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 Numbers1 + 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

123

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

alderaanalpheratzamberdaliescherpicassoreliablerenoiryukon

vangogh

How Many Questions? Question # Search space

0 100,0001 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

acrepotsstopcarepostsnap

acer: acreopst: potsopst: stopacer: careopst: postanps:snap

acer: acreacer: careanps:snapopst: potsopst: stopopst: post

sign sort

Anagram Program

acer: acre careanps: snapopst: pots stop post

merge

acer: acreacer: careanps:snapopst: potsopst: stopopst: 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

opstsign

lookup

acer: acre careanps: snapopst: 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