Discrete Structures &

Algorithms

EECE 320 : UBC : Spring 2009Matei Ripeanu

http://www.ece.ubc.ca/~matei/EECE320/

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•URL: http://www.ece.ubc.ca/~matei/EECE320/

•Instructor: Matei Ripeanu

• matei@...

• Office: KAIS 4033

• Office hours: after class (Thu 8:00pm -- …)

•Teaching assistants

• Sarah Motiee <hghasemi@>

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Sorting pancakes

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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.

The chefs at ubchop are sloppy: when they prepare 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).

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Treating pancakes as numbers

5

2

3

4

1

52341

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How do we sort this stack?

52341

How many flips do we need?

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4 flips suffice

12345

52341

43215

23415

14325

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What is the best way to sort?

52341

X = Smallest number of flips required to sort:

? X ?Upper Bound

Lower Bound 4

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4 flips are necessary

52341

41325

14325

Flip 2 must bring 4 to top (if it didn’t, we would spend more than 3)

Flip 1 has to put 5 on bottom

If we could do it in three flips:

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X = 4

Upper Bound

Lower Bound

4 X 4

X = 4

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Pancake numbers

MAX over all s { MIN # of flips to sort s }; s is a stack of 5 pancakes

P5 =

. . . . . .

13524

1

23145

2

X1 X2 X3 X119 X120

52341

4

12345

54321

Number of flips required to sort the worst case stack of 5

pancakes

P5 =

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The 5th pancake number

? P5 ?Upper Bound

Lower Bound 4

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Defining the pancake number

•Two ways to define Pn

•Pn = maximum over all s { minimum number of flips needed to sort stack s }; where s is one of n! stacks

•Pn = the number of flips required to sort the worst-case stack of n pancakes

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Pn for small n

•How do we reason about Pn for n=0, 1, 2, 3?

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Initial values of Pn

n 0 1 2 3

Pn

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Initial values of Pn

n 0 1 2 3

Pn 0 0 1 3

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P3 = 3

132

requires 3 flips, hence

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.

P3 ≥ 3

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The nth pancake number

Number of flips required to sort the worst case stack of n

pancakes.

Pn =

? Pn ?Upper Bound

Lower Bound

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? Pn ?

Try to find upper and lower bounds on Pn, for n > 3.

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Bracketing (bounding)

≤ f(x) ≤[ ]

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Procedure: Bring to top

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 Pn

•Use the bring to top method for n pancakes.

• If n=1: no work is needed

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

• Then: use the bring to top method for the remaining n-1 pancakes.

•Number of flips: 2(n-1)

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A better upper bound

•Using the bring to top method, we noted that we need at most 2(n-1) flips.

•When n=2, however, we need only 1 flip.

•Therefore, we need 2(n-2) flips for the biggest n-2 items and then 1 flip.

•Total number of flips: 2(n-2)+1 = 2n-3.

? Pn 2n-3

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Bring to top is not optimal

•Bring-to-top takes 5 flips,

•but we need only 4 flips.

32145

52341

23145

41325

14325

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Obtaining a lower bound

•The 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

2468..n135..

n-1

S

S contains n pairs that will S contains n pairs that will need to be broken apart during need to be broken apart during any sequence that sorts it.any sequence that sorts it.

Suppose n is even:

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Detail: This construction only works when n>2

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n Pn

1357..n246..

n-1

S

S contains n pairs that will S contains n pairs that will need to be broken apart during need to be broken apart during any sequence that sorts it.any sequence that sorts it.

Suppose n is odd:

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Detail: This construction only works when n>3

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n Pn 2n – 3 for n > 3

Bring-to-top is within a factor of 2 of the optimal sorting scheme!

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Starting from ANY stack we can get to the sorted stack

using no more than Pn flips.

n Pn 2n – 3 (for n 3)

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From ANY stack to the sorted stack in · Pn.

Reverse the sequences we use

to sort.

From the sorted stack to ANY stack in ·… ?

((( )))

Pn

How?

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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.

Q: From ANY stack to ANY stack in … ?

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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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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

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Starting from ANY stack we can get to the sorted stack

using no more than Pn flips.

Recap: n Pn 2n – 3 (for n 3)

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Known pancake numbersn Pn

1 02 13 34 45 56 77 88 99 1010 1111 1312 1413 15

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P14 is unknown

•1 x 2 x 3 x ... x 14 = 14! orderings for 14 pancakes.

•14! = 87,178,291,200.

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Computer engineering?

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Sorting by prefix reversal

Posed in American Mathematical Monthly, Vol. 82 (1), 1975,“Harry Dweighter” a.k.a. Jacob Goodman

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

William 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 (5n+5)/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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Pancake networks

•Assume a network with 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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Pancake network for n=3

123

321

213

312

231

132

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Pancake network for n=4

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Pancake network: routing

Pn

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

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Pancake network: reliability

If up to k nodes get hit by lightning does the network remain connected?

The pancake network is optimally reliable for its number of edges and nodes.

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Mutation distance

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One “simple problem”

A host of problems and applications at the frontiers of science.

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Motivating this course

•Computer science and engineering is not merely about computers and programming …

• … it is about (mathematically) modeling our world and about finding better ways to solve problemsways to solve problems

• Today’s example is a microcosm of this exercise.

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Mechanics for the course

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•eece320@ece.ubc.ca

•For questions and general discussion

•No discussion about assignments other than to clarify problem statements

Mailing list

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Assessment & grading

• 20% Assignments

• 20% Programming project

• 30% Quizzes

• 30% Final exam

• Participation bonus: up to 5%

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Homework & Quizzes

•6 homework assignments over the course of the term• Work in groups of three

• Typically assigned on a Thursday, due in a week

• Late policy: 33% penalty per late day

• Each homework is worth about 3.5%

•4 quizzes over the term• Individual

• 30 minutes or so, in class

• Each quiz is worth about 6%

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((( )))Ask questions

•Participation bonus: up to 5%

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Summary so far

• Pancake sorting

• A problem with many applications

• Bracketing (bounding a function)

• Proving bounds for pancake sorting

• You can make money solving such problems (Bill Gates!)

• Illustrated many concepts that we will learn in this course

• Proofs

• Sets

• Counting

• Performance of algorithms