Presentation is loading. Please wait.

Presentation is loading. Please wait.

Chapter 1: Urban Services Chapter at a Glance…

Similar presentations


Presentation on theme: "Chapter 1: Urban Services Chapter at a Glance…"— Presentation transcript:

1 Chapter 1: Urban Services Chapter at a Glance…
Management Science Optimal Solutions for Urban Services Euler Circuits Parking-Control Officer Problem Finding Euler Circuits Qualifications: Even Valence and Connectedness Beyond Euler Circuits Chinese Postman Problem Eulerizing a Graph Urban Graph Traversal Problems More practical applications and modifications

2 Chapter 1: Urban Services Management Science
Uses mathematical methods to help find optimal solutions to management problems…often called Operations Research. Optimal Solutions — The best (most favorable) solution Government, business, and individuals all seek optimal results. Optimization problems: Finish a job quickly Maximize profits Minimize costs Urban Services to optimize: Checking Parking Meters Delivering Mail Removing Snow Collecting Garbage

3 Chapter 1: Urban Services Application
Parking-Control Officer Problem Checking parking meters Our job is to find the most efficient route for the parking-control officer to walk as he checks the parking meters. Goals for Parking-Control Officer Must cover all the sidewalks without retracing any more steps than necessary. Should end at the same point at which he began. Street map for part of a town.

4 Chapter 1: Urban Services Terminology
Graph – A finite set of dots (vertices) and connecting links (edges). Graphs can represent a city map, air routes, etc. Vertex (pl. vertices) – A point (dot) in a graph where the edges meet. Edge – A link that joins two vertices in a graph (traverse edges). Graph Edge Vertex

5 Chapter 1: Urban Services Terminology
Valence – The number of edges touching that vertex (counting spokes on the hub of a wheel). Connectedness – You can reach any vertex by traversing the edges given in the graph.

6 Chapter 1: Urban Services Terminology
Path – A connected sequence of edges showing a route, described by naming the vertices traveled. ABC CBE Others? Circuit – A path that starts and ends at the same vertex. ABEDA BEB

7 Chapter 1: Urban Services Path vs. Circuit
Paths – start and end at any vertex using the edges given. Examples: NLB, NMRB, etc. Circuits – starts and ends at the same vertex. Examples: MRLM, LRBL, etc. Nonstop air routes

8 Chapter 1: Urban Services Euler Circuits
Street map for part of a town. Parking-Control Officer Problem Checking parking meters Our job is to find the most efficient route for the parking-control officer to walk as he checks the parking meters. Problem: Check the meters on the top two blocks. Goals for Parking-Control Officer Must cover all the sidewalks without retracing any more steps than necessary. Should end at the same point at which he began. Problem: Start and end at the top left-hand corner of the left-hand block. Euler circuit – A circuit that traverses each edge of a graph exactly once and starts and stops at the same point.

9 Chapter 1: Urban Services Euler Circuits
Simplified graph is superimposed on the streets with parking meters. Simplified graph is enlarged to show the points (vertices) labeled with letters A – F which are linked by edges.

10 Chapter 1: Urban Services Circuits vs. Euler Circuit
Both start and end at same vertex. Euler circuits travel each edge once and cover all edges. Circuits may retrace edges or not use all the edges.

11 Chapter 1: Urban Services Leonard Euler
Among other discoveries, he was credited with inventing the idea of a graph as well as the concepts of valence and connectedness. Leonard Euler (1707 – 1783)

12 Chapter 1: Urban Services Konigsburg Problem
Our story begins in the 18th century, in the quaint town of Königsberg, Prussia on the banks of the Pregel River. Königsberg became a very important city and trading center with its location strategically positioned on the river. The healthy economy allowed the people of the city to build bridges across the river. Within the town are two river islands connected to the banks by seven bridges. According to lore, the citizens of Königsberg used to spend Sunday afternoons walking around their beautiful city. While walking, the people of the city decided to create a game for themselves, their goal being to devise a way in which they could walk around the city, crossing each of the seven bridges only once. There are three bridges from each bank. Two bridges from each bank to one of the islands and one bridge from each bank to the other island. The islands are connected to each other by a bridge.

