Algorithms, Efficiency and Complexity Lecture of Week 2 on ...

furiouserectAI and Robotics

Nov 21, 2013 (3 years and 8 months ago)

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“Complexity”

P=NP? Who knows? Who cares?


Let’s revisit some questions from last
time


How many
pairwise

comparisons do I
need to do to check if a sequence of n
-
numbers is sorted?


If I have a procedure for checking
whether a sequence is sorted, is it
reasonable to sort a sequence of
numbers by generating permutations
and testing if any of them are sorted?


What major CS theorem did someone
claimed to have proved recently?

Intelligence is putting the


“test” part of



Generate&Test



into generate part…

Exactly when do we say an algorithm is
“slow”?


We kind of felt that O(N! * N) is a bit much
complexity


How about O(N
2
)? O(N
10
)? Where do we draw
the line?


Meet the Computer Scientist Nightmare


So “Polynomial” ~ “easy” & “exponential” ~ “hard”


2
n

eventually overtakes
any

n
k


however large k is..


How do we know if a problem is “really” hard to
solve or it is just that I am dumb and didn’t know
how to do better?

2
n

Classes P and NP

Class P


If a problem can be solved
in time polynomial in the
size of the input it is
considered an “easy”
problem


Note that
your

failure to solve
a problem in polynomial time
doesn’t mean it is not
polynomial (you could come
up with O(N* N!) algorithm
for sorting, after all



Class NP


Technically “if a problem can
be solved in polynomial time
by a non
-
deterministic
turing

machine, then it is in class NP”


Informally, if you can check the
correctness of a solution in
polynomial time, then it is in
class NP


Are there problems where even
checking the solution is hard?


Tower of Hanoi (or Brahma)


Shift the disks from the left
peg to the right peg


You can lift one disk at a
time


You can use the middle
peg to “park” disks


You can never ever have a
larger disk on top of a
smaller disk (or KABOOM)


How many moves to solve
a 2
-
disk version? A 3
-
disk
one? An n
-
disk one?


How long does it take (in
terms of input size), to
check if you have a
correct solution?

How to explain to your boss as to why
your program is so slow…

I can't find an efficient algorithm, I guess I'm just too dumb.

I can't find an efficient algorithm,

because no such algorithm is possible.

I can't find an efficient algorithm, but neither

can all these famous people.

The P=NP question


Clearly, all polynomial problems
are also NP problems


Do we know for sure that there
are NP problems that are
not

polynomial?


If we assume this, then we are
assuming P != NP


If P = NP, then some smarter
person can still solve a problem
that we thought can’t be solved
in
polytime


Can imply more than a loss of
face… For example, factorization
is known to be an NP
-
Complete
problem; and forms the basis for
all of cryptography.. If P=NP, then
all the cryptography standards
can be broken!

NP
-
Complete: “hardest” problems



in class NP


EVERY problem in class NP can


be reduced to an NP
-
Complete


problem in polynomial time



--
So you can solve that problem



by using an algorithm that solves



the NP
-
complete problem




Academic Integrity


What it means


Typical ASU policy


Homeworks


Exams


Take
-
Home Exams


Term papers



Scholarship Opportunities


General Scholarships


The FURI program


NSF
REU program

Is exponential complexity the worst?


After all, we can have 2
2



More fundamental question:

Can every computational
problem be solved in finite
time?


“Decidability”



--
Unfortunately not
guaranteed


[and Hilbert
turns in his grave]

2
n

Some Decidability Challenges


In First Order Logic, inference
(proving theorems) is
semi
-
decidable


If the theorem is true, you can show
that in finite time; if it is false you may
never be able to show it


In First Order Logic +
Peano

Arithmatic
, inference is
undecidable


There may be theorems that are
true

but you can’t prove them [
Godel
]

Reducing Problems…


Mathematician
reduces “mattress
on fire” problem


Make
Rao

Happy

Make Everyone in
ASU Happy

Make Little
Tommy Happy

Make his entire
family happy

General NP
-
problem

Boolean
Satisfiability

Problem

3
-
SAT

Practical Implications of Intractability


A class of problems is said to
be NP
-
hard as long as the
class contains at least one
instance that will take
exponential time..


What if 99% of the instances
are actually easy to solved?


--
Where then are the wild
things?


Satisfiability

problem


Given a set of propositions P
1

P
2


P
n




..and a set of “clauses” that the propositions must
satisfy


Clauses are of the form P
1

V P
7

VP
9

V P
12

V ~P
35




Size of a clause is the number of propositions it
mentions; size can be
anywhere from 1 to n



Find a T/F assignment to the propositions that respects
all clauses


Is it in class NP? How many “potential” solutions?



Canonical NP
-
Complete Problem.


3
-
SAT is where all clauses are of length 3



Example of a SAT problem


P,Q,R are propositions


Clauses


P V ~Q V R


Q V ~R V ~P


Is P=False, Q=True, R=False as solution?


Is Boolean SAT in NP?


Hardness of 3
-
sat as a function of

#clauses/#variables

#clauses/#variables

Probability that


there is a satisfying


assignment

Cost of solving


(either by finding


a solution or showing


there ain’t one)

p=0.5

You would

expect this

This

is what

happens!

~4.3

Phase Transition!

Phase Transition in SAT

Theoretically we only know that phase transition ratio


occurs between 3.26 and 4.596.

Experimentally, it seems to be close to 4.3

(We also have a proof that 3
-
SAT has sharp threshold)