Euler Circuit Problems

1. Notes 21 - Section 5.1 Euler Circuit Problems

2. Essential Learnings • Students will understand Euler Circuit Problems.

3. Euler Paths and Circuits

4. Euler Circuit Problems • Routing problems – problems concerned with finding ways to route the delivery of goods and/or services to an assortment of destinations. • Existence - is an actual route possible? • Optimization - which one is the optimal route?

5. Euler Circuit Problems • Euler circuit problems – a special class of routing problems. • The common thread in all Euler circuit problems is the exhaustion requirement – the requirement that the route must wind its way through…everywhere.

6. Euler Circuit Problems • In an Euler circuit problem, every single one of the streets, bridges, lanes, highways within a defined area must be covered by the route. • Exhaustive routes – Most common: mail delivery, police patrols, garbage collection, street sweeping, and snow removal.

7. Example - Walking the ‘Hood’ • After a rash of burglaries, a private security guard is hired to patrol the streets of the Sunnyside neighborhood. • The security guard’s assignment is to make an exhaustive patrol, on foot, through the entire neighborhood.

8. Example - Walking the ‘Hood’ • He leaves his car across from the school at S and must return to his car at the end of his route. • Is it possible to start and end at S, cover every block and pass through each block just once?

9. Example - Walking the ‘Hood’ If some of the blocks will have to be covered more than once, what is an optimal route that covers the entire neighborhood? • Optimal means “with the minimal amount of walking”.

10. Example - Delivering the Mail • A mail carrier has to deliver mail in the same Sunnyside neighborhood. • The mail carrier must pass through blocks twice with houses on both sides of the street and once with houses on one side of the street.

11. Example - Delivering the Mail • The mail carrier does not have to walk a block if there no houses on the street. • Find an optimal route that would allow the carrier to cover the neighborhood with the least amount of walking.

12. Example - The 7 Bridges • Can a walker take a stroll and cross each of the 7 bridges without crossing any of them more than once?

13. Example - The Bridges of Madison County • Madison County is a quaint old place, famous for its quaint old bridges. • A famous photographer is hired to take pictures of each of the 11 bridges for a national magazine. • The photographer needs to drive across each bridge once for the photo shoot. • There is a $25 toll to drive across a bridge. • The photographer wants to minimize the total cost of the trip and to re-cross the bridges only if it is absolutely necessary.

14. Example - The Bridges of Madison County

15. Example – Child’s Play • Unicursal tracings – trace each drawing without lifting the pencil or retracing any of the lines. • Closed unicursal tracing, you end at the same place you started. • Openunicursal tracing, you start and end in different places.

16. Example - Child’s Play Which can’t be traced (without cheating)? How can we tellif a unicursal tracing (open or closed) is possible?

17. Assignment CPA Review WS CPA - Thursday