Tower of Hanoi Final

8/8/2019 Tower of Hanoi Final

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The Towers of HanoiThe Towers of Hanoi

oror

ApocalypseApocalypse WhenWhen??


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A Legend (A Legend (CtdCtd))

The priests were to transfer the disks from the first needleThe priests were to transfer the disks from the first needleto the second needle, using the third as necessary.to the second needle, using the third as necessary.

But they couldBut they could only moveonly move one disk at a timeone disk at a time, and could, and couldnever put a larger disk on top of a smaller onenever put a larger disk on top of a smaller one..

When they completed this task,When they completed this task, the world would endthe world would end!!


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ExampleExample

For simplicity, suppose there were just 3 disks, and wellFor simplicity, suppose there were just 3 disks, and wellrefer to the three needles as A, B, and C...refer to the three needles as A, B, and C...

We then move the top disk from A to C.We then move the top disk from A to C.


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Example (Example (CtdCtd))


We then move the top disk from B to C.We then move the top disk from B to C.


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We then move the top disk from A to B.We then move the top disk from A to B.


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We then move the top disk from C to A.We then move the top disk from C to A.


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We then move the top disk from C to B.We then move the top disk from C to B.


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We then move the top disk from A to B.We then move the top disk from A to B.


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and were done!and were done!

The problem gets more difficult as the number of disksThe problem gets more difficult as the number of disksincreases...increases...


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Our ProblemOur Problem

Todays problem is to write a program that generates theTodays problem is to write a program that generates theinstructions for the priests to follow in moving the disks.instructions for the priests to follow in moving the disks.

While quite difficult to solve iteratively, this problem hasWhile quite difficult to solve iteratively, this problem hasa simple and eleganta simple and elegant recursiverecursive solution.solution.


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AnalysisAnalysis

For flexibility, lets allow the user to enter the number ofFor flexibility, lets allow the user to enter the number ofdisks for which they wish a set of instructions:disks for which they wish a set of instructions:

/* hanoi.cpp

* ...*/

void Move(int n, char src, char dest, char aux);

int main(){

cout numDisks;

Move(numDisks, A, B, C);}


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Analysis (Ctd)Analysis (Ctd)

Our task, then is to write function Move() that does allOur task, then is to write function Move() that does allthe work:the work:

/* hanoi.cpp

* ...*/

void Move(int n, char src, char dest, char aux);

int main(){

cout numDisks;

Move(numDisks, A, B, C);}


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DesignDesign

Basis: What is an instance of the problem that is trivial?Basis: What is an instance of the problem that is trivial?

p n == 1n == 1

Since this base case could occur when the disk is on anySince this base case could occur when the disk is on anyneedle, we simply output the instruction to move theneedle, we simply output the instruction to move thetop disk fromtop disk from srcsrc toto destdest..


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Design (Design (CtdCtd))

Induction Step: n > 1Induction Step: n > 1

pHow can recursion help us out?How can recursion help us out?

a.a. RecursivelyRecursively move nmove n--1 disks from1 disks from srcsrc toto auxaux..


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b. Move the one remaining disk fromb. Move the one remaining disk from srcsrc toto destdest..


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c.c. RecursivelyRecursively move nmove n--1 disks from1 disks from auxaux toto destdest......


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d. Were done!d. Were done!


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AlgorithmAlgorithm

We can combine these steps into the following algorithm:We can combine these steps into the following algorithm:

0.0.ReceiveReceive n, src, dest, auxn, src, dest, aux..

1.1.IfIf nn > 1:> 1:

a. Move(a. Move(nn--1, src, aux,dest1, src, aux,dest););b. Move(1,b. Move(1, src,dest, auxsrc,dest, aux););

c. Move(c. Move(nn--1, aux,dest, src1, aux,dest, src););

ElseElse

Display Move the top disk from ,Display Move the top disk from , srcsrc, to ,, to ,destdest..

End if.End if.


