1. Introduction (with applications) ADT & terminology
Graphs
Introduction (with applications)
ADT & terminology
March 21, 2018
Cinda Heeren / Geoffrey Tien

2. Cinda Heeren / Geoffrey Tien
Total order 1 2 5 4 7 6 3 A B
means A must go before B
March 21, 2018
Cinda Heeren / Geoffrey Tien

3. Cinda Heeren / Geoffrey Tien
Partial order
Getting dressed
pants
shirt
socks
coat
undies
belt
watch
boots
pocket
knife
March 21, 2018
Cinda Heeren / Geoffrey Tien

4. Cinda Heeren / Geoffrey Tien
Topological sort
Given a graph, 𝐺= 𝑉,𝐸 , output all vertices in V such that no vertex is output before any other vertex with an edge to it
Classic application: course prerequisites
CPSC 110
CPSC 121
MATH 100
MATH 101
CPSC 210
CPSC 221
STAT 241
MATH 200
MATH 221
CPSC 213
CPSC 310
CPSC 320
CPSC 313
March 21, 2018
Cinda Heeren / Geoffrey Tien

5. Cinda Heeren / Geoffrey Tien
Topological sort
Label each vertex's in-degree (# of inbound edges)
Initialize a queue to contain all vertices with in-degree zero
While there are vertices remaining in the queue
Pick a vertex 𝑣 from the queue and output it
Reduce the in-degree of all vertices adjacent to 𝑣
Put any of these with updated zero in-degree on the queue
Remove 𝑣 from the queue
March 21, 2018
Cinda Heeren / Geoffrey Tien

6. Cinda Heeren / Geoffrey Tien
Topological sort
Queue: 1 6
Vertex
In-degree 1 2 3 4 5 6 2 4 5 1 6 3
March 21, 2018
Cinda Heeren / Geoffrey Tien

7. Cinda Heeren / Geoffrey Tien
Topological sort
Queue: 1 6 2
Vertex
In-degree 1 2 3 4 5 6 2 4 5 1 6 3 1 1 6
March 21, 2018
Cinda Heeren / Geoffrey Tien

8. Cinda Heeren / Geoffrey Tien
Topological sort
Queue: 2 3
Vertex
In-degree 1 2 3 4 5 6 2 4 5 1 6 3 1 6 2
March 21, 2018
Cinda Heeren / Geoffrey Tien

9. Cinda Heeren / Geoffrey Tien
Topological sort
Queue: 3 4
Vertex
In-degree 1 2 3 4 5 6 2 4 5 1 6 3 1 1 6 2 3
March 21, 2018
Cinda Heeren / Geoffrey Tien

10. Cinda Heeren / Geoffrey Tien
Topological sort
Queue: 4 5
Vertex
In-degree 1 2 3 4 5 6 2 4 5 1 6 3 1 6 2 3 4 5
March 21, 2018
Cinda Heeren / Geoffrey Tien

11. Cinda Heeren / Geoffrey Tien
Graph applications
Graphs are used as representations of many types of problems
Computer network configuration
Airline flight booking
Pathfinding algorithms
Database dependencies
Task scheduling
Road traffic modeling …
March 21, 2018
Cinda Heeren / Geoffrey Tien

12. Cinda Heeren / Geoffrey Tien
Exam scheduling
CPSC 221
CPSC 210
BIOL 201
ECON 211
MATH 200
MATH 223
Vertices represent courses
Edges represent conflicts (at least one student is enrolled in both courses)
Colours represent time slots
Can we colour the graph (ideally using a limited number of colours) such that vertices joined by an edge do not have the same colour?
March 21, 2018
Cinda Heeren / Geoffrey Tien

13. Cinda Heeren / Geoffrey Tien
Finite state automata
Begin in the start vertex at the top
for each digit d in the given number, follow d blue (solid) edges in succession
after processing one digit, follow 1 red (dashed) edge
If you end up back at the start node, the number is divisible by 7
e.g. 3703
YES
March 21, 2018
Cinda Heeren / Geoffrey Tien

14. Cinda Heeren / Geoffrey Tien
Navigation
Find a driving route(s) from location A to B
March 21, 2018
Cinda Heeren / Geoffrey Tien

15. Cinda Heeren / Geoffrey Tien
Pathfinding
Movement/position of video game elements, around terrain/obstacles etc.
March 21, 2018
Cinda Heeren / Geoffrey Tien

