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1. Class 21: The Story So Far (Quicksort, Continuing Golden Ages) David Evans http://www.cs.virginia.edu/evans CS200: Computer Science University of Virginia Computer Science

2. InsertSort-tree (define (insertel-tree cf el tree) (if (null? tree) (make-tree null el null) (if (cf el (get-element tree)) (make-tree (insertel-tree cf el (get-left tree)) (get-element tree) (get-right tree)) (make-tree (get-left tree) (get-element tree) (insertel-tree cf el (get-right tree)))))) n = number of elements in tree (log n) n = number of elements in lst (define (insertsort-tree cf lst) (define (insertsort-worker cf lst) (if (null? lst) null (insertel-tree cf (car lst) (insertsort-worker cf (cdr lst))))) (extract-elements (insertsort-worker cf lst))) (n log n) CS 200 Spring 2004

3. extract-elements We need to make a list of all the tree elements, from left to right. (define (extract-elements tree) (if (null? tree) null (append (extract-elements (get-left tree)) (cons (get-element tree) (extract-elements (get-right tree)))))) CS 200 Spring 2004

4. Quicksort • C. A. R. (Tony) Hoare, 1962 • Divide the problem into: • Sorting all elements in the list where (cf (car list) el) is true (it is < the first element) • Sorting all elements in the list where (not (cf (car list) el) is true (it is >= the first element) • Will this do better? CS 200 Spring 2004

5. Quicksort (define (quicksort cf lst) (if (null? lst) lst (append (quicksort cf (filter (lambda (el) (cf el (car lst))) (cdr lst))) (list (car lst)) (quicksort cf (filter (lambda (el) (not (cf el (car lst)))) (cdr lst)))))) CS 200 Spring 2004

6. (define (quicksort cf lst) (if (null? lst) lst (append (quicksort cf (filter (lambda (el) (cf el (car lst))) (cdr lst))) (list (car lst)) (quicksort cf (filter (lambda (el) (not (cf el (car lst)))) (cdr lst)))))) How much work is quicksort? What if the input list is sorted? Worst Case: (n2) What if the input list is random? Expected: (n log2n) CS 200 Spring 2004

7. Comparing sorts > (testgrowth insertsort-tree) n = 250, time = 20 n = 500, time = 80 n = 1000, time = 151 n = 2000, time = 470 n = 4000, time = 882 n = 8000, time = 1872 n = 16000, time = 9654 n = 32000, time = 31896 n = 64000, time = 63562 n = 128000, time = 165261 (4.0 1.9 3.1 1.9 2.1 5.2 3.3 2.0 2.6) > (testgrowth quicksort) n = 250, time = 20 n = 500, time = 80 n = 1000, time = 91 n = 2000, time = 170 n = 4000, time = 461 n = 8000, time = 941 n = 16000, time = 2153 n = 32000, time = 5047 n = 64000, time = 16634 n = 128000, time = 35813 (4.0 1.1 1.8 2.7 2.0 2.3 2.3 3.3 2.2) Both are (n log2n) Absolute time of quicksort much faster CS 200 Spring 2004

8. Good enough for VISA? n = 128000, time = 35813 36 seconds to sort 128000 with quicksort (n log2n) How long to sort 800M items? > (log 4) 1.3862943611198906 > (* 128000 (log 128000)) 1505252.5494914246 > (/ (* 128000 (log 128000)) 36) 41812.57081920624 > (/ (* 128000 (log 128000)) 41812.6) 35.99997487578923 > (/ (* 800000000 (log 800000000)) 41812.6) 392228.6064130373 392000 seconds ~ 4.5 days CS 200 Spring 2004

9. Any other procedures we’ve seen that are more work than simulating the Universe? CS 200 Spring 2004

10. PS4: Break Lorenz Cipher Procedure • Try all possible wheel settings • How many possible wheel settings: 5 choices for first wheel * 5 choices for second wheel * 5 choices for third wheel • What is n? • The number of wheels • There are 5n possible wheel settings CS 200 Spring 2004

11. Lorenz Deciphering • For PS4: you had 3 wheels, each with 5 possible settings: 53 = 125 combinations • For WWII: Nazis has 12 wheels, each with more than 5 settings (up to 61 settings) 512 = 244 140 625 possible combinations • Bletchley Park’s cryptographers had to solve a problem that is 1 953 125 times harder than PS4! (and had to figure out structure of Lorenz machine themselves) CS 200 Spring 2004

