Pancakes With A Problem!Pancakes With A Problem!

Great Theoretical Ideas In Computer ScienceGreat Theoretical Ideas In Computer Science

Steven Steven RudichRudich

CS 15-251 Spring CS 15-251 Spring 20042004

Lecture 1Lecture 1 Jan 13, 2004Jan 13, 2004 Carnegie Mellon Carnegie Mellon UniversityUniversity

Magic Trick At 3:00pm Sharp!Magic Trick At 3:00pm Sharp!

Be punctual.Be punctual.

Sit close-up: some of the tricks Sit close-up: some of the tricks are hard to see from the back.are hard to see from the back.

Course StaffCourse Staff

Profs:Profs: Steven RudichSteven Rudich

Anupam GuptaAnupam Gupta

TAs:TAs:

Yinmeng ZhangYinmeng Zhang Bella VoldmanBella Voldman

Brendan JubaBrendan Juba Andrew GilpinAndrew Gilpin

Susmit SarkarSusmit Sarkar Adam WiermanAdam Wierman

Please feel free to ask questions!

((( )))

Course DocumentCourse DocumentYou must read this carefully.You must read this carefully.

1.1. Grading formula for the course.Grading formula for the course.1. 40% homework2. 30% quizes3. 30% final

2.2. Seven points a day late penalty.Seven points a day late penalty.3.3. Collaboration/Cheating PolicyCollaboration/Cheating Policy

1. You may NOT share written work.2. We reuse homework problems.

My Low Vision and You.My Low Vision and You.

I have a genetic retinal condition called I have a genetic retinal condition called Stargardt’s disease. My central vision is Stargardt’s disease. My central vision is going, one pixel at a time, to zero. I have going, one pixel at a time, to zero. I have working peripheral vision.working peripheral vision.

I can’t recognize faces – so please I can’t recognize faces – so please introduce yourself to me every time!introduce yourself to me every time!

I detect motion really well so please I detect motion really well so please move your hand when you raise it in move your hand when you raise it in class.class.

Pancakes With A Problem!Pancakes With A Problem!

Great Theoretical Ideas In Computer ScienceGreat Theoretical Ideas In Computer Science

Steven Steven RudichRudich

CS 15-251 Spring CS 15-251 Spring 20042004

Lecture 1Lecture 1 Jan 13, 2004Jan 13, 2004 Carnegie Mellon Carnegie Mellon UniversityUniversity

The chef at our place is sloppy, and when he prepares a stack of

pancakes they come out all different sizes.

Therefore, when I deliver them to a customer, on the way to the

table I rearrange them (so that the smallest winds up on top, and so

on, down to the largest at the bottom).

I do this by grabbing several from the top and flipping them over,

repeating this (varying the number I flip) as many times as necessary.

Developing A Notation:Developing A Notation:Turning pancakes into numbersTurning pancakes into numbers

Developing A Notation:Developing A Notation:Turning pancakes into numbersTurning pancakes into numbers

1

2

3

4

5

Developing A Notation:Developing A Notation:Turning pancakes into numbersTurning pancakes into numbers

2

3

4

5

1

Developing A Notation:Developing A Notation:Turning pancakes into numbersTurning pancakes into numbers

52341

How do we sort this stack?How do we sort this stack?How many flips do we need?How many flips do we need?

52341

4 Flips Are Sufficient4 Flips Are Sufficient

12345

52341

43215

23415

14325

Algebraic RepresentationAlgebraic Representation

52341

X = The smallest number of flips required to sort:

? X ?Upper Bound

Lower Bound

Algebraic RepresentationAlgebraic Representation

52341

X = The smallest number of flips required to sort:

? X 4Upper Bound

Lower Bound

4 Flips Are Necessary4 Flips Are Necessary

52341

41325

14325

Flip 1 has to put 5 on bottom

Flip 2 must bring 4 to top.

? X 4Lower Bound

Upper Bound

Lower Bound

4 X 4

X = 4

55thth Pancake Number Pancake Number

P5 = The number of flips required to sort the

worst case stack of 5 pancakes.