16. Cinda Heeren / Geoffrey Tien
Medical imaging
Automatic segmentation of anatomy in medical images
March 21, 2018
Cinda Heeren / Geoffrey Tien

17. Cinda Heeren / Geoffrey Tien
Graphics
Stanford bunny
March 21, 2018
Cinda Heeren / Geoffrey Tien

18. Cinda Heeren / Geoffrey Tien
Game theory
Game states and searching for winning sequence of moves
March 21, 2018
Cinda Heeren / Geoffrey Tien

19. Cinda Heeren / Geoffrey Tien
Graph ADT
Graphs are a formalism useful for representing relationships between things
A graph 𝐺 is represented as 𝐺= 𝑉,𝐸
𝑉 is a set of vertices: 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑛
E is a set of edges: 𝑒 1 , 𝑒 2 ,…, 𝑒 𝑚 where each 𝑒 𝑖 connects two vertices 𝑣 𝑖1 , 𝑣 𝑖2
𝑉= 𝑇𝑜𝑚, 𝑆ℎ𝑒𝑙𝑙𝑦, ℎ𝑎𝑚𝑠𝑡𝑒𝑟, 𝑝𝑜𝑝𝑐𝑜𝑟𝑛
𝐸= 𝑇𝑜𝑚, 𝑆ℎ𝑒𝑙𝑙𝑦 , 𝑆ℎ𝑒𝑙𝑙𝑦, 𝑇𝑜𝑚 , 𝑆ℎ𝑒𝑙𝑙𝑦,ℎ𝑎𝑚𝑠𝑡𝑒𝑟 , ℎ𝑎𝑚𝑠𝑡𝑒𝑟,𝑝𝑜𝑝𝑐𝑜𝑟𝑛
Tom
Shelly
hamster
popcorn
March 21, 2018
Cinda Heeren / Geoffrey Tien

20. Cinda Heeren / Geoffrey Tien
Graph ADT
Operations
Operations might include:
Creation (with a certain number of vertices)
Iterate through all vertices in the graph (traversal)
Iterating over vertices adjacent to a specific vertex
Asking whether an edge exists connecting two vertices
Ask about the weight/cost of an edge between two vertices
Inserting/removing edges
(other operations/algorithms that may be useful)
March 21, 2018
Cinda Heeren / Geoffrey Tien

21. Cinda Heeren / Geoffrey Tien
Graph terminology
Paths, vertices, edges
A graph consists of two sets, 𝑉 and 𝐸
Two vertices may be connected by a path
A sequence of edges that begins at one vertex and ends at the other
A simple path does not pass through the same vertex more than once
A cycle is a path that starts and ends at the same vertex
If a graph has v vertices, how many edges does it have?
If every vertex is connected to every other vertex, and we count each direction as two edges
v2 – v
If the graph is a tree
v – 1
Minimum number of edges
March 21, 2018
Cinda Heeren / Geoffrey Tien

22. Connected / unconnected graphs
A connected graph is one where every pair of distinct vertices has a path between them
A complete graph is one where every pair of vertices has an edge between them
A graph (usually) cannot have multiple edges between the same pair of vertices
A graph (usually) cannot have self edges
i.e. an edge from and to the same vertex
connected graph
complete graph
unconnected graph
March 21, 2018
Cinda Heeren / Geoffrey Tien

23. Cinda Heeren / Geoffrey Tien
Directed graphs
In a directed graph (or digraph) each edge has a direction and is called a directed edge
A directed edge can only be traveled in one direction
A pair of vertices in a digraph may have two edges between them, one in each direction
March 21, 2018
Cinda Heeren / Geoffrey Tien

24. Cinda Heeren / Geoffrey Tien
Weighted graphs
In a weighted graph each edge is assigned a weight
Edges are labeled with their weights
Each edge’s weight represents the cost to travel along that edge
The cost could be distance, time, money or some other measure
The cost depends on the underlying problem 1 2 4 3 5
March 21, 2018
Cinda Heeren / Geoffrey Tien

25. Readings for this lesson
Carrano & Henry
Chapter 20.1 – 20.2 (Graph terminology, ADT)
Next class:
Carrano & Henry, Chapter 20.2 – 20.3 (implementation, traversals)
March 21, 2018
Cinda Heeren / Geoffrey Tien