Induction

AE1APS Algorithmic Problem SolvingJohn Drake

Coursework 4 deadline – ◦ Thursday 29th November at 1pm

Same submission guidelines as before, hard-copy during lectures or soft-copy via e-mail

John.Drake@nottingham.edu.cn

Reminder

Invariants – Chapter 2 River Crossing – Chapter 3 Logic Puzzles – Chapter 5 Matchstick Games - Chapter 4

◦ Sum Games – Chapter 4 Induction – Chapter 6

◦ Tower of Hanoi – Chapter 8

Outline of the course

Induction is the name given to a problem solving technique based on using the solution to small instances of a problem to solve larger instances of a problem.

In fact we can solve problems of arbitrary size

Induction

We need to measure the "size" of an instance of a problem

E.g. with matchstick problems, we might call the size the number of matches

Size

It is not always obvious what the size should be for some types of problems

The size must be non–negative i.e. whole numbers 0, 1, 2, …

We call these numbers "natural numbers", as they are naturally used for counting

Size

Note that some mathematicians do not include 0 in the natural numbers

There are good reasons why we should include 0.Zero has important significance in maths (and the history of maths)

One reason in when counting the size of sets, the null set is defined to have size of 0

We will see examples of including 0 later

Zero

We solve the problem in two steps

First we solve the problem instance of size 0

This is usually very easy (often called trivial)

Then, we show how to solve, a problem of size n+1 (where n is an arbitrary natural number), given we have a solution to a problem instance of size n.

Two step process

The first step is called the base case of the basis of induction.

The second step is the induction step

Two step process

By this process, we can solve problems of size 0

We also know how to solve instances of size 1 (as it is one larger than 0)

We can apply the induction step to solve problems of size 2 and so on.

Sizes 0, 1, 2, …

The induction step allows us to solve problems of size one greater than the current size problem

By repeating this process we can solve problems of any size

Sizes 0, 1, 2, …

Straight lines are drawn across a sheet of paper, dividing the paper up into regions.Show that it is possible to colour each region black or white, so that no two adjacent regions have the same colour

Problem statement – cutting the plane.

P120

Cutting the plane

The number of lines is the size of a problem We have to show how to solve the problem

when there are zero lines We have to show how to solve a problem

when there are n+1 lines, assuming we can solve the problem with n lines

Let us call the situation when no two adjacent regions have the same colouring "a satisfactory colouring"

Outline

This is trivial.

With 0 lines, there is one region

This can be coloured black or white and will be a satisfactory colouring.

Zero case (base case)

Assume a number of lines (n) are drawn on the paper, and the regions have a satisfactory colouring

Induction step

Now we add an additional line (i.e. n + 1)

This will divide some existing regions into two new regions, which will have the same colour, therefore the colouring is not satisfactory.

Induction step

The task is to show how to modify the colouring so that it does become satisfactory.

Induction step

Inverting any satisfactory colouring will also give a satisfactory colouring.◦ i.e. changing black regions for white, and vice

versa will also be satisfactory

Let us call the two sides of the new line left and right. Inverting colours in either region will still give a satisfactory colouring in that half (left or right)

The key (or the trick)

Hence, inverting the colours in one half will guarantee a satisfactory colouring for the whole piece of paper

The key (or the trick)

The algorithm for colouring is non-deterministic in several ways◦ The initial colouring is not specified (could be black or

white)◦ The order in which the lines are added is unspecified◦ The naming of the left and right regions is not specified

(symmetry)

This means the final colouring can be arrived at in different ways, but it is guaranteed to be satisfactory

Non- deterministic

A square piece of paper is divided into a grid of size 2n by 2n, where n is a natural number

Individual squares are called grid squares and a single grid square is covered

E.g. n = 3

Triominoes – problem statement

A triomino is an "L" shape made of 3 grid squares. Show that it is possible to cover the remaining grid squares using triominoes

Triominoes – problem statement

The obvious size to use is n

The base case is when n=0, and the grid has size 1 by 1 (i.e. 20 by 20)

This square will of course be covered

The base case is solved

Size and Base case

We want to solve a grid of 2n+1 by 2n+1

We know how to cover a 2n by 2n grid, how do we use this to help us cover a grid size larger by one

We make the inductive hypothesis that this is possible

Inductive Step

A grid of size 2n+1 by 2n+1 can be divided into 4 grids of size 2n by 2n by drawing horizontal and vertical lines

These 4 regions can be called bottom-left, bottom-right, top-left and top-right

One square is already covered, we will assume this is in the bottom left square, why?

The key

The bottom left grid is of size 2n by 2n , of which one square has already been covered.By the induction hypothesis, the remaining squares in the bottom-left grid can be covered

The key

The bottom left grid is of size 2n by 2n , of which one square has already been covered.By the induction hypothesis, the remaining squares in the bottom-left grid can be covered

The key

This leaves us having to cover the remaining 3 grids

None of these have any squares covered

We can apply the inductive hypothesis if just one of the squares in each of the 3 remaining grids is covered

Induction

This can be done by placing a triomino at the junction of the 3 grids

Induction

What if we had chosen n = 1 to be our base case?

Our proof would have left out the 0 case, so we would still need to prove that separately

However including 0 provides a slightly deeper insight

"Humans" start counting at 1, computer scientists and mathematicians start counting at 0

If 1 was the base case.