? P5 ?Upper Bound

Lower Bound

55thth Pancake Number Pancake Number

P5 = The number of flips required to sort the

worst case stack of 5 pancakes.

4 P5 ?Upper Bound

Lower Bound

The 5The 5thth Pancake Number: Pancake Number: The MAX of the X’sThe MAX of the X’s

120

1199

32 . . . . . . .

X1 X2 X3 X119 X120

52341

4

P5 = MAX over s 2 stacks of 5 of MIN # of flips to sort s

120

1199

32 . . . . . . .

X1 X2 X3 X119 X120

52341

4

Pn

MAX over s stacks of n pancakes of

MIN # of flips to sort s

Or,

The number of flips required to sort a

worst-case stack of n pancakes.

Be Cool.

Learn Math-speak.

Pn = The number of flips required to sort a worst-case stack of n

pancakes.

What is Pn for small n?

Can you do n = 0, 1, 2, 3 ?

Initial Values Of Pn

n 0 1 2 3

Pn 0 0 1 3

PP33 = 3 = 3

11332 requires 3 Flips, hence 2 requires 3 Flips, hence PP33 ¸̧ 3. 3.

ANYANY stack of 3 can be done in stack of 3 can be done in 33 flips. flips.

Get the big one to the bottom (Get the big one to the bottom (·· 2 flips). 2 flips).

Use Use ·· 1 more flip to handle the top two. 1 more flip to handle the top two.

Hence, Hence, PP33 ·· 3. 3.

nth Pancake Number

Pn = Number of flips required to sort a worst case stack of n

pancakes.

? Pn ?Upper Bound

Lower Bound

? Pn ?

Take a few minutes to try

and prove bounds on Pn,

for n>3.

Bring To Top Method

Bring biggest to top. Place it on bottom. Bring next largest to

top. Place second from

bottom. And so on…

Upper Bound On PUpper Bound On Pnn::Bring To Top Method For n Bring To Top Method For n

PancakesPancakesIf n=1, no work - we are done.If n=1, no work - we are done.

Else: Else: flip pancake n to topflip pancake n to top and then and then flip it to position nflip it to position n..

Now use: Now use: Bring To Top Method Bring To Top Method For n-1 PancakesFor n-1 Pancakes

Total Cost: at most 2(n-1) = 2n –2 flips.

Better Upper Bound On PBetter Upper Bound On Pnn::Bring To Top Method For n Bring To Top Method For n

PancakesPancakesIf n=2, use one flip and we are done.If n=2, use one flip and we are done.

Else: flip pancake n to top and then Else: flip pancake n to top and then flip it to position n.flip it to position n.

Now use: Now use: Bring To Top Method Bring To Top Method For n-1 PancakesFor n-1 Pancakes

Total Cost: at most 2(n-2) + 1 = 2n –3 flips.

Bring to top not always optimal Bring to top not always optimal for a particular stackfor a particular stack

5 flips, but can be done in 4 flips5 flips, but can be done in 4 flips

32145

52341

23145

41325

14325

? Pn 2n - 3

What bounds can you prove on

Pn?

9

16

Breaking Apart ArgumentBreaking Apart Argument

Suppose a stack S contains a pair Suppose a stack S contains a pair of adjacent pancakes that will not of adjacent pancakes that will not be adjacent in the sorted stack. be adjacent in the sorted stack.

Any sequence of flips that sorts Any sequence of flips that sorts stack S must involve one flip that stack S must involve one flip that inserts the spatula between that inserts the spatula between that pair and breaks them apart. pair and breaks them apart.

916

Breaking Apart ArgumentBreaking Apart Argument

Suppose a stack S contains a pair Suppose a stack S contains a pair of adjacent pancakes that will not of adjacent pancakes that will not be adjacent in the sorted stack. be adjacent in the sorted stack.

Any sequence of flips that sorts Any sequence of flips that sorts stack S must involve one flip that stack S must involve one flip that inserts the spatula between that inserts the spatula between that pair and breaks them apart. pair and breaks them apart.

