MATHEMATICAL THOUGHT AND PRACTICE. Chapter 5: The Mathematics of Getting Around - Euler Paths and Circuits

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1 MATHEMATICAL THOUGHT AND PRACTICE Chapter 5: The Mathematics of Getting Around - Euler Paths and Circuits

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4 Is it possible for a person to take a walk around town in such a way that each of the seven bridges is crossed once, but only once?

5 Euler Paths and Circuits Euler laid the foundations for a totally new type of geometry: geometris situs or the geometry of location. These basic ideas developed into one of the most important and practical branches of modern mathematics: Graph Theory.

6 Routing Problems Routing problems - finding ways to route delivery of goods or services to a variety of destination Two Questions: 1. Can a route be found? 2. What is the optimal route?

7 Euler Circuit Problems Euler Circuit Problem - every single street (or bridge, lane, or highway) within a defined area must be covered by the route, also known as exhaustive routes. scotthull.com

8 Is it possible to start and end at S, cover every block of the neighborhood, and pass through each block just once? Euler Circuit Problems

9 If some of the blocks will have to be covered more than once, what is an optimal route that covers the entire neighborhood? Euler Circuit Problems

10 Euler Circuit Problems - Königsberg Is it possible for a person to take a walk around town in such a way that each of the seven bridges is crossed once, but only once?

11 The Bridges of Madison County The photographer needs to drive across each bridge once for a photo shoot and wants to cross as few times as possible to avoid toils.

12 Child s Play Open unicursal tracing - Tracing without lifting the pencil or retracing any lines, ending not where we started Closed unicursal tracing - Tracing without lifting the pencil or retracing any lines, ending where we started

13 What Is a Graph? Euler circuit problems can all be solved using a graph. Graphs are pictures made only of: 1. A set of vertices, or dots 2. A set of edges, or lines This graph has the vertex set of {A, B, C, D, E, and F} and edge set of {AB, BB, BC, CD, CD, DE, BE, AD}.

14 What Is a Graph? Loop Edge that loops back to starting vertex, BB. Multiple Edges More than one edge connecting the same two vertices, CD & CD Cross AD and BE cross each other AB = BA since edges have no direction

15 Facebook Friend Networks

16 Isolated Vertices Isolated vertices - vertices having no edges The edge set of a graph with no edges is the empty set (we can write it as E = { } or E = φ).

17 Drawing Graphs Draw this graph: The vertex set is V = {A, D, L, and R}, and the edge set is E = {AD, AL, AL, AR, AR, DL, DR}.

18 Pictures Optional

19 GRAPH A graph is a structure consisting of a set of objects (the vertex set) and relations (the edge set). Relations among objects include the possibility of an object being related to itself (a loop) as well as multiple relations between the same pair of objects (multiple edges). omegamom_graph

20 Terms Vertices in a graph are adjacent if joined by an edge. The degree of a vertex is the number of edges meeting at that vertex. A loop counts twice toward the degree. Odd vertices have odd degree, whereas even vertices have even degree.

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23 Terms Trip a sequence of adjacent edges where an edge can be traveled just once Path an open trip where the end doesn t come back to the beginning Circuit a closed trip where the end and beginning are the same

24 Terms A, B, E, D is a path from vertex A to vertex D. The length (number of edges) in A, B, E, D is 3. A, B, C, A, D, E is a path of length 5 from A to E. A, B, C, B, E is another path from A to E.

25 Terms A, B, C, B, E, E, D, A, C, B is not a path. B, C, B is a circuit of length 2. EE is a circuit of length 1.

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27 Terms A graph is connected (one piece) if you can get from any vertex to any other vertex along a path. An edge whose removal makes a connected graph disconnected is called a bridge. Do we have a bridge?

28 Disconnected Graphs A disconnected graph is made up of separate connected components. BF and FG were bridges, as is FH.

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31 Euler Paths and Circuits memoriesinthedark.files.wordpress.com An Euler path is a path in a connected graph that travels through all edges (only once). An Euler circuit is a circuit that travels through every edge. A connected graph has an Euler path or an Euler circuit or neither.

32 Euler Path Euler Circuit Neither

33 Modeling Euler rephrased problems as graph problems. Using a mathematical concept to describe and solve a real-life problem is one of the oldest and grandest traditions in mathematics: modeling. images.indiebound.com

34 The Seven Bridges of Königsberg This map is not entirely accurate the drawing is not to scale and the exact positions and angles of bridges are changed. Does it matter? An Euler circuit would cross each bridge once and end back at the start; an Euler path would cross each bridge once but not return to the start.

35 Walking the Hood Euler circuit? Euler path? Neither?

36 Delivering Mail Both Sides of Street Euler circuit? Euler path? Neither?

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40 Euler s Theorems

41 Euler Circuit?

42 Euler Path? Is it possible for a person to take a walk around town in such a way that each of the seven bridges is crossed once, but only once?

43 The Seven Bridges of Königsberg

44 Child s Play Euler Circuit or Path? Closed unicursal tracing = Euler circuit Open unicursal tracing = Euler path

45 Child s Play Euler Circuit or Path? Closed unicursal tracing = Euler circuit Open unicursal tracing = Euler path An open unicursal tracing is possible if we start it at K or I, and end it at the other.

46 Sum of Degrees

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56 Algorithms algorithm-art.com For small graphs, trialand-error works fine, but some applications involve thousands of vertices. We need a systematic strategy telling us how to create an Euler circuit or path: an algorithm - a set of procedural rules that, when followed, always lead to some solution.

57 Fleury s Algorithm for Circuits

58 Fleury s Algorithm for Paths

59 Implementing Fleury s Algorithm

60 Implementing Fleury s Algorithm

61 Implementing Fleury s Algorithm

62 Exhaustive Routes Exhaustive route a route that travels along the edges of a graph and passes through each and every edge of the graph at least once. Optimal exhaustive route an exhaustive route that passes through the fewest edges a second time. Eulerizing adding duplicate (must be duplicate) edges to odd vertices, making them even, in order to eliminate all odd vertices, creating an Euler circuit.

63 A 3 by 3 Street Grid The four duplicate edges (BC, EF, HI, and KL) are deadhead blocks, where a second pass is required.

64 A 4 by 4 Street Grid Only duplicate edges are allowed! Not optimal!

65 Open Routes Open route the route may start and end at different points. Semi-eulerization adding duplicate edges to odd vertices, making them even, in order to eliminate all but two odd vertices, creating an Euler path, an optimal exhaustive open route. We start at one of the two odd vertices and end at the other.

66 4 by 4 Street Grid Parade Route The fire department has stipulated that the parade starts at B and ends preferably far from B.

67 The Bridges of Madison County z.about.com

68 The Bridges of Madison County If the photographer needs to start and end his trip in the same place, an optimal route can be found for a cost of $325: $25 to cross each bridge, so the baseline is $275 and $25 for each recrossing.

69 The Bridges of Madison County If the photographer can choose any starting and ending points, an optimal open route (semieulerization) can be found for a cost of $300: $275 for the 11 bridges and $25 for just one recrossing.

70 The Bridges of Madison County If the photographer has to start at B and end at L, we can find a route with just two duplicate edges, for a cost of $325. We find a semi-eulerization where B and L remain odd and R and D go even. Hargrove Covered Bridge

71 The Exhaustive Patrol The odd vertices pair up beautifully, and the optimal eulerization requires only nine duplicate edges.

72 The Grateful No Deadhead All vertices are even, already! An Euler circuit, therefore, must already exist.

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74 Homework Online homework for Chapter 5 Paper homework for Chapter 5 Online quiz for Chapter 5