1. David Evans http://www.cs.virginia.edu/evans Class 19: Golden Ages and Astrophysics CS200: Computer Science University of Virginia Computer Science

2. 23 February 2004CS 200 Spring 20042 Astrophysics “If you’re going to use your computer to simulate some phenomenon in the universe, then it only becomes interesting if you change the scale of that phenomenon by at least a factor of 10. … For a 3D simulation, an increase by a factor of 10 in each of the three dimensions increases your volume by a factor of 1000.” How much work is astrophysics simulation (in  notation)? (n3)(n3) When we double the size of the simulation, the work octuples! (Just like oceanography octopi simulations)

3. 23 February 2004CS 200 Spring 20043 Orders of Growth bubblesort simulating universe insertsort-tree

4. 23 February 2004CS 200 Spring 20044 Astrophysics and Moore’s Law Simulating universe is  ( n 3 ) Moore’s law: computing power doubles every 18 months Tyson: to understand something new about the universe, need to scale by 10x How long does it take to know twice as much about the universe?

5. 23 February 2004CS 200 Spring 20045 ;;; doubling every 18 months = ~1.587 * every 12 months (define (computing-power nyears) (if (= nyears 0) 1 (* 1.587 (computing-power (- nyears 1))))) ;;; Simulation is  (n 3 ) work (define (simulation-work scale) (* scale scale scale)) (define (log10 x) (/ (log x) (log 10))) ;;; log is base e ;;; knowledge of the universe is log 10 the scale of universe ;;; we can simulate (define (knowledge-of-universe scale) (log10 scale)) Knowledge of the Universe

6. 23 February 2004CS 200 Spring 20046 (define (computing-power nyears) (if (= nyears 0) 1 (* 1.587 (computing-power (- nyears 1))))) ;;; doubling every 18 months = ~1.587 * every 12 months (define (simulation-work scale) (* scale scale scale)) ;;; Simulation is O(n^3) work (define (log10 x) (/ (log x) (log 10))) ;;; primitive log is natural (base e) (define (knowledge-of-universe scale) (log10 scale)) ;;; knowledge of the universe is log 10 the scale of universe we can simulate (define (find-knowledge-of-universe nyears) (define (find-biggest-scale scale) ;;; today, can simulate size 10 universe = 1000 work (if (> (/ (simulation-work scale) 1000) (computing-power nyears)) (- scale 1) (find-biggest-scale (+ scale 1)))) (knowledge-of-universe (find-biggest-scale 1))) Knowledge of the Universe

7. 23 February 2004CS 200 Spring 20047 > (find-knowledge-of-universe 0) 1.0 > (find-knowledge-of-universe 1) 1.041392685158225 > (find-knowledge-of-universe 2) 1.1139433523068367 > (find-knowledge-of-universe 5) 1.322219294733919 > (find-knowledge-of-universe 10) 1.6627578316815739 > (find-knowledge-of-universe 15) 2.0 > (find-knowledge-of-universe 30) 3.00560944536028 > (find-knowledge-of-universe 60) 5.0115366121349325 > (find-knowledge-of-universe 80) 6.348717927935257 Will there be any mystery left in the Universe when you die? Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. Albert Einstein

8. 23 February 2004CS 200 Spring 20048 Any Harder Problems? Understanding the universe is  ( n 3 ) Are there any harder problems?

9. 23 February 2004CS 200 Spring 20049 Who’s the real genius?

10. 23 February 2004CS 200 Spring 200410 All Cracker Barrel Games (starting with peg 2 1 missing) Pegs Left Number of Ways Fraction of Games IQ Rating 115500.01 “You’re Genius” 2206860.15 “You’re Purty Smart” 3627360.46 “Just Plain Dumb” 4467280.33 “Just Plain Eg-no-ra-moose” 556880.04 63740.0027 7820.00058 1020.00001

11. 23 February 2004CS 200 Spring 200411 Solving the Peg Board Game Try all possible moves Try all possible moves from the positions you get after each possible first move Try all possible moves from the positions you get after trying each possible move from the positions you get after each possible first move …

12. 23 February 2004CS 200 Spring 200412 Possible Moves Start Peg board game n = number of holes Initially, there are n-1 pegs. Cracker Barrel’s game has n = 15 Assume there are always exactly 2 possible moves, how many possible games are there?

13. 23 February 2004CS 200 Spring 200413 Cracker Barrel Game Each move removes one peg, so if you start with n-1 pegs, there are up to n-2 moves There are at most n choices for every move: n * n * n * n * … * n = n n-2 There are at least 2 choices for every move: 2 * 2 * 2 * … * 2 = 2 n-2

14. 23 February 2004CS 200 Spring 200414 How much work is our straightforward peg board solving procedure? Important Note: I don’t know if this is the best possible procedure for solving the peg board puzzle. So the peg board puzzle problem might not be harder than understanding the Universe (but it probably is.) O (n n ) upper bound is n n  (2 n ) lower bound is 2 n

15. 23 February 2004CS 200 Spring 200415 True Genius? “Genius is one percent inspiration, and ninety-nine percent perspiration.” Thomas Alva Edison “Genius is one percent sheer luck, but it takes real brilliance to be a true eg-no-ra-moose.” Cracker Barrel “80% of life is just showing up.” Woody Allen

16. 23 February 2004CS 200 Spring 200416 Orders of Growth Tuttlesort simulating universe insertsort-tree peg board game

17. 23 February 2004CS 200 Spring 200417 Orders of Growth TuttleSort simulating universe peg board game

18. Orders of Growth simulating universe peg board game “tractable” “intractable” I do nothing that a man of unlimited funds, superb physical endurance, and maximum scientific knowledge could not do. – Batman (may be able to solve intractable problems, but computer scientists can only solve tractable ones for large n )

19. 23 February 2004CS 200 Spring 200419 Any other procedures we’ve seen that are more work than simulating the Universe? (To be continued in Lecture 20)