12. Motivation Helps… Confronted with the prospect of defeat, the Allied cryptanalysts had worked night and day to penetrate German ciphers. It would appear that fear was the main driving force, and that adversity is one of the foundations of successful codebreaking. Simon Singh, The Code Book Having bombs dropping on you is at least 1 million times more motivating than getting a gold star! CS 200 Spring 2004

13. Recap • So far, we have been talking amount the work a procedure requires • In a few weeks, we will learn how to talk about the amount of work a problem requires • That is, how much work is the best possible sorting procedure? • For the general sorting problem, you can’t do better than quicksort: (n log2n) • But, VISA’s problem is simpler, so they can do much better: (n) CS 200 Spring 2004

14. The Endless Golden Age • Golden Age – period in which knowledge/quality of something doubles quickly • At any point in history, half of what is known about astrophysics was discovered in the previous 15 years! • Moore’s law today, but other advances previously: telescopes, photocopiers, clocks, etc. CS 200 Spring 2004

15. Endless Golden Age and “Grade Inflation” • Average student gets twice as smart and well-prepared every 15 years • You had grade school teachers (maybe even parents) who went to college! • If average GPA in 1970 is 2.00 what should it be today (if grading standards didn’t change)? CS 200 Spring 2004

16. Grade Inflation or Deflation? 2.00 average GPA in 1970 (“gentleman’s C”?) * 2 better students 1970-1985 * 2 better students 1985-2003 * 3 admitting women, non-whites (1971) * 1.54 population increase * 0.58 increase in enrollment Students 1970 11,000 Students 2002 18,848 (12,595 UG) Average GPA today should be: 21.4 CS200 has only the best of the best students, and only the best 26/31 of them stayed in the course after PS1, so the average grade in CS200 should be 21.4*2*2*31/26 = 102.06 CS 200 Spring 2004

17. Short Golden Ages • Golden Age – period in which knowledge/quality of something doubles quickly • Endless golden age: at any point in history, the amount known is twice what was known 15 years ago • Short golden age: knowledge doubles during a short, “golden” period, but only improves gradually most of the time CS 200 Spring 2004

18. Goal-den age Average Goals per Game, FIFA World Cups Changed goalkeeper passback rule

19. The Real Golden Rule? Why do fields like astrophysics, medicine, biology and computer science (?) have “endless golden ages”, but fields like • music (1775-1825) • rock n’ roll (1962-1973, or whatever was popular when you were 16) • philosophy (400BC-350BC?) • art (1875-1925?) • soccer (1950-1974) • baseball (1925-1950) • movies (1920-1940) have short golden ages? Thanks to Leah Nylen for correcting this (previously I had only 1930-1940, but that is only true for Hollywood movies). CS 200 Spring 2004

20. The Story so Far… CS 200 Spring 2004

21. From Lecture 1: The Liberal Arts language numbers Quadrivium (4 roads) Trivium (3 roads) Grammar Rhetoric Logic Arithmetic Music Geometry Astronomy CS 200 Spring 2004

22. From Lecture 1: Liberal Arts BNF replacement rules for describing languages, rules of evaluation for meaning • Grammar: study of meaning in written expression • Rhetoric: comprehension of verbal and written discourse • Logic: argumentative discourse for discovering truth • Arithmetic: understanding numbers • Geometry: quantification of space • Music: number in time • Astronomy Not yet… Interfaces between components (PS6-8), program and user (PS8) Trivium Rules of evaluation, if, recursive definitions Not much yet… wait until April Curves as procedures, fractals Quadrivium Yes, listen to “Hey Jude!” Yes: Neil deGrasse Tyson says so CS 200 Spring 2004

23. Charge If you want to do something important and be remembered, work in a field that has a short golden age from 2003-2018 • Shakespeare will be known a thousand years from now, but no one will have heard of any 21st century playwright • Bach will be known a thousand years from now, but no 20th century musician will be • Pele will be known a thousand years from now, but no one will remember Beckham, Ronaldo or Zidane CS 200 Spring 2004