Mathematical Ideas that Shaped the World

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Mathematical Ideas that Shaped the World

• Graphs and Networks

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Plan for this class

• What was the famous Königsberg bridge problem?
• What is a graph?
• Why was the 4-colour theorem controversial?
• How are soap bubbles and slime mould good at town
  planning?
• Why is it so hard for salesmen to be efficient?
• How does Google work?

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The Seven Bridges of Königsberg
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The Seven Bridges of Königsberg

• Once upon a time there was a city called
  Königsberg in Prussia.
• It was founded in 1255 by the Teutonic Knights,
  and was the capital of East Prussia until 1945.
• It was a centre of learning for centuries, being
  home to Goldbach, Hilbert, Kant and Wagner.

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The Seven Bridges of Königsberg

• Running through the city was the River Pregel.
• It separated the city into two mainland areas and
  two large islands.
• There were 7 bridges
  connecting the various
  areas of land.

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The Seven Bridges of Königsberg

• The residents of Königsberg wondered whether they
  could wander around the city, crossing each of
  the seven bridges once and only once.
• Can you find a way?

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Leonhard Euler (1707 1783)

• Born in Basel, Switzerland, and was expected to
  become a pastor like his father.
• Studied Hebrew and theology at university, but
  got private maths lessons from Johann Bernoulli.
• In 1727 got a job in the medical section at the
  Uni of St Petersburg

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Leonhard Euler (1707 1783)

• but in the chaos surrounding the death of
  Empress Catherine I, he managed to sneak into the
  maths department.
• Got married in 1733 and had 13 children, of whom
  5 survived to adulthood.
• In 1741 moved to Berlin, where he spent 25 years.

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Leonhard Euler (1707 1783)

• Published over 500 books and papers in his
  lifetime, with another 400 posthumously.
• Invented the notation i, p, e, sin, cos, f(x) and
  more!
• Lost sight in both eyes but became more
  productive, saying
• now I have fewer distractions

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Back to Königsberg

• In 1736 Euler turned his mind to the problem of
  the bridges of Königsberg.
• He realised that it didnt matter how you walked
  around the land, or where exactly the bridges
  were.
• It only mattered how many bridges there were
  between each bit of land, and in what order you
  crossed them.

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Reformulating the problem

• With this observation, we can re-draw the bridges
  of Königsberg as follows

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Conditions for a solution

• Eulers Eureka! moment was realising that
  whenever you cross into a bit of land, you also
  have to cross back out of it.
• Therefore, for a bridge tour to be possible,
  there must be an even number of bridges coming
  out of every bit of land.
• (Except for the starting and finishing points.)

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An impossible problem!

• If we look again at the map of Königsberg, we see
  that there are an odd number of bridges coming
  out of every bit of land, so such a walk around
  the city is impossible.

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Königsberg extra

• Look at your handout to learn about these
  characters.
• Can you make them all happy?

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Postscript on Königsberg

• Königsberg was heavily bombed during World War
  II.
• The city was taken over by Russia and re-named
  Kaliningrad.
• Two of the 7 bridges were destroyed

Question Is the bridge problem possible now?
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The beginning of graph theory

• By solving the problem the way he did, Euler
  invented the subject of graph theory.
• A graph is a collection of nodes and edges.

• It doesnt matter how long the edges are or where
  the nodes are it only matters which edges are
  connected to which nodes.

edge
node
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Examples of graphs
Train maps
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Examples of graphs
Social networks
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Examples of graphs
Chemical models
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The Four Colour Theorem
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The Four-Colour Problem

• Proposed by Francis Guthrie in 1852 and remained
  unsolved for more than a century.
• Can any map be coloured with 4 colours so that no
  two adjacent regions have the same colour?

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Example of a 4-colouring
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Why not 3 colours?

• A simple example shows that it impossible to
  always colour a map with only 3 colours.

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Why not 5 colours?

• It was proved by 1890 that every map can be
  coloured with at most 5 colours.
• The difficult part of the problem was to show
  that there was no map sufficiently complicated as
  to need 5 colours.
• Martin Gardner set the following graph as a
  problem to his readers. Can you colour it using
  only 4 colours?

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Martin Gardners map
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In terms of graphs

• The 4-colour problem can be phrased in terms of
  graphs.
• Each region of the map becomes a node, with two
  nodes being connected by an edge if and only if
  the regions are adjacent on the map.
• The problem becomes can you colour the nodes
  with 4 colours so that an edge never connects two
  nodes of the same colour?

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Maps to graphs example
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A proof?

• In 1976, two men called Kenneth Appel and
  Wolfgang Haken announced that they had a proof of
  the conjecture.

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A controversial result

• They had made a computer program to check the
  4-colouring of all possible examples (1,936 of
  them!).
• It was the first mathematical theorem to be
  proved with computer help, and aroused much
  controversy.

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An inelegant result

• One critic said
• A good mathematical proof is like a poem. This
  is a telephone directory!
• However, the proof is now widely accepted and
  computers are used in many areas of pure
  mathematics.

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Building efficient graphs
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Building the shortest graphs

• Very often we have a set of points and want to
  find the shortest collection of edges that
  connect them up. For example,
• Roads/railways connecting towns
• Telephone/internet cables
• Gas pipes
• Connections in electronic circuits
• Neurons connecting bits of your brain

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The shortest graph?

• Suppose we have 4 towns that we wish to connect
  up. Which of these do you think is shortest?

