INTRODUCTION TO

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INTRODUCTION TO COMPUTATIONAL COMPLEXITY
Instructor Stefan Dobrev E-mail
sdobrev_at_site.uottawa.ca Office SITE 5043 Office
hours by appointment
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Course Organization

• Course web page
• http//www.site.uottawa.ca/sdobrev/CSI5140-CC
• Textbook
• Christos Papadimitriu Computational Complexity
• Addison Wesley 1994, ISBN 0-201-53082-1
• (I will post additional reading on the course
  web-site)
• Grading
• 4 assignments 60
• test 40

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Algorithm/Problem Complexity

• Algorithm analysis
• how long will this algorithm execute on a given
  input?
• practically well motivated (we want to be sure
  it works efficiently)
• we can define (worst-case, average case) time
  complexity of an algorithm
• What if we cant find sufficiently efficient
  algorithm for a given computational problem?
• i.e. cant find O(n) sorting algorithm
• cant find polynomial algorithms for the TSP
• it makes sense to define also the complexity of
  the problem

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Problem Complexity

• How much resources must any algorithm solving
  this problem use?
• Usually, time is the resource, but we will also
  consider memory (space), non-determinism,
  randomness
• Are there computational problems that cannot be
  algorithmically solved regardless of how much
  time/memory is available?
• we need precise definition of what a
  computational problem is, and what is an
  algorithm

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Models of Computation

• What is an algorithm?
• several smart people in the first part of 20th
  century tried to formally capture the notion of
  algorithm/computation
• they came up with several notions
• Turing machines, recursive functions, lambda
  calculus,
• at the end, all of them turned out to be
  equivalent
• We will use Turing machines as our model of
  computation
• Lecture plan for the next 2 weeks - review the
  Turing machines and related results
  (undecidability)

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Complexity Classes

• What are the classes of problems that can be
  solved in
• O(n2) time?
• using polynomial memory?
• using polynomial time?
• deterministically and non-deterministically
• What are relationships between those classes?
• P NP?
• PSPACE NPSPACE?

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Reductions and Completeness
How to solve difficult problems An experiment
is being made. A physicist and a mathematician
are asked to boil the hot water in a kettle in
the living room. The physicist goes to the living
room, takes the kettle, returns to the kitchen,
puts it on the stove and boils the water. The
mathematician does the same. In the second
stage, the kettle is in the kitchen and the two
are again asked to boil hot water. The physicist
simply puts the kettle on the stove and boils the
water. However, the mathematician takes the
kettle, puts it in the living room and declares
"We have already solved this problem!" We will do
this again and again.
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Reductions and Completeness

• Are there representative problems that truly
  reflect the inherent difficulty of a complexity
  class?
• if we can solve this representative problem, we
  can solve (in about the same complexity) any
  other problem in the class?
• called complete problems
• can be used to show coincidence of classes
• if an NP complete problem is shown to be in P
  then PNP
• We will see complete problems for several
  complexity classes.
• We will study NP-complete problems in detail
• most natural problems are either in P or NP
  complete

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Function Problems

• Complexity classes are defined for decision
  problems
• In reality, we often need not only yes/no answer,
  but also the solution (i.e. for TSP, we want to
  know the path).
• Can we efficiently construct the solution from
  the decision problem? (self-reducibility)
• Some problems have trivial decision problem, but
  the function problem is very interesting
• factoring

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Randomized Computation

• How to deal with NP-complete problems?
• go to Monte Carlo/Las Vegas and have a drink
• Use randomized computation
• Monte Carlo algorithms
• yes indeed yes
• no probably no (with non-1 probability of
  error)
• we can run it several times to reduce the
  probability of error
• Las Vegas algorithms
• answers yes, no, I dont know - with non 1
  probability
• eventually it will produce the correct answer

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Inside P

• P represents problems efficiently solvable on
  sequential machines
• Unfortunately, we do not know how to efficiently
  parallelise all problems in P
• What is the class of problems that are
  efficiently parallelisable?
• NC - polylogarithmic parallel time, polynomial
  total work
• Open problems
• NC P?
• What is the power of non-determinism in (small)
  space?
• we will see that PSPACE NPSPACE, what about L
  NL?

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Other topics we will cover

• Circuit Complexity what kind of languages have
  polynomial circuits?
• Complexity vs security one way functions, trap
  door functions, interactive proofs
• Anytime algorithms have a (not necessarily
  perfect) answer ready any time the answer is
  refined as more time is allowed
• Counting complexity classes Given a boolean
  expression, compute the number of different truth
  assignments that satisfy it.
• PSPACE and its many alternative characterizations
  (Alternating Turing Machines, Interactive
  proofs)
• Satisfiability of quantified boolean
  expresssions is PSPACE complete