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Archive-Name: theory-faq-intro
Last-modified: July 22, 1999
Version: 0.81

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Comp.Theory FAQ

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Table of Contents

  1. Introduction
  2. Purely Theoretical Models of Computation
  3. Theoretical Models of More Practical Importance
  4. Kolmogorov Complexity
  5. Sorting and related topics
  6. Permutations and Random number generators
  7. P vs NP
  8. Complexity Theory
  9. Logic in Computer Science
 10. Algorithm Libraries
 11. Open Problems.
 12. Electronic Resources
 13. Bibliography

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  1. Introduction

     History of (Theoretical) Computer Science

          "[W]e so readily assume that discovering, like seeing or
          touching, should be unequivocally attributable to an
          individual and to a moment in time. But the latter
          attribution is always impossible, and the former often is as
          well. [...] discovering [...] involves recognizing both that
          something is and what it is."
          -- Thomas S. Kuhn, The Structure of Scientific Revolutions,
          2nd Ed, 1970.

     In the same manner, there is no particular time or person who should
     be credited with the discovery or creation of theoretical computer
     science. However, there are important steps along the way towards the
     consolidation of the study of systematic resolution of problems as a
     science.

     Timeline:

     EARLY ALGORITHMS

     300 B.C.
          Greatest Common Divisor Algorithm proposed by Euclid.

     250 B.C.
          Sieve of Eratosthenes for prime numbers.

          Compass and straight edge (ruler) constructions

          EARLY ANALYSIS

     1910
          H. C. Pocklington describes the complexity of an algorithm as
          polynomial on the number of bits.

          COMPUTABILITY

     1900
          David Hilbert proposes the famous tenth problem asking for a
          general procedure to solve diophantine equations (polynomial
          equations with integer unknowns), setting the background for a
          formal definition of computable and computability.

          LOGIC FOUNDATIONS

     1920-30
          Post proposes a simple unary model of computation known as the
          Post machine.

     1930-35
          Goedel proves that any set of axioms containing the axioms of
          integer numbers has undecidable propositions.

     1930
          Alonzo Church introduces lambda-calculus.

     1936
          Alan Turing publishes the seminal paper where he presents the
          Turing Machine, a formal model of computability which is also
          physically realizable.

     IEEE Computer has a timeline of the history of computing available on
     the web. References:

     E. Bach and J. Shallit. Algorithmic Number Theory : Efficient
     Algorithms. MIT Press, 1996.

  2. Purely Theoretical Models of Computation

        o The Turing Machine
        o Other equivalent models
          Lambda Calculus, Post Machines.
        o Weaker Models
          DFA's or Regular Languages

          NFA's or Regular Languages

          PDA's or Context Free Languages (CFLs)

          LBTM's or Context Sensitive Languages

        o Universal Turing Machine and Church's Thesis
        o Variations on the Turing Machine
          NDTMs, ATMs. Oracles.

  3. Theoretical Models of More Practical Importance

        o RAM
        o PRAM and other models of parallel computation (NC, AC).
        o Other Models (DNA, Quantum Computers)
        o Circuits

  4. Kolmogorov Complexity Motivation

     Consider the following sequences of coin tosses:

     1) head, head, tail, head, tail, tail, tail, head, head

     2) head, tail, head, tail, head, tail, head, tail, head

     3) tail, tail, tail, tail, tail, tail, tail, tail, tail

     Now, if you had bet a hunderd dollars on heads, it is likely that you
     would see outcomes (2) and (3) with suspicion, due to their
     regularity. However, standard probability theory argues that each of
     the three outcomes above is equally (un)likely, and thus there is no
     reason why you should complain.

     In the same manner, if the sequences above had been generated by a
     pseudo-random generator for, say, a Monte Carlo algorithm, you would
     consider (2) and (3) to be fairly poor random sequences.

     In other words, we have an intuitive notion of randomness --applicable
     to outcomes of random trials-- which is not properly captured by
     classic probability.

     Definition

     In words of A.N. Kolmogorov:

          In everyday language we call random those phenomena where we
          cannot find a regularity allowing us to predict precisely
          their results. gernerally speaking, there is no ground to
          believe that random phenomena should possess any definite
          probability. Therefore, we should distinguish between
          randomness proper (as absence of any regularity) and
          stochastic randomness (which is the subject of probability
          theory). There emerges the problem of finding reasons for
          the applicability of the mathematical theory of probability
          to the real world.

     As indicated above, we tend to identify randomness with lack of
     discernable patterns, or irregularity.

          Definition A sequence of numbers is non-stochastically
          random if it is irregular.

     Further, we can narrow down the definition of irregularity.

          Definition A sequence of numbers is irregular if there is no
          discernible pattern in it.

     While it might seem that this is not much progress, we are now in fact
     very close to a formal definition.

