ae1aps algorithmic problem solving john drake. invariants – chapter 2 river crossing – chapter...

Tower of Hanoi

AE1APS Algorithmic Problem SolvingJohn Drake

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

Breaking larger problems down into sub-problems

Using small instances of a problem to help us solve larger instances

Find a trivial base case where size = 0

Show for an arbitrary number n how to solve n + 1 (induction step)

A recap on induction

This problem is discussed in many maths texts, and is used in computer science and AI as an illustration of recursion and problem solving

You probably know the problem, but we will look at it slightly differently

Induction gives a systematic way of finding a solution

However this solution is undesirable A better solution is obtained by observing the

invariants of the inductive solution

Tower of Hanoi - Introduction

We will start at the end We begin with the solution (as you probably

already know it) and you can see where we are going

Another reason is to understand why a correct solution has been found, if no information about the solution method is provided

Begin with the solution

There is a temple in Bramah, where there are 3 giant poles fixed in the ground.

On the first pole, God placed 64 discs, each of different size in decreasing order of size

The Brahmin monks were given the task of moving the discs, one per day, from one pole to another

However, no disc must be above a smaller disc, when on the same pole

Problem specification

Tower of Hanoi

Almost every book (and webpage) I have seen draws the 3 poles in a line

We can call the poles A, B, C

However there is a simple symmetry which is being ignored here

Triangle representation

We draw them in different positions A better way is to draw and equilateral

triangle, and the symmetry is obvious. Now the moves of the discs can easily be

described as clockwise or anti-clockwise.

Triangle representation

There is a very easy solution to the Towers of Hanoi problem

It is easy to remember and execute We assume the problem is to move the

discs from one pole to the next in the clockwise direction

The days are numbered from 0, 1, 2… On day 0, the discs are placed in their initial

position, and the monks begin moving the discs on day 1.

Iterative solution (assumptions)

On every alternate day, the smallest disk is moved

The smallest disc should cycle around the poles.

The direction of rotation depends on the number of discs

It the total number is odd, it cycles clockwise, otherwise if the total is even, it cycles anticlockwise

Iterative solution 1

On every other day, another disc is moved (other than the smallest disc)

As no disc can be on a smaller disc, there is only one possible move for the disc

The algorithm terminates when no further moves are possible

That is, on a even numbered day when all the discs are on the same pole

Iterative solution 2

Day 0

Iterative solution – 4 discs

Day 1


Day 2


Day 3


Day 4


Day 5


Day 6


Day 7


Day 8


Day 9


Day 10


Day 11


Day 12


Day 13


Day 14


Day 15


Presenting the solution like this, provides us with no help in understanding how the solution was constructed

How would we give a formal mathematical verification of the correctness of the algorithm

By observing a number of invariants, we show how to derive the algorithm from the inductive solution.

Why?

Suppose the task is to move M discs from one pole to another specific pole.

Base case. When there are 0 discs, no steps are needed to complete the task

For the inductive step, we assume we can move n discs from A to B, and the problem is to show how to move n+1 discs from A to B

We get stuck with this approach

Inductive solution

First move n top discs from A to B

After doing this, all the discs are on B

We have no hypothesis about moving discs from this pole

i.e. The information that we can move n discs from A to B does not help us in trying to move n+1 from A to B

Stuck

Example with n = 3

A B C

Alternatively, we can move the smallest from pole A to pole C

We then move n discs from A to B

Once again we have exhausted all possibilities of using the inductive hypothesis

Because the n discs are on pole B and we have no hypothesis about moving discs from this pole.

Still stuck

We have been too specific about the inductive hypothesis

The way out of this is to introduce some parameters which model the start and final positions of the discs

Too Specific

We make a crucial decision

Rather than name the poles A B C, we observe that the problem exhibits a rotational symmetry

This is obvious when the poles are arranged in a triangle, but obscured when the poles are placed in a line.

Rotational symmetry

One additional parameter needs to be introduced, the direction of movement

We only need to say if a particular disc needs to be moved in a clockwise or anti- clockwise direction

The generalisation of the problem becomes, how to move n discs from one pole to the next in the direction d◦ (where d is either clockwise or anti-clockwise)

An additional parameter

The inductive hypothesis we use is that it is possible to move the n smallest discs from one pole to another in direction d beginning from any valid starting point

That is, a starting position in which the discs are distributed arbitrarily over the poles, but no disc is on top of a disc smaller than itself

This is a far more general definition

Inductive hypothesis

For n = 0, the sequence of moves is empty. In the case of n+1 discs, we assume we

have a method of moving the n smallest discs from one pole to one of the other two poles.

We must show how to move n+1 discs from one pole to another pole in direction d, where d is either clockwise or anticlockwise.

Assume the discs are numbered 1 upwards (with the smallest disc numbered 1)

Base and inductive case

There is little choice in exploiting the inductive hypothesis we can begin by moving the n smallest discs in the direction d, or direction ¬d.

The solution is to move the n discs in direction ¬d, then the n+1th disc can be moved in direction d

We can then use the inductive hypothesis again to move the n smallest discs in the direction ¬d.

This places the n discs above the n+1th disc, and all of the n+1 smallest discs have now been moved from their original position to a new pole in direction d

Using the inductive hypothesis

Example with n = 3

We can use the previous solution to get a solution for n+1 using the inductive step

For n+1 (i.e. 3+1 = 4)

Example with n = 4

The following code summarises the inductive solution to the problem

Hn(d) is the sequence of pairs <k,d’> where n in the number of discs, k is a disc number and d and d’ are directions

Directions are Boolean values, true representing clockwise, and false representing anti-clockwise

The pair <k, d’> means, move disc number k from its current position in the direction d’

The pair is therefore an action. We use the ; to separate movements.

Psuedocode

Hn(d) means the sequence to move the n smallest discs in direction d

Taking the pairs in order from left to right, the complete sequence Hn(d) prescribes how to move the n smallest discs, one by one, from one pole to another in direction d

Hn(d)

Hn(d)

H0(d) = []

– base case (i.e. 0 moves needed)

Hn+1(d) = Hn(¬d) ; [<n+1, d>] ; Hn(¬d)

– induction step (move n discs in a given direction, move the n+1th disc in the opposite direction then move n discs again in the same direction)

Psuedocode

Notice that the procedure name, H, occurs on both sides of the equation for Hn+1(d)◦ Hn+1(d) = Hn(¬d) ; [<n+1, d>] ; Hn(¬d)

For this reason we call this a recursive solution (i.e. the procedure calls itself, think of Java)

Psuedocode

This inductive procedure gives us a way to generate the solution to the problem for any size n

We simply use the rules as rewrite rules, until all occurrences of H have been eliminated◦ i.e. we are left with just a sequence of actions

(e.g. move disc 3 clockwise)

How to use

Let us use cw and aw to mean clockwise and anti-clockwise, rather than true and false to improve readability

Calculate H2(cw) i.e. how to move 2 discs, one pole in a clockwise direction

Equation (1)◦ H0(d) = []

Equation (2)◦ Hn+1(d) = Hn(¬d) ; [<n+1, d>] ; Hn(¬d)

H2(cw)

<1,aw>

Example with n = 2

<2,cw>

Example with n = 2

<1,aw>

Example with n = 2

Example with n = 2

Exercise – Can you do H3(aw)?

Can you see a pattern in the number of moves required for n discs? i.e. H0, H1, H2, H3…

Show this as a formula in terms of n?

If this is easy, try exercise 8.3 a) (without cheating and looking at the answer first)

H3(cw)?

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