UCT Algorithm Circle: An Introduction to Graph Theory · 2012-09-15 · A quick puzzleTerminologyRevisiting KonigsbergOnto some speciﬁcsApplications in computer scienceSimple searchesConclusion

A quick puzzle Terminology Revisiting Konigsberg Onto some specifics Applications in computer science Simple searches Conclusion

UCT Algorithm Circle: An Introduction toGraph Theory

Peter Faul

16 August 2012

Peter Faul Graph Theory


Outline

1 A quick puzzle

2 Terminology

3 Revisiting Konigsberg

4 Onto some specifics

5 Applications in computer science

6 Simple searches

7 Conclusion



The bridges of Konigsberg

Can you follow a path, walking over each bridge exactlyonce?


Konigsberg Diagram [1]


Outline

1 A quick puzzle

2 Terminology




6 Simple searches

7 Conclusion



Some terminology

Node: The fundamental unit out of which graphs areformed.Edge: Connects two nodes.Graph: Collection of Nodes and Edges.



Some terminology




Some terminology




Example of a graph


Graph Diagram[2]


More Terminology

Walk: An alternating sequence of nodes and edges, with abeginning and end node. Each node is connected to thenode that came before it, and the node that comes after it.Trail: A walk where all edges are distinct.Path: A trail where all nodes (except maybe the first andlast) are distinct.



More Terminology




More Terminology




Example of a path



Example of a path



Example of a path



Example of a path



Example of a path



Example of a path



Even more terminology

Eulerian Path: A trail where each edge is visited once.Hamiltonian path: A path where each node is visited once.Degree of node: The number of edges attached to thenode.











Outline

1 A quick puzzle

2 Terminology




6 Simple searches

7 Conclusion



Konigsberg

How can we simplify this problem?



Turn it into a graph

Konigsberg bridges in the form of a graph.



What now?

So how has this helped?We can restate the problem now; perhaps it will be easierto prove in this form.



What now?

So how has this helped?We can restate the problem now; perhaps it will be easierto prove in this form.



Restating the problem

Does the following graph have an Eulerian path?



Euler’s proposition

Any graph with more than two nodes of odd degree doesnot have an Eulerian path.The bridges of Konigsberg graph, has 4 nodes of odddegree.Hence it has no Eulerian path











Outline

1 A quick puzzle

2 Terminology




6 Simple searches

7 Conclusion



Classification of graphs

Graphs may have the following attributes:Connected: From any one node, there is a walk to anyother node.Cyclic: A graph that contains some number of nodesconnected in a closed chain.Directed: A graph in which the edges have a directionassociated with them.Tree: An undirected graph where any two nodes areconnected by one unique path.



















Disconnected Graph

A graph that is not connected:



Cyclic Graph

A cyclic graph, note the part in red:



Directed Graph

Up until now we have been considering undirected graphswhere you are traverse edges in both directions. Edgesmay also be directed, in which case edges may only betraversed along the direction in which they point



Tree Graph

An example of a tree:



Outline

1 A quick puzzle

2 Terminology




6 Simple searches

7 Conclusion



Justification

So why is this being taught in a computer science lecture?Many problems in computer science can be modelled interms of graphs



Justification

So why is this being taught in a computer science lecture?Many problems in computer science can be modelled interms of graphs



Examples:

Path finding in video games.The link structure of a website.Notorious travelling salesman problem.Many more...



Examples:




Examples:




Examples:




Representing a graph

How do we represent a graph on a computer?Adjacency ListAdjacency Matrix











Adjacency List

A representation of a graph of n nodes, using an array of nlinked lists.List A contains node b if node a is connected to node b.



Adjacency List

A representation of a graph of n nodes, using an array of nlinked lists.List A contains node b if node a is connected to node b.



Adjacency List



Adjacency List



Adjacency Matrix

A matrix representation of the graph (Use nested arrays).The entry aij in the matrix A is equal to the number ofedges from node i to node j



Adjacency Matrix

A matrix representation of the graph (Use nested arrays).The entry aij in the matrix A is equal to the number ofedges from node i to node j



Adjacency Matrix



Adjacency Matrix

0 1 0 0 1 01 0 1 0 1 00 1 0 1 0 00 0 1 0 1 11 1 0 1 0 00 0 0 1 0 0



An interesting property

Let A be the adjacency matrix for some graph G.Then the entry kij in An represents the number of differentwalks from node i to node j.Remember how matrix multiplication is defined to makesense of this.











A comparison

So which representation is best?That depends.Adjacency Matrices take up O(n2) space, however thereare constant time lookups, insertions and deletions.Adjacency Lists take up O(n+e) space, however they haveslower lookup times etc.So really your choice should depend on your requirementsand on the graph.



A comparison




A comparison




A comparison




A comparison




Outline

1 A quick puzzle

2 Terminology




6 Simple searches

7 Conclusion



Searching Algorithms

Often we might want to see if one node is attached toanother nodeOr find the shortest path from one node to anotherWe use searching algorithms to accomplish this.We will look at two simple searching algorithms.















Breadth First Search

Systematically searches through all nodesDoes so by inspecting node then adding all nodes linked,to a queue.



Breadth First Search

Systematically searches through all nodesDoes so by inspecting node then adding all nodes linked,to a queue.



Breadth First Properties

Breadth First search guarantees the shortest path.However it uses more memory than a depth first search onaverage.



Breadth First Properties

Breadth First search guarantees the shortest path.However it uses more memory than a depth first search onaverage.



Depth First Search

Also searches through all nodesDoes so by inspecting node then adding all nodes linked,to a stack.



Depth First Search

Also searches through all nodesDoes so by inspecting node then adding all nodes linked,to a stack.



Depth First properties

For tree traversal, depth first uses less memory generallyas it need only store the nodes on the current path.While you shouldn’t count on it, you may hit the targethaving only explored a fraction of the tree.This however is not necessarily the best solution.There is also the potential for the search to get stuckexploring an infinite path.















Summary

So your choice of algorithm largely depends on your graphAnd on your own requirements.



Summary

So your choice of algorithm largely depends on your graphAnd on your own requirements.



Outline

1 A quick puzzle

2 Terminology




6 Simple searches

7 Conclusion



Conclusion

So there you have it, an introduction to graphs.Very important in many disciplinesAny Questions?



Conclusion




Conclusion




Licensing

These lecture slides were created by Peter Faul for UCTAlgorithm Circle and are licensed under the Creative CommonsAttribution-ShareAlike 3.0 Unported License.

I made use of the following third-party content also licensed underCC-BY-SA 3.0:

1 Konigsberg Diagram: http://en.wikipedia.org/wiki/File:Red-black_tree_example.svg.

2 Tree Diagram:http://en.wikipedia.org/wiki/File:Tree_graph.svg

3 Graph Diagram:http://en.wikipedia.org/wiki/File:6n-graf.svg


http://algorithm.cs.uct.ac.za/

http://algorithm.cs.uct.ac.za/

http://creativecommons.org/licenses/by-sa/3.0/



http://en.wikipedia.org/wiki/File:Red-black_tree_example.svg

http://en.wikipedia.org/wiki/File:Red-black_tree_example.svg

http://en.wikipedia.org/wiki/File:Tree_graph.svg

http://en.wikipedia.org/wiki/File:6n-graf.svg

Documents

UCT Algorithm Circle: An Introduction to Graph Theory · 2012-09-15 · A quick puzzleTerminologyRevisiting KonigsbergOnto some speciﬁcsApplications in computer scienceSimple searchesConclusion