David Evanshttp://www.cs.virginia.edu/evans

Class 19: Golden Ages and Astrophysics CS200: Computer Science

University of VirginiaComputer Science

23 February 2004 CS 200 Spring 2004 2

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)When we double the size of the simulation, the work octuples! (Just like oceanography octopi simulations)

23 February 2004 CS 200 Spring 2004 3

Orders of Growth

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

logn n

nlogn n̂ 2

n̂ 3

bubblesort

simulatinguniverse

insertsort-tree

23 February 2004 CS 200 Spring 2004 4

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?

23 February 2004 CS 200 Spring 2004 5

;;; 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))

Knowledge of the Universe

23 February 2004 CS 200 Spring 2004 6

(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

23 February 2004 CS 200 Spring 2004 7

> (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

23 February 2004 CS 200 Spring 2004 8

Any Harder Problems?

• Understanding the universe is (n3)

• Are there any harder problems?

23 February 2004 CS 200 Spring 2004 9

Who’s the real genius?

23 February 2004 CS 200 Spring 2004 10

All Cracker Barrel Games(starting with peg 2 1 missing)

Pegs Left

Number of Ways

Fraction of Games

IQ Rating

1 1550 0.01 “You’re Genius”2 20686 0.15 “You’re Purty Smart”3 62736 0.46 “Just Plain Dumb”4 46728 0.33

“Just Plain

Eg-no-ra-moose”

5 5688 0.04

6 374 0.0027

7 82 0.00058

10 2 0.00001

23 February 2004 CS 200 Spring 2004 11

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

• …

23 February 2004 CS 200 Spring 2004 12

Possible Moves

StartPeg board gamen = number of holesInitially, there are n-1 pegs.

Cracker Barrel’s game hasn = 15

Assume there arealways exactly 2 possible moves,how many possiblegames are there?

23 February 2004 CS 200 Spring 2004 13

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

23 February 2004 CS 200 Spring 2004 14

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 (nn) upper bound is nn

(2n) lower bound is 2n

23 February 2004 CS 200 Spring 2004 15

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

23 February 2004 CS 200 Spring 2004 16

Orders of Growth

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

logn n

nlogn n̂ 2

n̂ 3 2̂ n

Tuttlesort

simulatinguniverse

insertsort-tree

peg board game

23 February 2004 CS 200 Spring 2004 17

Orders of Growth

0

10000

20000

30000

40000

50000

60000

70000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

logn n

nlogn n̂ 2

n̂ 3 2̂ n

TuttleSort

simulatinguniverse

peg board game

Orders of Growth

0

200000

400000

600000

800000

1000000

1200000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

logn n

nlogn n̂ 2

n̂ 3 2̂ n

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)

23 February 2004 CS 200 Spring 2004 19

Any other procedures we’ve seen that are more work than

simulating the Universe?

(To be continued in Lecture 20)