Pancakes With A Problem!

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Pancakes With A Problem!
Great Theoretical Ideas In Computer Science Great Theoretical Ideas In Computer Science Great Theoretical Ideas In Computer Science
Anupam Gupta CS 15-251 Fall 2006
Lecture 1 Aug 29th, 2OO6 Carnegie Mellon University
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Fall 06 Course Staff

• Profs Anupam Gupta
• John Lafferty
• TAs Albert Sheu
• Andreas Krause
• Todd Phillips
• Webpage http//www.cs.cmu.edu/15251
• Bboard academic.cs.15-251
• Please check the webpages and Bboards regularly!

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Read the Course Document!
Lowest homework grade is dropped
Lowest test grade is worth half
If Suzie gets 60,90,80 in her tests, how many
total test points will she have in her final
grade?
((½ 60) 90 80) (30/250) 24
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Homework
Homework will go out every Tuesday and is due the
Tuesday after.
Seven points per day late penalty
No homework will be accepted more than seven days
late
Any written answers must be typset and given as a
PDF file.
HOMEWORK 1 IS OUT TODAY!
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Extra Credit Tokens
Each correct solution to an Extra Credit problem
gives you a token
Each token is worth
A) One problem (up to 20 points) on a subsequent
assignment.
or B) One late submission up to 3 days late on
a subsequent assignment without any penalty.
You have to explicitly tell us when you are
using your token.
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Clarifications?
To use option A, your HW submission must say I
am using one token for problem XYZ (worth at most
20 pts).

• To use option B, your HW submission must still be
  at most 7 days late, and must say something like
  I am using one token for this late submission,
  which is N days late. We then only take off
  (N-3) 7 points.

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Collaboration Cheating
You may NOT share written work
You may NOT use Google, or solutions to previous
years homework
You MUST read and sign the class honor code
(last page of Course Document).
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Textbook
There is NO textbook for this class
Students from previous years tell us that its
imperative to come to class
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((( )))
Feel free to ask questions
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Pancakes With A Problem!Lecture 1 Aug 29th, 2OO6
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The chef at our place is sloppy when he prepares
pancakes, they come out all different sizes
When the waiter delivers them to a customer, he
rearranges them (so that smallest is on top, and
so on, down to the largest at the bottom)
He does this by grabbing several from the top and
flipping them over, repeating this (varying the
number he flips) as many times as necessary
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Developing A NotationTurning pancakes into
numbers
5
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2
3
4
1
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How do we sort this stack?How many flips do we
need?
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How many flips suffice?
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4 Flips Are Sufficient
12345
52341
43215
23415
14325
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Best Way to Sort
52341
X Smallest number of flips required to
sort
Lower Bound
4
? ? X ? ?
Upper Bound
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Four Flips Are Necessary
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41325
14325
Flip 1 has to put 5 on bottom
Flip 2 must bring 4 to top (if it didnt, we
would spend more than 3)
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4 ? X ? 4
Upper Bound
Lower Bound
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Our First Theorem

• One can sort in 4 flips.

52341
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Food for thought

• Can every stack of 5 pancakes be sorted in 4
  flips?

How difficult can the cook make things for the
waiter?
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Two adversaries

• If the cook chose the worst possible stack of 5
  pancakes
• and
• the waiter sorted it in thefewest possible
  flips,
• how many flips would it take?

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5th Pancake Number
The least number of flips required to sort the
worst case stack of 5 pancakes.
P5
Upper Bound
? ? P5 ? ?
Lower Bound
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5th Pancake Number
The least number of flips required to sort the
worst case stack of 5 pancakes.
P5
Upper Bound
4 ? P5 ? ?
Lower Bound
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The 5th Pancake Number The MAX of the Xs
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54 3 2 1
1 2 3 4 5
j
120
1
199
3
2
. . . . . . .
X120
X1
Xj
X2
X3
X119
the fewest number of flips to sort stack 1 (best
strategy for waiter given this stack)
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P5 MAX over s 2 stacks of 5 of MIN of flips
to sort s
12345
5 4 3 2 1
120
199
3
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. . . . . . . . . .
X120
0
4
1
X3
X119
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Pn The least number of flips required to sort
the worst-case stack of n pancakes.
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Pn The least number of flips required to sort
the worst-case stack of n pancakes.
a, not the
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What is Pn for small n?
Can you do n 0, 1, 2, 3 ?
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Initial Values of Pn
n
3
0
1
2
Pn
0
0
1
3
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P3 3
requires 3 Flips, hence
P3 3
ANY stack of 3 can be done by getting the big one
to the bottom ( 2 flips), and then using 1
flips to handle the top two
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BracketingWhat are the best lower and upper
bounds that I can prove?
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? ? Pn ? ?
Take a few minutes to try and prove upper and
lower bounds on Pn,for n gt 3.
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Bring-to-top Method
Bring biggest to top Place it on bottom Bring
next largest to topPlace second from bottom And
so on
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Upper Bound On PnBring-to-top Method For n
Pancakes

• If n1, no work required we are done!
• Otherwise, flip pancake n to top and then flip
  it to position n
• Now use

Bring To Top Method For n-1 Pancakes
Total Cost at most 2(n-1) 2n 2 flips
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Better Upper Bound On PnBring-to-top Method For
n Pancakes

• If n2, at most one flip and we are done!
• Otherwise, flip pancake n to top and then flip
  it to position n
• Now use

Bring To Top Method For n-1 Pancakes
Total Cost at most 2(n-2) 1 2n 3 flips
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? ? Pn ? 2n-3
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• Given any stack, does Bring-to-Topdo the fewest
  flips possiblefor this given stack?

