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Great Theoretical Ideas in Computer Science

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Administrative details

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Course Staff Prof. Danny Sleator Prof. Ryan O’Donnell TAs A: JD Nir B: Tim Wilson C: Mark Kai D: Adam Blank E: Sang Tian F: Jasmine Peterson G:Andrew Audibert Shashank Singh Office hours start Wednesday.

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Web Sites https://colormygraph.ugrad.cs.cmu.edu = Announcements, Calendar, Notes, Homeworks, … Bulletin Board: Questions, Comments, Announcements, …

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Textbook There is no textbook. Slides will be posted on the website. Some supplementary notes will also be posted.

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Grading 28%Homework (12, lowest one dropped) 5% Quizzes (12, lowest two dropped) 2%Participation 35%Midterms (3, lowest one counts half) 30%Final

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Homework Homeworks out/due each Thursday at 11:59pm. They are hard. Must be typeset in LaTeX, submitted online. 5 “late days” for the semester, ≤ 3 per HW. Turned in any time Friday: costs 1 late day Turned in any time Saturday: costs 2 late days Turned in any time Sunday:costs 3 late days After Sunday: grade of 0.

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Collaboration On each HW, you may work in a group of ≤ 4 people. You must report who you worked with. You must think about the problems yourself for ≥ 1 hour before discussing them with others. You must write up all solutions by yourself.

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Cheating Handled in accordance with university policy: You MAY NOT: You MAY: You MUST: sign and return the Honor Code today. Share written work. (Erase all whiteboards.) Get help from anyone besides your HW collaborators, staff. Refer to solutions/materials from earlier versions of 251. Search the web/books for specific homework problems. Get help from others/books/web on ‘general topics’ (“induction”, “random variable”, “diagonalization”) Use tools like Mathematica/Maple/Wolfram Alpha.

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Quizzes Every Tuesday (a few exceptions), beginning of class The quiz is DUE AT 3:07pm. Therefore, do NOT be late to class. Tested on material from the prior week. These are designed to be easy, assuming you are keeping up with the lectures.

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Midterms Designed to be doable in 1 hour. You will have 3 hours. “Semi-cumulative.” Held Wednesdays, 6pm – 9pm at locations TBD. February 15, March 21, April 18 Mark these dates on your calendar now!

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Automatic A There are 4 “Automatic A” problems on website. Solve one, get an A in the course. (Actually +2 letter grades.) All standard homework rules apply (except groups can submit jointly written solution). At most one submission per person/group per semester. They are super-hard… but some easier versions may appear on homeworks or tests…

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15-251: Great Theoretical Ideas in Computer Science Pancakes With A Problem Lecture 1

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Feel free to ask questions

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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 chef in our place is sloppy; when she prepares pancakes they come out all different sizes. When the waiter delivers them to a customer, he rearranges them (so that the smallest is on top, and so on, down to the largest at the bottom).

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

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2 flips sufficient 2 flips necessary

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

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Developing Notation: Turning pancakes into numbers

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

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4 flips are sufficient

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Best way to sort this stack? Let X be the smallest number of flips that can sort this specific stack. ? ≤ X ≤ ? Upper Bound Lower Bound

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Is 4 a lower bound? What would it take it show that? ? ≤ X ≤ ? Upper Bound Lower Bound 4 A convincing argument that every way of sorting the stack uses at least 4 flips

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Four Flips Are Necessary Second flip has to bring 4 to the top, because… First flip has to put 5 on bottom, because… If we could do it in three flips:

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Best way to sort? Let X be the smallest number of flips that can sort this specific stack. ? ≤ X ≤ ? Upper Bound Lower Bound X = 4

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Matching upper and lower bounds! ? ≤ X ≤ ? Upper Bound Lower Bound 44 X = 4

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?????????? P5P5

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5 th Pancake Number Number of flips required to sort when your worst enemy gives you a stack of 5 pancakes P 5 = MAX over all 5-stacks S of MIN # of flips to sort S P 5 =

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? ≤ P 5 ≤ ? Upper Bound Lower Bound 4 5 th Pancake Number Fact: P 5 = 5 1. Show a specific 5-stack. 2. Argue that every way of sorting this stack uses at least 5 flips. Give a way of sorting every 5-stack using at most 5 flips. To show P 5 ≥ 5 ?To show P 5 ≤ 5 ?

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What is P n for small n? Can you do n = 0, 1, 2, 3 ?

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Initial Values of P n n3012 PnPn 0013

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P 3 = 3 Biggest one to bottom using ≤ 2 flips. Smallest one to top using ≤ 1 flip. 1. Show a specific 3-stack. 2. Argue that every way of sorting this stack uses at least 3 flips. To show P 3 ≥ 3 ? Give a way of sorting every 3-stack using at most 3 flips. To show P 3 ≤ 3 ?

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? ≤ P n ≤ ? Lower Bound n th Pancake Number Upper Bound “What is the best upper bound and lower bound I can prove?”

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? ≤ P n ≤ ? n th Pancake Number Upper Bound Let’s start by thinking about an upper bound.

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???????????????????? n Upper Bound on P n Fix the biggest pancake.

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???????????????????? Upper Bound on P n n ???????????????????? n ???????????????????? n n−1 Fix the biggest pancake.

