and it never changes)
Let u be the vertex which first ‘discovered’ v
Lemma In BFS, the vertices are marked in
distance order (smallest first)
Let x be the previous vertex on this path
Assume for a contradiction that there is a path
from s to v with length < dist(u) + 1
x has distance < dist(u) so was marked before u
(by the Lemma) and ‘discovered’ v first
Contradiction!
BFS sets dist(v) = dist(u) + 1
length < dist(u) length = 1
s
v
x

209. Conclusion
Depth First Search (DFS)Breadth First Search (BFS)
- the traversal order depends on the type of bag
TRAVERSE visits every vertex in a connected graph in O(|E|) time
with a Queue the algorithm is called with a Stack the algorithm is called
Applications Applications
Shortest paths in unweighted graphs
Max-Flow
Testing whether a graph is bipartite
Finding (strongly) connected components
Topicologically sorting a Directed Acyclic Graph
Testing for planarity
(with an O(1) time bag)
take O(|V | + |E|) time using BFS
Question
What does TRAVERSE do on an
unconnected graph?(works for directed graphs too)
(and it works for directed graphs too)
V is the set of vertices
|V | is the number of vertices
E is the set of edges
|E| is the number of edges