13 Chapter 1: Urban Services Konigsburg Problem
Directions: Use the construction paper given to you to set up the problem and reenact the people of Königsberg walking around the city, trying to cross each of the seven bridges only once. Use the hint on the right to help you set it up. Can you take a walk through the town, visiting each part of the town and crossing each bridge only once? *Include answer and reenactment below*

14 Chapter 1: Urban Services Konigsburg Problem

15 Chapter 1: Urban Services Drawing Graphs

16 Task: Represent the street network of stores to be serviced for delivery as a graph. The stars represent stores.

17 Solution: Start with a basic street network
Solution: Start with a basic street network. Without the stores, it looks like this:

18 Solution: Now replace each intersection or corner with a vertex
Solution: Now replace each intersection or corner with a vertex. Represent these with circles like this:

19 Solution: By replacing each row of stores with an edge, the graph is made!

20 Task: Is this graph an Euler Circuit? Draw an Euler Circuit.

21 Solution: The stores as an Euler Circuit!

22 Task: Consider the following floor plan
Task: Consider the following floor plan. The open space represents a door that one can pass through. Represent the floor plan as a graph.

23 Solution:

24 Solution: Comparison

25 Task: Is this graph an Euler Circuit? Draw an Euler Circuit.

26 Solution: OABEDACDCO

27 Chapter 1: Urban Services Konigsburg Problem

28 Chapter 1: Urban Services Finding Euler Circuits
Two Ways to Find an Euler Circuit Trial and error Keep trying to create different paths to find one that starts and ends at the same point and does not retrace steps. Mathematical approach (better method) An Euler circuit exists if the following statements are true: All points (vertices) have even valence. The graph is connected.

29 Chapter 1: Urban Services Finding Euler Circuits
Valence – The number of edges touching that vertex (counting spokes on the hub of a wheel). Connectedness – You can reach any vertex by traversing the edges given in the graph. Euler circuit – Has even-valent vertices and is connected. If vertices have odd valence, it is not an Euler circuit. An Euler circuit starting and ending at A

30 Chapter 1: Urban Services Finding Euler Circuits
Is there an Euler Circuit? Does it have even valence? (Yes) Is the graph connected? (Yes) Euler circuit exists if both “yes.” Create (Find) an Euler Circuit Pick a point to start (if none has been given to you). Number the edges in order of travel, showing the direction with arrows. Cover every edge only once, and end at the same vertex where you started.

31 Chapter 1: Urban Services Beyond Euler Circuits
Chinese Postman Problem In real life, not all problems will be perfect Euler circuits. If no Euler circuit exists (odd valences), you want to minimize the length of the circuit by carefully choosing the edges to be retraced. For our purposes, we assume all edges have the same length—simplified Chinese postman problem. Chinese mathematician Meigu Guan first studied this problem in 1962, hence the name. The blue dots indicate parking meters along the street.

32 Task: Represent the street network of parking meters as a graph
Task: Represent the street network of parking meters as a graph. The blue dots represent parking meters.

33 Solution: Is this graph an Euler Circuit?

34 Chapter 1: Urban Services Beyond Euler Circuits
Eulerize the Graph to Solve Chinese Postman Problem For graphs that are connected but have vertices with odd valence, we will want to reuse (duplicate) the minimum number of edges until all vertices appear to have even valence. Only existing edges can be duplicated (or added). Each edge that is duplicated (added) will later be the edge that will be reused during eulerization. A circuit is made by reusing the edge CG. Below, the graph is eulerized (starts and stops at same point and covers all “edges” once — including reused ones. The edge CG is reused, which would make all vertices appear to have even valence. 34

