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Explore the relationship between simulating the universe and the orders of growth in astrophysics. Understand the impact of scaling on computing power and knowledge of the universe.
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David Evans http://www.cs.virginia.edu/evans Class 19: Golden Ages and Astrophysics CS200: Computer Science University of Virginia Computer Science
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)? When we double the size of the simulation, the work octuples! (Just like oceanography octopi simulations) (n3) CS 200 Spring 2004
Orders of Growth simulating universe bubblesort insertsort-tree CS 200 Spring 2004
Astrophysics and Moore’s Law • Simulating universe is(n3) • 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? CS 200 Spring 2004
Knowledge of the Universe ;;; 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 (n3) 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)) CS 200 Spring 2004
Knowledge of the Universe (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))) CS 200 Spring 2004
> (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 Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. Albert Einstein Will there be any mystery left in the Universe when you die? CS 200 Spring 2004
Any Harder Problems? • Understanding the universe is (n3) • Are there any harder problems? CS 200 Spring 2004
Who’s the real genius? CS 200 Spring 2004
All Cracker Barrel Games(starting with peg 2 1 missing) CS 200 Spring 2004
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 • … CS 200 Spring 2004
Possible Moves Peg board game n = number of holes Initially, there are n-1 pegs. Cracker Barrel’s game has n = 15 Start Assume there are always exactly 2 possible moves, how many possible games are there? CS 200 Spring 2004
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 = nn-2 • There are at least 2 choices for every move: 2 * 2 * 2 * … * 2 = 2n-2 CS 200 Spring 2004
How much work is our straightforward peg board solving procedure? O(nn) upper bound isnn (2n)lower bound is 2n ImportantNote: 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.) CS 200 Spring 2004
“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 True Genius? “80% of life is just showing up.” Woody Allen CS 200 Spring 2004
Orders of Growth simulating universe peg board game Tuttlesort insertsort-tree CS 200 Spring 2004
Orders of Growth peg board game simulating universe TuttleSort CS 200 Spring 2004
Orders of Growth peg board game “intractable” “tractable” simulating universe 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)
Any other procedures we’ve seen that are more work than simulating the Universe? (To be continued in Lecture 20) CS 200 Spring 2004