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No. Bring-to-top not always optimal for a
particular stack
32145
52341
23145
41325
14325

• Bring-totop takes 5 flips,
• but we can do in 4 flips

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? ? Pn ? 2n-3
What other bounds can you prove on Pn?
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Breaking Apart Argument
Suppose a stack S has a pair of adjacent pancakes
that will not be adjacent in the sorted stack
Any sequence of flips that sorts stack S must
have one flip that inserts the spatula between
that pair and breaks them apart
Furthermore, this is true of the pair formed by
the bottom pancake of S and the plate
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n ? Pn
S
2468..n135..n-1
Suppose n is even
S contains n pairs that will need to be broken
apart during any sequence that sorts it
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Detail This construction only works when ngt2
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n ? Pn
S
1357..n246..n-1
Suppose n is odd
S contains n pairs that will need to be broken
apart during any sequence that sorts it
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Detail This construction only works when ngt3
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n ? Pn ? 2n 3 for n gt 3
Bring-to-top is within a factor of 2 of optimal!
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From ANY stack to sorted stack in Pn
From sorted stack to ANY stack in Pn ?
Reverse the sequences we use to sort
Hence, from ANY stack to ANY stack in 2Pn
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From ANY stack to ANY stack in 2Pn.
Can you find a faster way than 2Pn flips to go
from ANY to ANY?
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ANY Stack S to ANY stack T in Pn
3,5,1,2,4
1,2,3,4,5
new T
Rename the pancakes in S to be 1,2,3,..,n
Rewrite T using the new naming scheme that you
used for S
The sequence of flips that brings the sorted
stack to the new T will bring S to T
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The Known Pancake Numbers
Pn
n

• 01345

12345678910111213
7891011131415
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P14 is Unknown
1?2?3?4??13?14 14! orderings of 14 pancakes
14! 87,178,291,200
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Is This Really Computer Science?
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Sorting By Prefix Reversal
Posed in Amer. Math. Monthly 82 (1) (1975),
Harry Dweighter a.k.a. Jacob Goodman
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(17/16)n ? Pn ? (5n5)/3
William H. Gates and Christos Papadimitriou.
Bounds For Sorting By Prefix Reversal. Discrete
Mathematics, vol 27, pp 47-57, 1979.
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(15/14)n ? Pn ? (5n5)/3
H. Heydari and H. I. Sudborough. On the Diameter
of the Pancake Network. Journal of Algorithms,
vol 25, pp 67-94, 1997.
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How many different stacks of n pancakes are there?
n! 1 x 2 x 3 x x n
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Permutation

• Any particular ordering of all n elements of an n
  element set S, is called a permutation on the set
  S.
• Example S 1, 2, 3, 4, 5
• One possible permutation 5 3 2 4 1
• 54321 120 possible permutations on S

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Permutation

• Any particular ordering of all n elements of an n
  element set S, is called a permutation on the set
  S.
• Each different stack of n pancakes is one of the
  permutations on 1..n.

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A Permutation is a NOUN.A permutation can also
be a VERB.

• I start with a permutation S of pancakes.
• I continue to use a flip operation to permute my
  current permutation, so as to obtain the sorted
  permutation.

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There are n! permutations of n elements.
n! 1 x 2 x 3 x x n
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Pancake NetworkDefinition For n! Nodes
For each node, assign it the name of one of the
n! stacks of n pancakes
Put a wire between two nodes if they are one flip
apart
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Network For n3
213
123
312
321
132
231
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Network For n4
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Pancake NetworkMessage Routing Delay

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

Pn
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Pancake NetworkReliability

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

The Pancake Network is optimally reliable for its
number of edges and nodes.
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Mutation Distance
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High Level Point

• Computer Science is not merely
• about computers and programming,
• it is about mathematically modeling
• our world, and about finding better
• and better ways to solve problems.
• This lecture is a microcosm of this exercise.

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One Simple Problem
A host of problems and applications at the
frontiers of science
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Study Bee

• Please read the course document carefully.
• You must hand in asigned collaboration policy
  page.

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Study Bee

• Definitions of
• nth pancake number
• lower bound
• upper bound
• permutation
• Proof of
• ANY to ANY in Pn
• Important Technique
• Bracketing

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References

• 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
  he Pancake Network. Journal of Algorithms, vol
  25, pp 67-94, 1997