Redraw this PCB in such a way that no wires cross. Review: Mini-Quiz Planarity Redraw this PCB in such a way that no wires cross. B C A D E *** Notice that B and C do not have to connect. Slide 1

Class Greeting

Objective: The student will be able to solve Cross Topic Mathematical Problems.

What’s the problem?

You are choosing your courses… You want to study Math, Physics, French and Philosophy You also want to attend Basketball Club Each of these is available in one or more of 6 different option blocks Subject Option Blocks Maths A, B, F Physics D, E, F French A, B Philosophy C, E Basketball Club C

if your combination of courses is possible? How do you decide if your combination of courses is possible?

Let’s summarise the problem We have a set of subjects and a set of option blocks with each subject available in one or more of the option blocks.

We want to match up our chosen subjects with possible option blocks in a systematic method.

This is an example of a Matching Problem

Matching Problems We have two distinct sets of vertices with some edges between the two sets but no edges within a set

We want to find as large as possible a subset of the edges with no two edges sharing a vertex.

Problem: A student at the beginning of year M6 is choosing his course options. He wants to study Math, Physics, French and Philosophy, and he also wants to attend Basketball Club. The College timetable is divided into 6 option blocks labelled A to F. The table below shows which of the student’s chosen courses is available in each option block. Subject Option Blocks Maths A, B, F Physics D, E, F French A, B Philosophy C, E Basketball Club C

  Draw a bipartite graph to represent the availability of subjects. Explain why it is not possible to find a complete matching. The student would like to avoid having lessons in Option E because this is taught on a Friday afternoon. Explain why this is not possible. Having accepted that he cannot avoid Option E, the student would like to try to avoid Option A, which is taught on a Monday morning. Find a maximal matching that does not include Option A. The student suddenly realises that the College also offers guitar lessons, available only in Option D. Even though it will mean that he has a full timetable (and has to get out of bed on a Monday morning) he would like to also attend these. With your maximal matching from (d) as a starting point, use the Alternating Path Algorithm to find a complete matching

Solution: Draw a bipartite graph to represent the availability of subjects. a) The bipartite graph is shown below. MATHS PHYSICS FRENCH PHILOSOPHY BASKETBALL A B C D E F   It is only possible to find a complete matching when each set has the same number of nodes. Explain why it is not possible to find a complete matching.

The student would like to avoid having lessons in Option E because this is taught on a Friday afternoon. Explain why this is not possible. MATHS PHYSICS FRENCH PHILOSOPHY BASKETBALL A B C D E F Basketball Club is only available in Option C and Philosophy only in C and E. The student will therefore have to do Basketball in C and Philosophy in E.   

Having accepted that he cannot avoid Option E, the student would like to try to avoid Option A, which is taught on a Monday morning. Find a maximal matching that does not include Option A. MATHS PHYSICS FRENCH PHILOSOPHY BASKETBALL A B C D E F A maximal matching is   Maths - F Physics – D French – B Philosophy – E Basketball – C

An alternating path is: Guitar – D = Physics – F = Maths – A The student suddenly realises that the College also offers Guitar Lessons, available only in Option D. Even though it will mean that he has a full timetable (and has to get out of bed on a Monday morning) he would like to also attend these. With your maximal matching from (d) as a starting point, use the Alternating Path Algorithm to find a complete matching MATHS PHYSICS FRENCH PHILOSOPHY BASKETBALL A B C D E F GUITAR An alternating path is: Guitar – D = Physics – F = Maths – A This gives a complete matching of:   Maths – A Physics – F French – B Philosophy – E Basketball – C Guitar – D

Another example where this is used British Airways scheduling staff: deciding who will work, when, and where…

Lesson Summary: Objective: The student will be able to solve Cross Topic Mathematical Problems.

Preview of the Next Lesson: Objective: The student will be able to solve Cross Topic Mathematical Problems.

Homework Draw your PCB and HW 7