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COM1721: Freshman Honors Seminar A Random Walk Through Computing Rajmohan Rajaraman Tuesdays, 5:20 PM, 149 CN

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Introduction Explore a potpourri of concepts in computing 1: a mixture of flowers, herbs, and spices that is usually kept in a jar and used for scent 2: a miscellaneous collection Etymology: French pot pourri, literally rotten pot Theory, examples, and applications Readings: Handouts and WWW Grading: Quizzes, homework, and class participation

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Sample Concepts Abstraction Modularity Randomization Recursion Representation Self-reference …

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

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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

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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

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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

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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

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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

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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

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Illustration: Representation Problem: Derive an expression for the sum of the first n natural numbers 1 + 2 + 3 + … + n-2 + n-1 + n = ?

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Sum of First n Natural Numbers 1 + 2 + 3 + … + 98 + 99 + 100 = S 100 + 99 + 98 + … + 3 + 2 + 1 = S 101 + 101 + 101 + … + 101 + 101 = 2S S = 100*101/2 S = n(n+1)/2

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A Different Representation 1 2 3

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A “Geometric Derivation”

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Other Equalities Sum of first n odd numbers 1 + 3 + 5 + … + 2n-1 = ? Sum of first n cubes 1 + 4 + 9 + 16 + … + n^3 = ?

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Representation and Programming Representation is the essence of programming Brooks, “The Mythical Man-Month” Data structures

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

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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?

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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

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Twenty Questions Compare with 50,000 th 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

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How Many Questions? ajuma alderaan alpheratz amber dali escher picasso reliable renoir yukon vangogh

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How Many Questions? Question # Search space 0 100,000 1 50,000 2 25,000 3 12,500 5 3,125 10 100 15 4 17 1

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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

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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

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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()

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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

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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:acer deposit posited topside dopiest: deiopst subessential suitableness: abeeilnssstu

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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

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Anagram Program acer: acre care anps: snap opst: pots stop post merge acer: acre acer: care anps:snap opst: pots opst: stop opst: post

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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

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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

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References Programming Pearls, by Jon Bentley, Addison-Wesley Great Ideas in Theoretical Computer Science, Steven Rudich A course at CMU

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