COM1721: Freshman Honors Seminar

A Random Walk Through Computing Rajmohan Rajaraman

Tuesdays, 5:20 PM, 149 CN

Introduction Explore a potpourri of concepts in

computing1: a mixture of flowers, herbs, and spices that is usually kept in a jar and used for scent2: a miscellaneous collectionEtymology: French pot pourri, literally rotten pot

Theory, examples, and applications

Readings: Handouts and WWW Grading: Quizzes, homework, and

class participation

Sample Concepts

Abstraction Modularity Randomization Recursion Representation Self-reference …

Sample Topics

Dictionary search Structure of the Web Self-reproducing programs Undecidability Private communication Relational databases Quantum computing, bioinformatics,

…

Abstraction A view of a problem that extracts the

essential information relevant to a particular purpose and ignores inessential details

Driving a car: We are provided a particular abstraction of the car

in which we only need to know certain controls Building a house:

Different levels of abstraction for house owner, architect, construction manager, real estate agent

Related concepts: information hiding, encapsulation, representation

Modularity

Decomposition of a system into components, each of which can be implemented independent of the others

Foundation for good software engineering

Design of a basic processor from scratch

Representation

To portray things or relationship between things

Knowledge representation: model relationship among objects as an edge-labeled graph

Data representation: bar graphs, histograms for statistics

Querying a dictionary; Web as a graph

Randomization

An algorithmic technique that uses probabilistic (rather than deterministic) selection

A simple and powerful tool to provide efficient solutions for many complex problems

Has a number of applications in security Cryptography and private

communication

Recursion

A way of specifying a process by means of itself Complicated instances are defined in

terms of simpler instances, which are given explicitly

Closely tied to mathematical induction

Fibonacci numbers

Self-reference A statement/program that refers to itself Examples:

“This statement contains five words” “This statement contains six words” “This statement is not self-referential” “This statement is false”

Important concept in computing theory Undecidability of the halting problem, self-

reproducing programs Gödel Escher Bach: an Eternal Golden Braid,

Douglas Hofstader

Illustration: Representation

Problem: Derive an expression for the sum of the first n natural numbers

1 + 2 + 3 + … + n-2 + n-1 + n = ?

Sum of First n Natural Numbers

1 + 2 + 3 + … + 98 + 99 + 100 = S100 + 99 + 98 + … + 3 + 2 + 1 = S

101 + 101 + 101 + … + 101 + 101 = 2S

S = 100*101/2

S = n(n+1)/2

A Different Representation

1

2

3

A “Geometric Derivation”

54

1)n(n S2

Other Equalities

Sum of first n odd numbers 1 + 3 + 5 + … + 2n-1 = ?

Sum of first n cubes 1 + 4 + 9 + 16 + … + n^3 = ?

Representation and Programming

Representation is the essence of programming Brooks, “The Mythical Man-

Month” Data structures

Dictionary

A collection of words with a specified ordering Dictionary of English words Dictionary of IP addresses Dictionary of NU student names

Searching a Dictionary

Suppose we have a dictionary of 100,000 words

Consider different operations Search for a word List all anagrams of a word Find the word matching the largest

prefix What representation (data

structure) should we choose?

Search for a Word

Store the words in sorted order in a linear array

Unsuccessful search: compare with 100,000 words

Successful search: on average, compare with 50,000

words

Twenty Questions

Compare with 50,000th word If match, then done If further in dictionary order, search right

half If earlier in dictionary order, search left

half Until word found, or search space empty Recursion Binary search

How Many Questions?

ajuma

alderaan

alpheratz

amber

dali

escher

picasso

reliable

renoir

yukon

vangogh

How Many Questions?

Question # Search space0 100,0001 50,0002 25,0003 12,5005 3,12510 10015 417 1

Anagrams

An anagram of a word is another word with the same distribution of letters, placed in a different order

Input: deposit Output: posited, topside, dopiest Anagrams: subessential

suitableness

Detecting Anagrams

How do you determine whether two words X and Y are anagrams? Compare the letter distributions Time proportional to number of

letters in each word Suppose this subroutine

anagram(X,Y) is fast

Listing Anagrams of a Word

Dictionary of 100,000 English words List all anagrams of least How should we represent the

dictionary? Linear array

Loop through dictionary: if anagram(X,least), include X in list

Running time = 100,000 calls to anagram()

A Different Data Structure

If X and Y are anagrams of each other, they are equivalent; the list of anagrams of X is same as the list for Y

This indicates an equivalence class of anagrams!

deposit posited topside dopiest race care acre adroitly dilatory idolatry

Anagram Signatures Would like to store anagrams in the

same class together How do we identify a class? Assign a signature!

Sort all the letters in the anagram word(s) Same for each word in a class!acre race care: acerdeposit posited topside dopiest: deiopst subessential suitableness:

abeeilnssstu

Anagram Program

acre

pots

stop

care

post

snap

acer: acre

opst: pots

opst: stop

acer: care

opst: post

anps:snap

acer: acre

acer: care

anps:snap

opst: pots

opst: stop

opst: post

sign sort

Anagram Program

acer: acre care

anps: snap

opst: pots stop post

merge

acer: acre

acer: care

anps:snap

opst: pots

opst: stop

opst: post

Listing Anagrams for Given Word X

Compute sign(X) and lookup sign(X) in dictionary using binary search

List all words in list adjacent to sign(X)post

opst

sign

lookup

acer: acre care

anps: snap

opst: pots stop post

Efficiency of Anagram Program

Once dictionary has been stored in new representation: Lookup takes at most 17 queries Listing time is proportional to number of

anagrams in the class What about the cost of new

representation? Sign each word, sort, and merge Expensive, but need to do it only once!

Preprocessing

References

Programming Pearls, by Jon Bentley, Addison-Wesley

Great Ideas in Theoretical Computer Science, Steven Rudich

A course at CMU