Graph Theory: Week 1

Transcription

1 Graph Theory: Week Introduction to Graph Theory John Quinn August 30, 00

2 The Tokyo subway

3 Week overview What are graphs? Why study graph theory? Examples of graphs in the real world Different types of graphs Example problem: finding the shortest path

4 What is a graph? C B D F A E A graph G consists of a set of vertices and a set of edges. G = {V,E} in this example, V = {A,B,C,D,E,F}, E = {AB, BC, BD, CD, DF, DE,EF}. Any graph can be drawn on paper in many ways the important thing is which vertices are connected (adjacent) to each other.

5 Why study graph theory? Useful set of techniques for solving real-world problems particularly for different kinds of optimisation. Graph theory is useful for analysing things that are connected to other things, which applies almost everywhere. Some difficult problems become easy when represented using a graph. There are lots of unsolved questions in graph theory: solve one and become rich and famous. maybe

6 Graph example: Gnucleus peer connections Source: cybergeography.org

7 Graph example: Structure of the internet Source: Internet Mapping Project

8 Weighted graphs C B 4 D F A E Can extend graphs by associating a weight with each edge. Might represent e.g. the cost of travelling between two points.

9 Directed graphs (digraphs) C B 4 D F A E Can also make edges directional. This might now represent, for example, a network of one-way streets.

10 Bipartite graphs In this type of graph, the vertices are divided into two sets V = A B. There are no edges between vertices in the same set.

11 Shortest path problems How would you go about finding the shortest path from one place to another on a graph? A useful algorithm for doing this is Dijkstra s algorithm. Extra terminology: Walk An alternating, connected, sequence of vertices and edges. Path A walk in which all the vertices are unique. Cycle A path which starts and ends in the same place.

12 Dijkstra s algorithm to find shortest distance from vertex to all other vertices Start with a weighted graph G={V,E}, where a(i, j) is the distance from vertex i to vertex j. L(i) is the shortest distance from vertex to vertex i. L (i) is a temporary upper bound on L(i). P V is the set of permanently labelled vertices. T is the complement of P. Initially, P={}, L()=0 and L (j)=a(,j). Step : Find a vertex k in T with the smallest upper bound L (k). Add k to P, and set L(k) = L (k). Step : Set L (j) = min[l (j), L(k)+a(j, k)] Stop when P=V.

13 How could you prove that this algorithm always gives the shortest path? What is the complexity of this algorithm? If you tried it on a computer and it took one second to find a shortest path in a graph with a million vertices, how long would it take for a graph with two million vertices?

14 Another shortest path problem C B 4 D F A E What s the shortest path between A and F, using Dijkstra s algorithm?

15 Facility location problems Given a map of a town, where should town planners put a new school or police station? In the case of a school, it might be best to put it such that the average distance from all buildings or houses is minimised (minsum). In the case of a police station, it might be best to put it such that the maximum distance to any building is minised (minimax). We can calculate both of these from the shortest distance matrix (a matrix for which d(i, j) is the shortest distance from i to j).

16 New concepts this week Graphs as sets of edges and vertices Different types of graphs: directed, undirected, bipartite Dijkstra s algorithm to find shortest paths Facility location

17 Programming exercises We ll be using the Python language to work on applications of graph theory. Windows installation files for Python and other required libraries are on muele. Install Python, then the NetworkX and Matplotlib libraries. On Linux, installation is even easier as Python should already be installed. To set up the extra libraries on Ubuntu: sudo apt-get install python-networkx python-matplotlib