1 / 32

COM1721: Freshman Honors Seminar

COM1721: Freshman Honors Seminar. A Random Walk Through Computing Rajmohan Rajaraman Tuesdays, 5:20 PM, 149 CN. Introduction. Explore a potpourri of concepts in computing. Theory, examples, and applications Readings: Handouts and WWW Grading: Quizzes, homework, and class participation.

louis
Download Presentation

COM1721: Freshman Honors Seminar

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Introduction • Explore a potpourri of concepts in computing • Theory, examples, and applications • Readings: Handouts and WWW • Grading: Quizzes, homework, and class participation 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

  3. Sample Concepts • Abstraction • Modularity • Randomization • Recursion • Representation • Self-reference • …

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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

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

  12. 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

  13. A Different Representation 1 2 3

  14. A “Geometric Derivation”

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

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

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

  18. 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?

  19. 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

  20. 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

  21. How Many Questions? vangogh

  22. 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

  23. 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

  24. 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

  25. 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()

  26. 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

  27. 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

  28. Anagram Program sign sort

  29. Anagram Program merge

  30. 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 sign opst lookup

  31. 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

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

More Related