1. Comp 245 Data Structures
Graphs

2. Introduction to the Graph ADT
A graph can be defined as a set of points connected by lines.
The graph ADT can be defined this way also; the set of points will be called nodes or vertices and the set of lines will be called edges.
A tree is a graph but a graph is not necessarily a tree!
A tree is a subset of a graph!

3. An Abstract View of Graphs
A tree imposes a hierarchical view of data, a graph does not have
to impose any kind of hierarchy.

4. Graph Terminology Undirected Directed
If a graph is undirected an edge essentially goes in both
directions, in a directed graph an edge travels in one
direction.

5. Graph Terminology Connected Disconnected
A graph is connected if every node can reach every other
node!

6. Graph Terminology Complete
A graph is complete if each node can reach every other
node directly.

7. Graph Terminology Other Terms Weighted Path Cycle Adjacent C B A F D20 10
100 40 25 30
Other Terms
Weighted
Path
Cycle
Adjacent

8. Implementing a Graph Adjacency MatrixB F C E G H
Adjacency Matrix 1 2 3 4 5 6  0  1 0  1 
1  
Map A B C E F G H 1 2 3 4 5 6

9. Implementing a Graph Adjacency ListB F C E G H A B C E F G H

10. Graph Traversals A graph traversal can start at any node
How many nodes can it reach; this is a measure of “connectedness”
Utilize the “backtracking” idea to do a traversal

11. Can We Connect to a Node? (Traverse from a given node)B F C E G H
From E can we get to H (E->H)?
PUSH all moves from E onto stack
ZERO OUT E column (Breadcrumbing)
If stack is EMPTY, node is disconnected
Otherwise, POP stack to get next move
If this is node desired then finished, otherwise repeat process 1 2 3 4 5 6  0  1 0  1 
1   A B C E F G H 1 2 3 4 5 6

12. Dijkstra’s Shortest Path Algorithm
Used with a weighted, possibly directed graph.
Will calculate the shortest path, least cost, or minimum distance from an “origin” node in a graph to every other node it is “connected” to in the graph.

13. Dijkstra’s Algorithm A is “origin” nodeB C D E F G H 3 1 2 4 5

14. Dijkstra’s Algorithm Shortest Path TableV
Known dv Pv A 1 - B 3 C 4 D 2 E F G 5 H 6 A B C D E F G H 3 1 2 4 5
V – visited node
Known – indicates we “know” the shortest path
dv – distance (weight, cost, etc…) from “origin” node to this node
Pv – previous node

15. Dijkstra’s Algorithm Shortest Path Table - InterpretingV
Known dv Pv A 1 - B 3 C 4 D 2 E F G 5 H 6 A B C D E F G H 3 1 2 4 5
Path To:
Distance
Path B 3
A - B C 4
A - B - C D 2
A - D E
A - E F
A - D - F G 5
A - D - F - G H 6
A - D - F - H

16. Practice Create the Shortest Path TableD C F G E 30 50 40 20 5 10 60

17. Practice Create the Shortest Path TableV
Known dv Pv A 1 - B 30 C 50 D 20 E 40 F G A B D C F G E 30 50 40 20 5 10 60
Path To:
Distance
Path B 30
A - B C 50
A - C D 20
A - D E 40
A - B - E F
A - D - F G
A - D - G