Transcript Class 37: P = NP ? CS200: Computers, Programs and Computing University of Virginia David Evans Computer Science http://www.cs.virginia.edu/evans.

Class 37:
P = NP ?
CS200: Computers, Programs and Computing
University of Virginia
David Evans
Computer Science
http://www.cs.virginia.edu/evans
Complexity and Computability
• We’ve learned how to measure the
complexity of any procedure ()
• We’ve learned how to show some
problems are undecidable
• But…we haven’t learned how to
measure the complexity of
problems
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Permuted Sorting
• A (possibly) really dumb way to sort:
– Find all possible orderings of the list
(permutations)
– Check each permutation in order, until you
find one that is sorted
• Example: sort (3 1 2)
All permutations:
(3 1 2) (3 2 1) (2 1 3) (2 3 1) (1 3 2) (1 2 3)
is-sorted?
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is-sorted? is-sorted?
is-sorted?
CS 200 Spring 2004
is-sorted?
is-sorted?
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permute-sort
(define (permute-sort cf lst)
(car
(filter (lambda (lst) (is-sorted? cf lst))
(all-permutations lst))))
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is-sorted?
(define (is-sorted? cf lst)
(or (null? lst)
(= 1 (length lst))
(and (cf (car lst) (cadr lst))
(is-sorted? cf (cdr lst)))))
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all-permutations
(define (all-permutations lst)
(flat-one
(map
(lambda (n)
(if (= (length lst) 1)
(list lst) ; The permutations of (a) are ((a))
(map
(lambda (oneperm)
(cons (nth lst n) oneperm))
(all-permutations (exceptnth lst n)))))
(intsto (length lst)))))
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> (time (permute-sort <= (rand-int-list 5)))
cpu time: 10 real time: 10 gc time: 0
(4 14 14 45 51)
> (time (permute-sort <= (rand-int-list 6)))
cpu time: 40 real time: 40 gc time: 0
(6 29 39 40 54 69)
> (time (permute-sort <= (rand-int-list 7)))
cpu time: 261 real time: 260 gc time: 0
(6 7 35 47 79 82 84)
> (time (permute-sort <= (rand-int-list 8)))
cpu time: 3585 real time: 3586 gc time: 0
(4 10 40 50 50 58 69 84)
> (time (permute-sort <= (rand-int-list 9)))
Crashes!
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How much
work is
permute-sort?
(define (permute-sort cf lst)
(car
(filter (lambda (lst) (is-sorted? cf lst))
(all-permutations lst))))
• We evaluated is-sorted? once for each
permutation of lst.
• How much work is is-sorted??
(n)
• How many permutations of the list are
there?
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Number of permutations
(map
(lambda (n)
(if (= (length lst) 1) (list lst)
(map (lambda (oneperm) (cons (nth lst n) oneperm))
(all-permutations (exceptnth lst n)))))
(intsto (length lst)))
• There are n = (length lst) values in the first map,
for each possible first element
• Then, we call all-permutations on the list without
that element (length = n – 1)
• There are n * n – 1 * … * 1 permutations
• Hence, there are n! lists to check: (n!)
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Procedures and Problems
• So far we have been talking about
procedures (how much work is permutesort?)
• We can also talk about problems: how
much work is sorting?
• A problem defines a desired output for a
given input. A solution to a problem is a
procedure for finding the correct output for
all possible inputs.
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The Sorting Problem
• Input: a list and a comparison function
• Output: a list such that the elements
are the same elements as the input
list, but in order so that the
comparison function evaluates to true
for any adjacent pair of elements
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Problems and Procedures
• If we know a procedure that is that is (f (n))
that solves a problem then we know the
problem is O(f (n)).
• The sorting problem is O (n!) since we know
a procedure (permute-sort) that solves it in
 (n!)
• Is the sorting problem is (n!)?
No, we would need to prove there is no
better procedure.
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Problem Complexity
• How complex is the pegboard puzzle?
