2. Complejidad 2012-I

8/2/2019 2. Complejidad 2012-I

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Inefficienc and Intractabilit


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Introduction

There are problems for which: are solved by different algorithms.

forever (and ever)!

there are no known algorithms that solve them.


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Towers of Hanoi (I)

What is the time complexity of the best algorithm

that solves it?


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Towers of Hanoi (II)

The time complexity of the best knownalgorithm is 2N -1.

moves is also 2N -1.

The solution is optimal!


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Towers of Hanoi (III)

The original problem had 64 rings. If onering could be moved every 10 seconds, it

would take over 5 trillion ears to solve the

problem.

Even moving 1 million rings a second takes

over 1/2 million years


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Decision Problems

Perhaps part of the long running time of analgorithm is that we are looking for a

solution.

What if we just asked, Does a solution

exist?

A decisionproblem is one which has a yesorno answer.


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Most problems have a decision problemversion.

For example: What is the minimum grade I

can get on the final and still get an A in the

class?

The decision problem version is: If I get a(some number) on the final, will I still get

an A in the class?


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The Monkey Puzzle (I)


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The Monkey Puzzle (II)

Would you please give a naive algorithm tosolve this problem?


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The Monkey Puzzle (III)

Given N cards, a brute-force solution takesN! steps.

takes over 490 billion years for 25 cards.

For N=36, it would take far far longer than

the time elapsed since the Big Bang!


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The Monkey Puzzle (IV)

Arrangements that are easier to solve: All N cards are either of:


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The Monkey Puzzle (V)

The worst case is what matters. The performance of the most efficient

naive one.

Can there be an algorithm that takes

reasonable time?


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NKvs N! N! vs NN

Different Functions of TimeComplexity (I)

2N vs NK

What is the order?


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Different Functions of TimeComplexity (II)

Algorithms whose run time is bounded fromabove by a polynomial function of N are

called ol nomial-time al orithms.

The other algorithms are calledpolynomial-

time algorithms. Well call them

exponential algorithms.


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Different Functions of TimeComplexity (III)


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Different Functions of TimeComplexity (V)


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Reasonable Algorithms

Examining the previous three slides we findthat some algorithms might be reasonable

and others not reasonable.

Polynomial-time algorithms are said to bereasonable.


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Unreasonable Algorithms

An algorithm which has an exponential orsuper-exponential upper bound is said to be

unreasonable.

Note that an intractable algorithm has a

solution. It just takes an unreasonably long

time to find it.


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Reasonable Vs UnreasonableAlgorithms (I)

Consider N1000 and N!.. Which one is better

for values lower than1500?


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Reasonable Vs UnreasonableAlgorithms (II)

However, the vast majority of reasonablealgorithms that take NKhave a K no greater

that 5 or 6.

Thus, most of the reasonable algorithms are

useful and most of the unreasonable

algorithms are useless.


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Tractable and IntractableProblems


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More on the Monkey Puzzle

1. Computers are becoming faster by theweek. Over the last 10 years or so

com uter s eed has increased rou hl b a

factor of 50. Perhaps obtaining a practicalsolution to the problem is just a question of

awaiting an additional improvement in

computer speed.


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2. Doesnt the fact that we have not found abetter algorithm for this problem indicate

our incompetence at devising efficient

algorithms? Shouldnt computer scientistsbe working at trying to improve the

situation rather than s endin their time

teaching classes about it?

3. Havent people tried to look for an

exponential-time lower bound on the

problem, so that we might have aproof that

no reasonable algorithm exists?


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4. Maybe the whole issue is not worth theeffort, as the monkey puzzle problem is

just one specific problem. It might be a

colorful one, but it certainly doesnt looklike a very important one.


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Reply to point 4: It so happens that theMonkey Puzzle problem is not alone.

There are close to 1000 diverse algorithmic

problems like this one.

They have the amazing property that if we

,


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Problems NP-Complete (I)

Problems which have intractable solutions.But, none are known not to have tractable

solutions.

No one has proven they require super-polynomial time!!!

The best-known lower bounds are

(N).


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Problems NP-Complete (II)

They are complete. That means that if wesolve one of them, we can solve all of them.

,

exponential lower bound, that applies for allof them!


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Some NPC Problems

Generalized jigsaw or arrangement Graph problems

Hamiltonian path

Scheduling and Matching problem

Determining Logical Truth Map Colorings

Graph Coloring


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Airline puzzle

Can they form a rectangle?


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Ordinary Puzzle

If just one piece can fit in a given place, thetime complexity is cuadratic.

NPC because partial matches may not leadto a complete solution.


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Traveling-Salesman Problem

What is the naive solution? What is its

complexity?


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Hamiltonian Path

What is its complexity?


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Other problems

Determining Logical Truth Map Colorings

Gra h Colorin


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Nondeterminism

The notion of nondeterminism is rather non-intuitive.

algorithm as an abstract notion, not as arealistic goal.


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Nondeterminism is important to thedevelopment of the theory and the

explanation of the existence of this class of

problems. A nondeterministic algorithm has, in

deterministic algorithm, a very powerfuloracle.


