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

AI and Robotics

Nov 21, 2013 (4 years and 5 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

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

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

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)