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CodingCoding// ...

void Move(int n, char src, char dest, char aux){if (n > 1){

Move(n-1, src, aux, dest);Move(1, src, dest, aux);Move(n-1, aux, dest, src);}elsecout


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TestingTestingThe Hanoi Towers

Enter how many disks: 1Move the top disk from A to B


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Testing (Testing (CtdCtd))The Hanoi Towers

Enter how many disks: 2Move the top disk from A to CMove the top disk from A to BMove the top disk from C to B


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Testing (Testing (CtdCtd))

The Hanoi Towers

Enter how many disks: 3Move the top disk from A to BMove the top disk from A to CMove the top disk from B to CMove the top disk from A to BMove the top disk from C to AMove the top disk from C to BMove the top disk from A to B


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Testing (Testing (CtdCtd))

The Hanoi Towers

Enter how many disks: 4

move a disk from needle A to needle B

move a disk from needle C to needle B

move a disk from needle A to needle C

move a disk from needle B to needle Amove a disk from needle B to needle C




move a disk from needle C to needle A

move a disk from needle B to needle Amove a disk from needle C to needle B





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AnalysisAnalysis

Lets see how many moves it takes to solve this problem,Lets see how many moves it takes to solve this problem,as a function ofas a function of nn, the number of disks to be moved., the number of disks to be moved.

nn Number of diskNumber of disk--moves requiredmoves required

11 11

22 33

33 77

44 1515

55 3131

......

ii 22ii--11

6464 226464

--1 (a big number)1 (a big number)


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Analysis (Analysis (CtdCtd))

How big?How big?

Suppose that our computer andsuperSuppose that our computer andsuper--printer canprinter cangenerate and print 1,048,576 (2generate and print 1,048,576 (22020) instructions/second.) instructions/second.

How long will ittake toHow long will ittake to printprint the priests instructions?the priests instructions?

There are 2There are 26464 instructions to print.instructions to print.

Then it will take 2Then it will take 26464/2/22020 = 2= 24444 secondsseconds to printthem.to printthem.

1 minute == 60 seconds.1 minute == 60 seconds.

Lets take 64 = 2Lets take 64 = 266 as an approximation of 60.as an approximation of 60.

Then it will takeThen it will take $224444/ 2/ 266 = 2= 23838 minutesminutes to printthem.to printthem.


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Hmm. 2Hmm. 23838 minutes is hardto grasp. Lets keep going...minutes is hardto grasp. Lets keep going...

1 hour == 60 minutes.1 hour == 60 minutes.


Then it will takeThen it will take $223838/ 2/ 266 = 2= 23232 hourshours to printthem.to printthem.

1 day == 24 hours.1 day == 24 hours.


Then it will takeThen it will take $223232/ 2/ 255 = 2= 22727 daysdays to printthem.to printthem.


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Hmm. 2Hmm. 22727 days is hardto grasp. Lets keep going...days is hardto grasp. Lets keep going...

1 year == 365 days.1 year == 365 days.


Then it will takeThen it will take $222727/ 2/ 299 = 2= 21818 yearsyears to printthem.to printthem.

1 century == 100 years.1 century == 100 years.


Then it will takeThen it will take $221818/ 2/ 277 = 2= 21111 centuriescenturies to printthem.to printthem.


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Hmm. 2Hmm. 21111 centuries is hardto grasp. Lets keep going...centuries is hardto grasp. Lets keep going...

1 millenium == 10 centuries.1 millenium == 10 centuries.


Then it will takeThen it will take $221111/ 2/ 244 = 2= 277 == 128 millenia128 millenia

justtojustto printprint the priests instructions (assuming our computerthe priests instructions (assuming our computerdoesnt crash, in which case we have to start all over again).doesnt crash, in which case we have to start all over again).

How fast can the priests actuallyHow fast can the priests actually movemove the disks?the disks?Ill leave itto you to calculate the data ofthe apocalypse...Ill leave itto you to calculate the data ofthe apocalypse...


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SummarySummary

Recursion is a valuable tool that allows some problems toRecursion is a valuable tool that allows some problems tobe solved in an elegant and efficient manner.be solved in an elegant and efficient manner.

Functions can sometimes require more than one recursiveFunctions can sometimes require more than one recursive

call in order to accomplish their task.call in order to accomplish their task.

There are problems for which we can design a solution,There are problems for which we can design a solution,but the nature of the problem makes solving itbut the nature of the problem makes solving iteffectively uncomputableeffectively uncomputable..

Tower of Hanoi Final

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