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???????????????????? Upper Bound on P n Fix pancake #n ≤ 2 flips n ???????????????????? n ???????????????????? n Fix pancake #n−1 ≤ 2 flips n-1 Fix pancake #n−2 ≤ 2 flips n-2 Fix pancake #2 ≤ 2 flips n-3 3 Fix pancake #3 ≤ 2 flips 2 1 ≤ 2n−2

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???????????????????? Upper Bound on P n Fix pancake #n ≤ 2 flips n ???????????????????? n ???????????????????? n Fix pancake #n−1 ≤ 2 flips n-1 Fix pancake #n−2 ≤ 2 flips n-2 Fix pancake #2 ≤ 1 flip n-3 3 Fix pancake #3 ≤ 2 flips ? ? ≤ 2n−3

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Upper Bound on P n Fix pancake #n ≤ 2 flips Fix pancake #n−1 ≤ 2 flips Fix pancake #n−2 ≤ 2 flips Fix pancake #2 ≤ 1 flip Fix pancake #3 ≤ 2 flips ≤ 2n−3 Bring-To-Top Method shows P n ≤ 2n−3 assuming n ≥ 2.

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? P n 2n−3 Let’s think about a lower bound for P n 1. Show a specific n-stack. 2. Argue that every way of sorting this stack uses a lot of flips.

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This stack seems pretty painful! Lower Bound on P n

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Breaking-Apart Argument Suppose the stack has an adjacent pair which should not be adjacent in the end. Spatula must go between them at least once. (“Adjacent pair” includes bottom pancake and the plate.) Each flip can achieve at most 1 “break-apart”.

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Proof of P n ≥ n n n S S contains n adjacent pairs which need to be broken apart, each necessitating at least one flip. Case 1: n is even. Detail: Assuming n >

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Proof of P n ≥ n n n S S contains n adjacent pairs which need to be broken apart, each necessitating at least one flip. Case 2: n is odd. Detail: Assuming n >

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Upper and lower bounds are within a factor of 2. Upper Bound Lower Bound n P n 2n−3 (for n > 4)

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From any n-stack to sorted n-stack in ≤ P n Slight Digression From sorted n-stack to any n-stack in ≤ P n ? Reverse the sequence of flips used to sort! Hence: any n-stack to any n-stack in ≤ 2P n Is there a better way?

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Any Stack S to Any Stack T in ≤ P n S: 4,3,5,1,2T: 5,1,4,3,2 3,4,1,2,5 1,2,3,4,5 Rename the pancakes in T to be 1,2,3,…,n Rewrite S using the new naming scheme In ≤ P n flips can sort “new S”. The same sequence of flips also brings S to T. “new S”

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The Known Pancake Numbers n PnPn

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P 20 is unknown ⋅⋅⋅ 21 = 20! possible 20-stacks 20! = 2.43 × (2.43 exa-pancakes) Brute-force analysis is impossible! It is either 23 or 24, we don’t know which.

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Is This Really Computer Science?

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Posed in Amer. Math. Monthly 82(1), 1975, by “Harry Dweighter” (haha). AKA Jacob Goodman, a computational geometer.

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In 1977, the observations we have made so far were published by Mike Garey, David Johnson, & Shen Lin

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(17/16)n ≤ P n ≤ (5/3)n+5/3 Discrete Mathematics 27(1), 1979 Bounds For Sorting By Prefix Reversal by: William H. Gates (Microsoft, Albuquerque NM) Christos Papadimitriou (UC Berkeley) upper bound also by Győri & Turán

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(15/14)n ≤ P n ≤ (5/3)n+5/3 “On the Diameter of the Pancake Network” Journal of Algorithms 25(1), 1997 by Hossain Heydari and Hal Sudborough

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(15/14)n ≤ P n ≤ (18/11)n “An (18/11)n Upper Bound For Sorting By Prefix Reversals” Theoretical Computer Science 410(36), 2009 by B. Chitturi, W. Fahle, Z. Meng, L. Morales, C.O. Shields, I.H. Sudborough, W. UT Dallas

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(3/2)n−1 ≤ B P n ≤ 2n+3 Burnt Pancakes

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(3/2)n ≤ B P n ≤ 2n−2 Burnt Pancakes “On The Problem Of Sorting Burnt Pancakes” Discrete Applied Math. 61(2), 1995 by David S. Cohen and Manuel Blum X.

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Worst Case: There is an algorithm using ≤ (18/11)n flips, even when your worst enemy gives you stack of n pancakes Average Case: There is an algorithm using ≤ (17/12)n flips on average when given a random stack of n pancakes (Josef Cibulka, 2009)

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Applications

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“The Pancake Network” on n! nodes Nodes are named after the n! different stacks of n pancakes Put a link between two nodes if you can go between them with one flip

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Pancake Network, n =

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Pancake Network, n = 4

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PnPn Pancake Network: Message Routing Delay What is the maximum distance between two nodes in the pancake network?

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If up to n−2 nodes get hit by lightning, the network remains connected, even though each node is connected to only n−1 others The Pancake Network is optimally reliable for its number of nodes and links Pancake Network: Reliability

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Computational Biology Transforming Cabbage Into Turnip: polynomial algorithm for sorting signed permutations by reversal by S. Hannenhalli & P. Pevzner Of Mice And Men: algorithms for evolutionary distances between genomes with translocation by J. Kececioglu and R. Ravi

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One “Simple” Puzzle A host of problems and applications at the frontiers of science

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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. Today’s lecture is a microcosm of this.

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Definitions of: n th pancake number upper bound lower bound Proof of: Bring-To-Top Breaking-Apart ANY to ANY in ≤ P n Study Guide

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Homework 1: Puzzle Hunt. Begins now. Turn in your signed Honor Code. Go to HW1 on the course web site. Special homework rules exceptions: 1. Work in randomly pre-selected teams of 4 2. May use the Internet as much as you want

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