graph theory application - not only for engineering problems

Graph Theory Application - Not Only for Engineering Problems

Zuzana Kubínová, HTS, 11-9-2012

2 HONEYWELL - CONFIDENTIAL

About The Speaker

•Master's degree in Mathematical Engineering obtained at Brno University of Technology

•Spent six months studying abroad in 2008 - in UK Nottingham Trent University

•Diploma thesis Analysis of Data Flow in the Fly-by-Wire System written for Honeywell

•Systems Engineer for Honeywell Technical Solutions (HTS CZ) in Brno, Czech Republic since May 2009


Why Is This Relevant To Me?

•Finishing school is never the end of learning.

•Every company needs innovation. Not only innovation of products but also innovation of processes. Be the one with out of the box ideas and bring your company growth. Become a competitive advantage for your company!

•Graphs describe relations. In many parts of your life and work there are relations, therefore there is a good chance to use graphs and graph algorithms.

•There are ways to bake a cake and there are ways to analyze and make decisions. Let's explore!


Introduction

•The Graph Theory started in a paper published in 1736 by the Swiss mathematician Leonhard Euler

•Problem known as Seven bridges of Koningsberg 1

1 Picture from www.uh.edu


Introduction

•Graphs versus Charts

• Charts

• Graphs from a mathematical point of view

• G=(N,E,ε), where N is a set of nodes, E is a set of edges and ε:E N x N is a mapping

• In accordance with characteristics of ε we can model several types of graphs - e.g. simple graph, multigraph or directed graph


Basic Types Of Graphs

•Simple graph • Multigraph • Directed graph (digraph)


Graph – Graphical And Non-Graphical Representation

•Graphical representation 1

• Non-Graphical representation

• it is what makes the Graph Theory so powerful

• allows handle large data effectively

• Edge list, Adjacency list, Adjacency matrix

1 Both graphs from www.graphviz.org


A House By One Line

•Do you know how to draw a house by one line?

•Do you know if it is possible at all?


A House By One Line

•How about these?

• Euler path – it exists if and only if each node has even number of edges or there are exactly two nodes with odd number of edges.


Back To Seven Bridges Of Koningsberg

•So, is it possible to cross all the bridges only once and get back home?

•How would you modify it?


Students And Graph Coloring

•Problem:Arrange the students’ sitting in a classroom such that they can’t copy from each other during a test.



•Graph Coloring

• Any two adjacent nodes are assigned a different color.

• If you use k colors, we call the graph k-colorable.

• Goal is to minimize k – use as little colors as possible.



Graph Coloring


Path Search Algorithms•Used for variety of tasks – e.g. shortest path search, search for

existing path, search in timetables, find steps to solve the Rubik’s Cube…


Path Search Algorithms

… and their use in Honeywell Aero Flight Control division


Path Search AlgorithmSystem is really complicated …

• Using search path algorithms saves lots of time, money and sanity of engineers


Summary

•The Graph Theory is a fascinating branch of mathematics suitable for many engineering applications.

•Mr Euler's story shows that even amusing little brainteasers can give rise to a new and very useful theory.

•Appropriate use of graph algorithms can be very benefiting.


Q & A


Thank you for your attention.