Furthermore, this same principle is Furthermore, this same principle is true of the “pair” formed by the true of the “pair” formed by the bottom pancake of S and the plate.bottom pancake of S and the plate.

n n P Pnn

Suppose n is even. Suppose n is even.

Such a stack S contains n Such a stack S contains n pairs that must be broken pairs that must be broken apart during any apart during any sequence that sorts stack sequence that sorts stack S.S.

2468..n1357..

n-1

S

n n P Pnn

2

1

S

Detail: This construction only works when n>2

Suppose n is even. Suppose n is even.

Such a stack S contains n Such a stack S contains n pairs that must be broken pairs that must be broken apart during any apart during any sequence that sorts stack sequence that sorts stack S.S.

n n P Pnn1357..n2468..

n-1

S

Suppose n is odd. Suppose n is odd.

Such a stack S contains n Such a stack S contains n pairs that must be broken pairs that must be broken apart during any apart during any sequence that sorts stack sequence that sorts stack S.S.

n n P Pnn

1

3

2

S

Detail: This construction only works when n>3

Suppose n is odd. Suppose n is odd.

Such a stack S contains n Such a stack S contains n pairs that must be broken pairs that must be broken apart during any apart during any sequence that sorts stack sequence that sorts stack S.S.

n ≤ Pn ≤ 2n – 3 (for n ≥ 3)

Bring To Top is within a factor

of two of optimal!

So starting from ANY stack we can get to

the sorted stack using no more than Pn

flips.

n ≤ Pn ≤ 2n – 3 (for n ≥ 3)

From ANY stack to sorted stack in · Pn.

Reverse the sequences we use

to sort.

From sorted stack to ANY stack in · Pn ?

((( )))

From ANY stack to sorted stack in · Pn.

From sorted stack to ANY stack in · Pn.

Hence,

From ANY stack to ANY stack in · 2Pn.

From ANY stack to ANY stack in · 2Pn.

Can you find a faster way

than 2Pn flips to go from

ANY to ANY?

((( )))

From ANY Stack S to ANY stack T in · Pn

Rename the pancakes in S to be 1,2,3,..,n. Rewrite T using the new naming scheme that you used for S. T will be some list: (1),(2),..,(n). The sequence of flips that brings the sorted stack to (1),(2),..,(n) will bring S to T.

S: 4,3,5,1,2

T: 5,2,4,3,1

1,2,3,4,5 3,5,1,2,4

The Known Pancake NumbersThe Known Pancake Numbers

112233445566778899

1010111112121313

0011334455

n Pn

778899

10101111131314141515

PP1414 Is Unknown Is Unknown

14! Orderings of 14 pancakes.14! Orderings of 14 pancakes.

14! = 87,178,291,20014! = 87,178,291,200

Is This Really Computer Science?Is This Really Computer Science?

Posed in Amer. Math. Monthly 82 (1) (1975), “Harry Dweighter” a.k.a. Jacob Goodman

(17/16)n Pn (5n+5)/3

Bill Gates & Christos

Papadimitriou:

Bounds For Sorting By Prefix

Reversal.

Discrete Mathematics,

vol 27, pp 47-57, 1979.

(15/14)n Pn (5n+5)/3

H. Heydari & Ivan H. Sudborough.

On the Diameter of the Pancake

Network.

Journal of Algorithms, vol 25, pp 67-94,

1997.

PermutationPermutation

Any particular ordering of all n Any particular ordering of all n elements of an n element set S is elements of an n element set S is called a called a permutationpermutation on the set S. on the set S.

Example:Example: S = {1, 2, 3, S = {1, 2, 3, 4, 5}4, 5}

Example permutation:Example permutation: 5 3 2 4 1 5 3 2 4 1

120 possible permutations on S120 possible permutations on S

PermutationPermutation

Any particular ordering of all n Any particular ordering of all n elements of an n element set S is elements of an n element set S is called a called a permutationpermutation on the set S. on the set S.