35 Chapter 1: Urban Services Beyond Euler Circuits
Steps to Eulerizing a Graph Locate all of the vertices with an odd valence. Add one edge at each such vertex, matching up the new edge with an existing edge in the original graph. After Eulerization, each vertex has an even valence. Create an Euler circuit by numbering each edge and indicating which direction. Final step is to “squeeze our Euler circuit onto the original graph that indicates reuse. **If you add the new edges correctly, the number of reuses of the edges equals the number of edges added during eulerization**

36 Chapter 1: Urban Services Beyond Euler Circuits
Hints for Eulerizing a Graph For the most efficient eulerization, look for the fewest edges to add to make all vertices even. Typically, locate odd valence vertices and try to reuse (add) the connecting edge between the vertices. Sometimes vertices are more than one edge apart; in this case, reuse edges between vertices (see graph below). Remember: Only duplicate (add to) the existing edges. Odd vertices, X and Y, are more than one edge apart. This is not allowed — must only reuse existing edges. Reuse existing edges between the odd vertices.

37 Chapter 1: Urban Services Beyond Euler Circuits
A Better Eulerization Only reuse (add) edge BC. Squeeze the eulerized circuit onto the graph.

38 Task: Eulerize the following graph.

39 Task: Now “squeeze” the graph.

40 Solution: Is this optimal?

41 Chapter 1: Urban Services Beyond Euler Circuits
Rectangular Networks – This is the name given to a street network composed of a series of rectangular blocks that form a large rectangle made up of so many blocks high by so many blocks wide. Eulerizing rectangular networks: “Edge Walker” Start in upper left corner (at A). Travel (clockwise) around the outer boundary. As you travel, add an edge by the following rules: If the vertex is odd, add an edge by linking it to the next vertex. If this next vertex becomes even, skip it (just keep “walking”). If this next vertex becomes odd, (on a corner) link it to the next vertex. Repeat this rule until you reach the upper left corner again.

42 Chapter 1: Urban Services Urban Graph Traversal Problem
Euler Circuits and Eulerizing Graphs: Practical Applications Checking parking meters (discussed) Collecting garbage Salting icy roads Inspecting railroad tracks Special Requirements May Need to Be Addressed Traffic directions Number of streets/lanes (divided routes) Parking time restriction

43 Chapter 1: Urban Services Urban Graph Traversal Problem
Theory Modifications Can Address Special Requirements A digraph (directed graph) is used to show one-way street. Arrows show restriction in traversal possibilities (not part of circuits). Territories may need to be divided into multiple routes. Salt-spreading route, where each west-west street has two traffic lanes in the same directions Appropriate digraph model

44 Chapter 1: Urban Services Urban Graph Traversal Problem
Dr. Doug Shaw is usually blamed for spreading mathematical rumors. After investigating the rumor sources we have the following: Basho T. heard the rumor from Jordan M. Wayland M. heard it from Ren W. & Jordan M. Doug Shaw heard it from Laurel S. and Wayland M. Ren W. heard it from Laurel S. Laurel S. heard it from Doug Shaw Make a digraph to investigate!

45 Chapter 1: Urban Services Urban Graph Traversal Problem
Can one conclude that Doug Shaw is the true culprit? Who is the only one who could not have started the rumor? If Laurel S. incorrectly stated that Doug Shaw was the source of the rumor she heard, then what can one conclude?

46 Suppose Amina needs to spread a written message among friends
Suppose Amina needs to spread a written message among friends. The paper can only be given to one person at a time. Because of a restriction in the flow of communication, her 6 friends can pass the message as follows: Friend Can Deliver to Heidi Nadia, Amina Nadia Heidi, Adam Ali Faiz, Bara Faiz Bara, Ali Amina Heidi, Faiz Bara Faiz, Ali Adam Heidi, Nadia

47 Create a digraph.


Download ppt "Chapter 1: Urban Services Chapter at a Glance…"

Similar presentations


Ads by Google