Tractable and intractable problems for parallel computers

1
Tractable and intractable problemsfor parallel
computers

• COMP 308

2
Complete problems

• A computational problem is complete for a
  complexity class when it is, in a formal sense,
  one of the "hardest" or "most expressive"
  problems in the complexity class.
• Complexity classes are sets of all of the
  problems that can possibly be solved with at most
  a certain amount of some computational resource,
  and may include problems that actually require
  far less resources.
• The complete problems, however, are the most
  resource-intensive problems in the class.

3
Complete problems encode all of the difficulty
of the complexity class

• Besides being very resource-intensive, complete
  problems are very expressive in some sense, they
  "encode" all of the expressiveness or difficulty
  of the complexity class.
• This is because, given a method of efficiently
  solving a complete problem, we can efficiently
  solve any other problem in the complexity class.
• So, for example, many of the problems in EXPTIME,
  the exponential time class, are very hard, and we
  could never hope to find efficient solutions for
  them. But if we were given a magic box that
  quickly solved any EXPTIME-complete problem
  (called an oracle machine for that problem), we
  could quickly solve any other problem in EXPTIME

4
Hard and complete problems

• When a problem has the property that it allows
  you to quickly solve any problem in a complexity
  class, we say that it is hard for that class.
• Equivalently, we can say that there is a
  reduction from any problem in the class to the
  hard problem.
• When a problem is both hard for a class, and a
  member of the class, it is complete for that
  class.

5
P (complexity)

• In computational complexity theory, P is the
  complexity class containing decision problems
  which can be solved by a deterministic Turing
  machine using a polynomial amount of computation
  time, or polynomial time.
• P is known to contain many natural problems,
  including linear programming, calculating the
  greatest common divisor, and finding a maximum
  matching.
• In 2002, it was shown that the problem of
  determining if a number is prime is in P.

6
P-complete class

• In complexity theory, the complexity class
  P-complete is a set of decision problems and is
  useful in the analysis of which problems can be
  efficiently solved on parallel computers.
• A decision problem is in P-complete if it is
  complete for P, meaning that it is in P, and that
  every problem in P can be reduced to it in
  polylogarithmic time on a parallel computer with
  a polynomial number of processors.
• In other words, a problem A is in P-complete if,
  for each problem B in P, there are constants c
  and k such that B can be reduced to A in time
  O((log n)c) using O(nk) parallel processors.

7
Motivation

• The class P, typically taken to consist of all
  the "tractable" problems for a sequential
  computer, contains the class NC, which consists
  of those problems which can be efficiently solved
  on a parallel computer. This is because parallel
  computers can be simulated on a sequential
  machine.
• It is not known whether NCP. In other words, it
  is not known whether there are any tractable
  problems that are inherently sequential.
• Just as it is widely suspected that P does not
  equal NP, so it is widely suspected that NC does
  not equal P.

8
P-complete problems

• The most basic P-complete problem is this
• Given a Turing machine, an input for that
  machine, and a number T (written in unary), does
  that machine halt on that input within the first
  T steps?
• It is clear that this problem is P-complete if
  we can parallelize a general simulation of a
  sequential computer, then we will be able to
  parallelize any program that runs on that
  computer.
• If this problem is in NC, then so is every other
  problem in P.

9

• This problem illustrates a common trick in the
  theory of P-completeness. We aren't really
  interested in whether a problem can be solved
  quickly on a parallel machine.
• We're just interested in whether a parallel
  machine solves it much more quickly than a
  sequential machine. Therefore, we have to reword
  the problem so that the sequential version is in
  P. That is why this problem required T to be
  written in unary.
• If a number T is written as a binary number (a
  string of n ones and zeros, where nlog(T)), then
  the obvious sequential algorithm can take time
  2n. On the other hand, if T is written as a unary
  number (a string of n ones, where nT), then it
  only takes time n. By writing T in unary rather
  than binary, we have reduced the obvious
  sequential algorithm from exponential time to
  linear time. That puts the sequential problem in
  P. Then, it will be in NC if and only if it is
  parallelizable.

10
P-complete problems

• Many other problems have been proved to be
  P-complete, and therefore are widely believed to
  be inherently sequential. These include the
  following problems, either as given, or in a
  decision-problem form
• In order to prove that a given problem is
  P-complete, one typically tries to reduce a known
  P-complete problem to the given one, using an
  efficient parallel algorithm.

11
Examples of P-complete problems

• Circuit Value Problem (CVP) - Given a circuit,
  the inputs to the circuit, and one gate in the
  circuit, calculate the output of that gate
• Game of Life - Given an initial configuration of
  Conway's Game of Life, a particular cell, and a
  time T (in unary), is that cell alive after T
  steps?
• Depth First Search Ordering - Given a graph with
  fixed ordered adjacency lists, and nodes u and v,
  is vertex u visited before vertex v in a
  depth-first search?

12
Problems not known to be P-complete

• Some problems are not known to be either
  NP-complete or P. These problems (e.g. factoring)
  are suspected to be difficult.
• Similarly there are problems that are not known
  to be either P-complete or NC, but are thought to
  be difficult to parallelize.
• Examples include the decision problem forms of
  finding the greatest common divisor of two binary
  numbers, and determining what answer the extended
  Euclidean algorithm would return when given two
  binary numbers.

13
Conclusion

• Just as the class P can be thought of as the
  tractable problems, so NC can be thought of as
  the problems that can be efficiently solved on a
  parallel computer.
• NC is a subset of P because parallel computers
  can be simulated by sequential ones.
• It is unknown whether NC P, but most
  researchers suspect this to be false, meaning
  that there are some tractable problems which are
  probably "inherently sequential" and cannot
  significantly be sped up by using parallelism
• The class P-Complete can be thought of as
  "probably not parallelizable" or "probably
  inherently sequential".