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

Class 19:
Golden Ages and
Astrophysics
CS200: Computer Science
University of Virginia
Computer Science
David Evans
http://www.cs.virginia.edu/evans
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
3
(n )
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simulation, the work octuples!
(Just like oceanography octopi
simulations)
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Orders of Growth
1200
1000
simulating
universe
logn
n
nlogn
n^2
800
600
400
n^3
200
bubblesort
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2
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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?
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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))
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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))))
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(knowledge-of-universe
(find-biggest-scale
1)))
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> (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
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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?
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Any Harder Problems?
• Understanding the
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universe is (n )
• Are there any harder
problems?
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Who’s the real genius?
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All Cracker Barrel Games
(starting with peg 2 1 missing)
Pegs
Left
Number
of Ways
1
1550
2
20686
3
62736
4
46728
5
5688
6
374
7
82
10
2
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Fraction of
Games
IQ Rating
0.01 “You’re Genius”
0.15 “You’re Purty Smart”
0.46 “Just Plain Dumb”
0.33
0.04
“Just Plain
0.0027
Eg-no-ra-moose”
0.00058
0.00001
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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
• …
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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?
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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
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How much work is our straightforward
peg board solving procedure?
O

n
(n )
n
(2 )
n
n
upper bound is
n
lower bound is 2
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.)
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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
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Orders of Growth
1200
1000
simulating
universe
logn
n
nlogn
n^2
800
600
peg board
game
400
n^3
2^n
200
Tuttlesort
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Orders of Growth
70000
peg board
game
60000
logn
n
nlogn
n^2
n^3
2^n
50000
40000
30000
20000
10000
simulating
universe
0
1
2
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10 11 12 13 14 15 16 TuttleSort
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Orders of Growth
1200000
1000000
logn
n
nlogn
n^2
peg
board
game
800000
“intractable”
600000
400000
n^3
2^n
“tractable”
200000
0
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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)
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