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An unexpected solution

• If were restricted to roads between towns, then
  the first graph is the shortest.
• But there is a better solution, which we can find
  using a bit of perspex and some soap bubbles

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Soap bubbles know best

• So the best solution is to create two ghost towns!

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How do they do it?

• We currently have no (fast) algorithm for finding
  the shortest Steiner graph between a given number
  of points.
• Nature, on the other hand, is quite good at it.
• http//www.youtube.com/watch?v0lpsLCgCp2Q

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Slime mould is better than politicians

• Scientists studied slime mould growing in a
  region shaped like Tokyo, placing food sources
  where the major regional cities would be.
• The resulting slime mould network was remarkably
  similar to the Tokyo train network.
• In some respects it was actually better!

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Slime mould networks
Tokyo train network
Slime mould
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Finding the shortest route
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The Chinese postman problem

• Now suppose that the towns and roads are fixed,
  and we know the distances between them.
• The Chinese postman problem asks what is the
  shortest route that travels over every road at
  least once and returns to the start?

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3
5
9
8
8
6
4
9
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The Chinese postman problem

• Heres how to solve the problem
• If the graph is Eulerian (i.e. an even number of
  edges out of every node) then each edge can be
  walked exactly once, so we are done.
• If not, find the shortest distances between the
  nodes with odd numbers of edges, and add extra
  edges to turn it into an Eulerian graph.

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Chinese postman example
B
C
5
5
3
9
8
A
D
8
4
9
6
F
E
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The travelling salesman problem

• If, instead, you are a travelling salesman, you
  wish to find the route that allows you to visit
  each town exactly once (and then return to the
  start).
• This problem was posed as long
  ago as 1800 by the Irish
  mathematician Hamilton, and
  rose drastically in popularity
  in the 1950s and 60s.

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The Icosian Game
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(Or the travel version!)
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• This is the poster for a contest run by Proctor
  Gamble in 1962.
• There were 33 cities in this problem.

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A tantalising problem

• Unlike the Chinese postman problem, nobody has
  ever found a fast algorithm for solving the
  Travelling Salesman Problem (TSP).
• Deciding whether there is a route shorter than a
  given length is an NP-complete problem.
• Finding a good algorithm is currently worth 1
  million!

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Methods of solving the TSP

• Brute force try all possible routes and pick
  the fastest one.
• Caveat using todays fastest supercomputer,
  solving the 33-city problem using this method
  would take about 100 trillion years!

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Methods of solving the TSP

• Branch and bound algorithms divide the problem
  into smaller graphs and try to eliminate edges
  that cant be part of the solution.
• The record set with this kind of exact method is
  85,900 cities, which took over 126 CPU years to
  compute in 2006.

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Methods of solving the TSP

• Heuristics find good solutions which are
  highly likely to be close to the perfect
  solution. For example,
• The nearest neighbour algorithm lets the salesman
  pick the nearest unvisited city every time.
• Find any route, then rearrange edges to find a
  shorter one.

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Methods of solving the TSP

• Heuristic algorithms can find solutions to TSP
  with millions of cities in a fairly short amount
  of time.
• Caveat these solutions may not always be the
  best possible.

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Have a go!

• Humans are surprisingly good at finding solutions
  to TSP quite quickly.
• Play the following game online to see how good
  you are!
• http//www.tsp.gatech.edu/games/tspOnePlayer.html

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Applications

• The Travelling Salesman Problem has lots of
  applications in our lives
• Logistics of delivering goods
• Drilling holes in circuit boards
• Genome sequencing
• Programming space telescopes like Hubble
• Collecting post from postboxes every day
• Making travel itineraries
• X-ray crystallography

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How does Google work?
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The internet to Google

• Google sees the internet is a giant graph.
• Each webpage is a node, and two pages are joined
  by an edge if there is a link from one page to
  the other.
• Note the edges in the internet graph have a
  direction.
• The algorithm that Google uses to rank its
  searches is called PageRank.

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How does PageRank work?

• Idea the more links a page has pointing to it,
  the more important it is.
• Second idea if an important page links to your
  page, this is worth more than if an unimportant
  page links to you.
• For example, Wikipedia referencing you is worth
  more than Haggis The Sheep referencing you.

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Example
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Using PageRank to make money!

• People who understand PageRank can make it very
  lucrative for them.
• For example, businesses or individuals with a
  high page rank can sell links to those wishing to
  boost their page rank.
• Businesses have also used similar algorithms to
  rank universities in the job market.

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Social networking

• Graphs are also important to social networking
  sites like Facebook.
• By analysing the preferences of your friends
  and pages that you like, Facebook can target
  its advertising very effectively.

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Making recommendations

• Similarly, shops like Amazon use graphs to make
  suggestions for future shopping.
• In 2009 the US company Netflix awarded 1 million
  to the people who best improved their
  recommendation algorithm.
• (One of the problems was that they could never
  predict whether someone would like the film
  Napoleon Dynamite!)

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Lessons to take home

• That Graph Theory is an incredibly important part
  of modern-day life.
• That a solution to a single graph theory problem
  can have many different real-world applications.
• That slime mould is often cleverer than humans.
• That problems in graph theory can be worth a lot
  of money!

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References

• Here are some good places to read more about the
  subjects in todays lecture.
• Four Colour Theorem http//nrich.maths.org/6291
• Four Colours Suffice by Robin Wilson, Penguin
  Books, 2003
• A comprehensive website about the Travelling
  Salesman problem http//www.tsp.gatech.edu/
• A New York Times article about the NetFlix prize
  http//www.nytimes.com/2008/11/23/magazine/23Netfl
  ix-t.html