Each different stack of n pancakes is Each different stack of n pancakes is one of the permutations on [1..n].one of the permutations on [1..n].

Representing A PermutationRepresenting A Permutation

We have many choices of how to specify a We have many choices of how to specify a permutation on S. Here are two methods:permutation on S. Here are two methods:

1)1) List a sequence of all elements of [1..n], List a sequence of all elements of [1..n], each one written exactly once. each one written exactly once. Ex: 6 4 5 2 1 3Ex: 6 4 5 2 1 3

2)2) Give a function Give a function on S s.t. on S s.t. (1) (1) (2) (2) (3) .. (3) .. (n) (n) is a sequence listing [1..n], each one exactly is a sequence listing [1..n], each one exactly once.once.

Ex: Ex: (1)=6 (1)=6 (2)=4 (2)=4 (3) = 5 (3) = 5 (4) = 2 (4) = 2 (4) = 1 (4) = 1 (6) =3(6) =3

A Permutation is a NOUNA Permutation is a NOUN

An ordering S of a stack of pancakes An ordering S of a stack of pancakes is a permutation. is a permutation.

An ordering S of a stack of pancakes is An ordering S of a stack of pancakes is a permutation.a permutation.

We can permute S to obtain a new We can permute S to obtain a new stack S’.stack S’.

PermutePermute also means to rearrange so also means to rearrange so as to obtain a permutation of the as to obtain a permutation of the original.original.

A Permutation is a NOUNA Permutation is a NOUNA Permutation can also be a A Permutation can also be a

VERBVERB

Permute A Permutation.Permute A Permutation.

I start with a I start with a permutationpermutation S of S of

pancakes. pancakes. I continue to use a flip I continue to use a flip operation to operation to permutepermute

my current my current permutationpermutation, , so as to obtain the so as to obtain the

sorted sorted permutationpermutation..

Ultra-Useful FactUltra-Useful Fact

There are There are n! = 1*2*3*4*…*nn! = 1*2*3*4*…*n permutations on permutations on nn elements. elements.

Proof: in the first counting lecture.Proof: in the first counting lecture.

Pancake NetworkPancake Network

This network has n! nodes This network has n! nodes

Assign each node the name of one of Assign each node the name of one of the possible n! stacks of pancakes.the possible n! stacks of pancakes.

Put a wire between two nodes Put a wire between two nodes if they are one flip apart.if they are one flip apart.

Network For n=3Network For n=3

123

321

213

312

231

132

Network For n=4Network For n=4

Pancake Network: Routing Delay Pancake Network: Routing Delay

What is the maximum distance What is the maximum distance between two nodes in the pancake between two nodes in the pancake network?network?

Pn

Pancake Network: ReliabilityPancake Network: Reliability

If up to n-2 nodes get hit by If up to n-2 nodes get hit by lightning the network remains lightning the network remains connected, even though each node connected, even though each node is connected to only n-1 other is connected to only n-1 other nodes. nodes.

The Pancake Network is optimally reliable for its number of edges and

nodes.

Mutation DistanceMutation Distance

One “Simple” Problem

A host of problems and applications

at the frontiers of

science

Study Bee

You must read the You must read the course document course document carefully.carefully.

You must hand-in the You must hand-in the signed cheating policy signed cheating policy page.page.

Study Bee

Definitions of:Definitions of:

nth pancake nth pancake numbernumber

lower boundlower bound

upper boundupper bound

permutationpermutation

Proof of:Proof of:

ANY to ANY in ANY to ANY in ·· P Pnn

High Level PointHigh Level Point

This lecture is a microcosm of This lecture is a microcosm of mathematical modeling and mathematical modeling and optimization.optimization.

ReferencesReferences

Bill Gates & Christos Papadimitriou: Bounds For Sorting By Prefix Reversal. Discrete Mathematics, vol 27, pp 47-57, 1979.

H. Heydari & H. I. Sudborough: On the Diameter of the Pancake Network. Journal of Algorithms, vol 25, pp 67-94, 1997