– We know a solution that is (2n)
– So, the problem is O(2n)
• But, we can’t say it is (2n) unless we
know there is no better solution
• We can say it is (n) since the we know
we can’t do better than a solution that
considers n moves
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Deterministic Computing
• All our computing models so far are
deterministic: you know exactly what to do
every step
– TM: only one transition rule can match any
machine configuration, only one way to follow
that transition rule
– Lambda Calculus: always -reduce the
outermost redex
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Nondeterministic Computing
• Allows computer to try different options at
each step, and then use whichever one
works best
• Make a Finite State Machine nondeterministic: doesn’t change computing time
• Make a Turing Machine non-deterministic:
might change computing time
• No one knows whether it does or not
• This is the biggest open problem in Computer
Science
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Making FSM’s Nondeterministic
0
0
0
1
1
2
1
3
#
HALT
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Nondeterministic FSM
0, 1
0
1
Input Read
2
3
Possible States
0
00
001
{ 1, 2 }
{ 1, 2 }
{ 1, 3 }
0010
{ 1, 2 }
00101
{ 1, 3 }
00101#
{ HALT }
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1
#
HALT
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Power of NFSM
• Can NFSMs computing anything FSMs
cannot compute?
No – you can always convert an NFSM to a FSM by
making each possible set of states in the NFSM into one
state in the new FSM. (Number of states may increase as 2s)
• Can NFSMs compute faster than FSMs?
No – both read input one letter at a time, and transition
automatically to the next state. (Both are always (n) where
n is the number of symbols in the input.)
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Nondeterministic TMs
• Multiple transitions possible at every step
• Follow all possible steps:
– Each possible execution maintains its own
state and its own infinite tape
– If any possible execution halts (in an
accepting state), the NTM halts, and the tape
output of that execution is the NTM output
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PS7 Community Prize
http://www.people.virginia.edu/~en9j/ps7/user-page.html
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With Great Communities,
Comes Great Responsibility
• Evong will collect the course evaluations:
– SEAS official evaluation (bubble form with
general comments space) used for determining
whether or not to offer CS200 again and
whether or not to fire me
– My own course improvement survey (lots of
specific questions for improving future CS200s)
• Make sure to give her your form on the last
day of class or earlier
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From the Course Pledge
I will provide useful feedback. I realize
this is an evolving course and it is
important that I let the course staff know
what they need to improve the course. I
will not wait until the end of the course to
make the course staff aware of any
problems. … I will fill out all course
evaluation surveys honestly and
thoroughly.
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Nondeterministic TMs
• Multiple transitions possible at every step
• Follow all possible steps:
– Each possible execution maintains its own
state and its own infinite tape
– If any possible execution halts (in an
accepting state), the NTM halts, and the tape
output of that execution is the NTM output
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Power of NTM
• Can NTMs computing anything TMs
cannot compute?
No – you can simulate a NTM with a regular TM:
keep track of all possible states on the tape
keep track of all possible tapes on the tape
(really long, but its an infinitely long tape)
• Can NTMs compute faster than TMs?
Maybe! No one has proved it one way or the other.
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This is amazing!
• An NTM is a machine that every time it has to
make a decision can just make both and use
whichever one turns out to be right later
• Equivalently: whenever an NTM has to make a
decision, it always makes the right one!
• We don’t know if a computer that always makes
the right decision is really better than one that
has to try all possible decisions
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Complexity Class P
Class P: problems that can be solved in
polynomial time by a deterministic TM.
O (nk) for some constant k.
Easy problems like sorting, making a
photomosaic using duplicate tiles,
simulating the universe are all in P.
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Complexity Class NP
Class NP: problems that can be solved in
polynomial time by a nondeterministic TM
If we could try all possible solutions at once, we
could identify the solution in polynomial time.
All problems in P are in NP.
Hard problems like the smiley puzzle and
the pegboard puzzle sorting are all in NP
(but might not be in P).
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Is the Pegboard Puzzle in P?
No one knows!
We can’t find a O(nk) solution.
We can’t prove one doesn’t
exist.
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Intractable Problems
1E+30
1E+28
time
since
“Big
Bang”
n!
1E+26
2n
1E+24
1E+22
1E+20
1E+18
1E+16
1E+14
P
1E+12
1E+10
2022
today
1E+08
1E+06
n2
n log n
10000
100
1
2
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64
128
log-log scale
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P = NP?
• Is there a polynomial-time solution to the
“hardest” problems in NP?
• No one knows the answer!
• The most famous unsolved problem in
computer science and math
• Listed first on Millennium Prize Problems
– win $1M if you can solve it
– (also an automatic A+ in this course)
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Charge
• PS8 is due less than one week from now!
• Next time:
– How to prove a problem is in NP
– How to prove a problem is one of the hardest
problems in NP
– Why the peg board puzzle, perfect
photomosaic problem, drug discovery problem,
traveling salesperson problem, map coloring
problem, etc. are all really the same problem
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