     Three key observations need to be made:

       1. Some sequences seem more regular than others, which suggests the
          need for a measure of randomness (as opposed to a strict yes/no
          criterion).
       2. We can use Church's Thesis to define "discernible" as anything
          that can be computed or discovered by a computer, such as a
          Turing Machine.
       3. A "pattern" is anything that allows us to make a succint
          description of a sequence.

     For example, a sequence such as 00000000000000000000000000000000 can
     be described by a short program such as

      for i=1,30 print 0

     while a sequence with no pattern can only be described by itself (or
     some equally long sequence), e.g. 01001101110111011000001011

      print 01001101110111011000001011

          Definition The Kolmogorov Complexity K(x)of a string x is
          the length of the shortest program that outputs it.

     One can actually show that the choice of programming language affects
     the Kolmogorov Complexity by at most a constant additive
     factor (i.e. such factor does not depend on x) for all but a finite
     number of strings x.

          Definition A string is said to be incompressible (i.e.
          random or irregular) if
          K(x) > length(x)+constant

     Reference:

     Li, M and Vitanyi, P. An Introduction to Kolmogorov Complexity and its
     applications. New York, Springer-Verlag, 1993.

  5. Sorting and related topics

     (a) What is the fastest sort?

     The answer to this question depends on (i) what you are sorting and
     (ii) which tools you have.

     To be more precise, the parameters for (i) are the (a) size of the
     universe from which you select the elements to be sorted, (b) the
     number of elements to be sorted, (c) whether the comparison function
     can be applied over parts of the keys, (d) information on the
     distribution of the input.

     For (ii) we have the amount of available primary and secondary storage
     as a function of (i.a) and (i.b), and the power of the computational
     model (sorting network, RAM, Turing Machine, PRAM).

     To illustrate with a simple example, sorting n different numbers in
     the interval [1,n] can be done trivially in linear time on a RAM.

     A well studied case is sorting n elements from an infinite universe
     with a sequence of comparators which only accept two whole elements
     from the universe as input and produce as output the sorted pair.

     These comparators may be arranged in a predetermined manner or the
     connections can be decided at run time.

     It turns out that this apparently unrealistic setting (after all we
     sort with von Neumann RAM machines running programs which use
     arithmetic operations) models a large class of sorting algorithms for
     RAMs which are used in practice, and thus the importance of the next
     section.

     (b) n log n information bound

          Theorem. Any comparison based sorting program must use at
          least ceil(lg N!) > N lg N - N/ ln 2 comparisons for some
          input.

     The main two reasons for using this model are that (a) it is amenable
     to study and (b) it produces bounds and timings that were generally
     sufficiently close to practical applications.

     However, nowadays servers come routinely equipped with up to 1
     Gigabyte of memory. Under this configuration some methods which are
     memory intensive (such as radix sort or bucket sort) become practical.
     In fact, a method such as bucket sort on N = 10,000,000 records and
     100,000 buckets takes time 27 N. In contrast, the best comparison
     based sorting algorithms take time ~ 40 N.

     Recently, Andersson has proposed a promising algorithm that takes time
     O(n log log n) which takes the advantage of the fact that RAM
     computers can operate on many bits (usally 32 or 64) bits on a single
     instruction. You can find more information in Stefan Nilsson's Home
     Page.

          Theorem. It is possible to sort n keys each occupying L
          words in O(nL) time using indirect addressing on a RAM
          machine.

          Theorem. A set of n keys of length w = word size can be
          sorted in linear space in O(n log n/log log n) time.

     Other sorts:

        o Adaptive sorting
        o Sorting Networks

     References:

     R. Sedgewick, P. Flajolet. An introduction to the anlysis of
     algorithms. Addison Wesley, 1996.

     A. Andersson. Sorting and Searching Revisited. Proceedings of the 5th
     Scandinavian Workshop on Algorithm Theory. Lecture Notes in Computer
     Science 1097. Springer-Verlag, 1996.

     Electronic Resources:

     Sedgewick's Shell Sort home page.

     Pat Morin's sorting Java Applets

  6. Permutations and Random number generators The pLab page on the "Theory
     and Practice of Random Number Generation" is a very good start

  7. P vs NP

     Historical Context

     If one goes back a couple of hundred years, we can see that the
     historical motivation for the study of complexity of algorithms is the
     desire to identify, under a formal framework, those problems that can
     be solved "fast".

     To achieve this, we need to formally define what we mean by "problem",
     "solve" and "fast".

     Let's postpone the issue of what "problem" and "solve" is by
     restricting ourselves to well-defined mathematical problems such as
     addition, multiplication, and factorization.

     One of the first observations that can be made then, is that even some
     "simple" problems may take a long time if the question is long enough.
     For example, computing the product of two numbers seems like a fast
     enough problem, Nevertheless one can easily produce large enough
     numbers that would bog down a fast computer for a few minutes.

     We can conclude then that we must consider the size of the "question"
     as a parameter for time complexity. Using this criterion, we can
     observe that constant time answers as a function of the size of the
     question are fast and exponential time are not. But what about all the
     problems that might lie in between?