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Problems solved by nondeterministic

algorithms have the property that they cantake an unacceptably long time to produce a

solution. If a solution is arrived at,

however, there is an easy way to convince

someone that it is a solution.


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Polynomial-Time Reduction (I)

Given two NP-Complete problems, apolynomial-time reduction is an algorithm

that runs in ol nomial time and reduces

one problem to the other. If problem X can be polynomically reduced

to problem Y. It means that X is no worse

than Y.


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Polynomial-Time Reduction (II)

Example


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Polynomial-Time Reduction (III)


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Exercise

How would you polynomically reduce the3-coloring problem to the satisfiability

roblem?

N countries C1, C2, , CN

Colors R (red), B (blue), Y (yellow)

Propositional sentence F


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Solution

3N assertions why?

What is the time of the reduction?


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Showing that a problem is NP-Complete (I)

Cooks remarkable theorem proved thatthere exists NP-completeproblems. In

articular he demonstrated the SAT was

NP-complete (1971). Cooks theorem makes showing a problem

is NP-complete easier.


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Showing that a problem is NP-Complete (II)

New problem R You must prove that you can

Reduce R to a NP-Complete problem Q (or

show a certificate/non-deterministic magical

algorithm).

Reduce a NP-Complete problem S to R (or use

a theorem)


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Where are the NP-Completeproblems in the sphere?


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Class P

The class P is comprised by the set ofproblems that are tractable, i.e. problems

that can be solved b a deterministic-

polynomial time algorithm.


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Where are the P problems in thesphere?


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Class NP (I)

NP is the set of all decision problems for which

the instances where the answer is "yes" haveefficiently verifiable proofs of the fact that the

answer s n ee yes.

More precisely, these proofs have to be

verifiable in polynomial time.

The hardest problems in NP are NP-Complete

problems.


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Class NP (II)

P = NP ? First asked in 1971

Is there NP-Intermediate?


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P = NP? (I)

Its clear that is P

NP, but P====

NP ? No one has yet demonstrated a problem that

To prove P = NP requires us to show that

every problem in NP can be solved by a

deterministic polynomial time algorithms.

Nobody has done this either!


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NP-Intermediate Problems (I)

Set of problems that are in the complexityclass NPbut are neither in the class P nor

NP-com lete.

The class of such problems is called NPI.


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NP-Intermediate Problems (II)

Ladner's theorem, shown in 1975, is a

result asserting that:

IfP NP, then NPI is not empty; that is, NP

con a ns pro ems a are ne er n nor -

complete.

P = NP if and only if NPI is empty.

NPI Candidates: graph isomorphism problem

computing the discrete logarithm.


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NP-Hard Problems (I)

At least as hard as the hardest problems inNP.

-

is an NP-complete problem L that ispolynomically reducible to X.

This means that no problem in NP is harder

than X.


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NP-Hard Problems (II)


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NP-Hard Problems (III) A problem X is called an NP-complete

problem if (1) it belongs to NP and (2) it is NP-hard.

o e a , a -comp e epro em can e

solved by a deterministic polynomial

algorithm, then P = NP.

Also, if any NP-hardproblem is shown to bein P, then P = NP.


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Optimizations Problems that areNP-Complete

NP-completeproblems are usually taken as thedecisionproblem (yes/no) version of

o timization roblems. E. . TSP.

Finding an optimal tour cannot be tractable

without the yes/no version being tractable too.

The original problem is also said to be NP-Complete.


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Imperfect Solutions to NP-Complete Problems (I)

But, humans manage to solve theseproblems all the time in short time!

Approximation algorithms find solutionsless than perfect, yet of practical value.

Some of them are guaranteed to be close the

optimal solution.


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Imperfect Solutions to NP-Complete Problems (II)

Some of them are guaranteed to be close theoptimal solution.

Ex: Cubic algorithm that always produces a

result at most 1.5 worse than the optimal.

Some of them produce a solution very close

to the optimal solution almost always. Ex: Algorithm that parts the graph and

combines local solutions.


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Provably Intractable Problems

We are not able toprove that problems inNP have no tractable solution

ere are pro ems, owever, suc as t e

Towers Of Hanoi, that can be shown to

have an exponential lower bound and that

have no certificate.


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Class Co-NP

Class of problems for which efficientlyverifiable proofs ofno instances, sometimes

called counterexam les .

Example ofNP-Complete: Given a finite set ofintegers is there a non-empty subset which

sums to zero?

Example ofco-NP: Given a finite set ofintegers, does every non-empty subset have a

nonzero sum?

Complexity Classes


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Complexity Classes

Theoretical Computer Scientists talk about a

hierarchy of complexity classes.

These classes are for both time and space

requirements.


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Research on Complexity Classes

Some theorists work with questions such as:Is P = NP? orIs NP = co-NP? orCan we

re ine the com lexit classes?

Others work to develop approximate

solutions or heuristic solutions or alternate

models of computation: genetic algorithms,neural networks, etc..

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2. Complejidad 2012-I