     It turns out that even though digital computers have only been around
     for fifty years, people have been trying for at least thrice that long
     to come up with a good definition of "fast". (For example, Jeff
     Shallit from the University of Waterloo, has collected an impressive
     list of historical references of mathematicians discussing time
     complexity, particularly as it relates to Algorithmic Number Theory).

     As people gained more experience with computing devices, it became
     apparent that polynomial time algorithms were fast, and that
     exponential time were not.

     In 1965, Jack Edmonds in his article Paths, trees, and flowers
     proposed that "polynomial time on the length of the input" be adopted
     as a working definition of "fast".

     So we have thus defined the class of problems that could be solved
     "fast", i.e. in polynomial time. That is, there exists a polynomial
     p(n) such that the number of steps taken by a computer on input x of
     length n is bounded from above by p(n). This class is commonly denoted
     by P.

     By the late 1960s it had become clear that there were some seemingly
     simple problems that resisted polynomial time algorithmic solutions.
     In an attempt to classify this family of problems, Steve Cook came up
     with a very clever observation: for a problem to be solved in
     polynomial time, one should be able --at the very least-- to verify a
     given correct solution in polynomial time. This is called certifying a
     solution in polynomial time.

     Because, you see, if we can solve a problem in polynomial time and
     somebody comes up with a proposed solution S, we can always rerun the
     program, obtain the correct solution C and compare the two, all in
     polynomial time.

     Thus the class NP of problems for which one can verify the solution in
     polynomial time was born. Cook also showed that among all NP problems
     there were some that were the hardest of them all, in the sense that
     if you could solve any one of those in polynomial time, then it
     followed that all NP problems can be solved in polynomial time. This
     fact is known as Cook's theorem, and the class of those "hardest"
     problems in NP is known as NP-complete problems. This result was
     independently discovered by Leonid Levin and published in the USSR at
     about the same time.

     In that sense all NP-complete problems are equivalent under polynomial
     time transformation.

     A year later, Richard Karp showed that some very interesting problems
     that had eluded polynomial time solutions could be shown to be
     NP-complete, and in this sense, while hard, they were not beyond hope.
     This list grew quite rapidly as others contributed, and it now
     includes many naturally occuring problems which can not yet be 
     solved in polynomial time.

     References

     S. Cook. ``The complexity of theorem-provin procedures'', Proceedings
     of the 3rd Annual Symposium on the Theory of Computing, ACM, New York,
     151-158.
     J. Edmonds. Paths, trees, and flowers.Canadian Journal of Mathematics,
     17, pp.449-467.
     M. R. Garey, D. S. Johnson. Computers and Intractability, W.H.Freeman
     &Co, 1979.
     L. Levin. Universal Search Problems, Probl.Pered.Inf. 9(3), 1973. A
     translation appears in B. A. Trakhtenbrot. A survey of Russian
     approaches to Perebor (brute-force search) algorithms, Annals of the
     History of Computing, 6(4):384-400, 1984

     (a) What are P and NP?

     First we need to define formally what we mean by a problem. Typically
     a problem consists of a question and an answer. Moreover we group
     problems by general similarities.

     Again using the multiplication example, we define a multiplication
     problem as a pair of numbers, and the answer is their product. An
     instance of the multiplication problem is a specific pair of numbers
     to be multiplied.

     Problem. Multiplication
     Input. A pair of numbers x and y
     Output. The product x times y

     A clever observation is that we can convert a multiplication problem
     into a yes/no answer by joining together the original question and the
     answer and asking if they form a correct pair. In the case of
     multiplication, we can convert a question like
     4 x 9 = ??

     into a yes/no statement such as

     "is it true that 4 x 9 = 36?" (yes), or
     "is it true that 5 x 7 = 48?" (no).

     In general we can apply this technique to most (if not all) problems,
     simplifying formal treatment of problems.

          Definition A decision problem is a language L of strings
          over an alphabet. A particular instance of the problem is a
          question of the form "is x in L?" where x is a string. The
          answer is yes or no.

     The rest of this section was written by Daniel Jimenez

     P is the class of decision problems for which we can find a solution
     in polynomial time.

          Definition A polynomial time function is just a function
          that can be computed in a time polynomial in the size of its
          parameters.

          Definition P is the class of decision problems (languages) L
          such that there is a polynomial time function f(x) where x
          is a string and f(x)=True (ie. yes) if and only if x is in
          L.

     NP is the class of decision problems for which we can check solutions
     in polynomial time.

          Definition NP is the class of decision problems (languages)
          L such that there is a polynomial time function f(x,c) where
          x is a string, c is another string whose size is polynomial
          in the size of x, and f(x,c)=True if and only if x is in L.

     c in the definition is called a "certificate", the extra information
     needed to show that x is indeed in the language. NP stands for
     "nondeterministic polynomial time", from an alternate, but equivalent,
     definition involving nondeterministic Turing machines that are allowed
     to guess a certificate and then check it in polynomial time. (Note: A
     common error when speaking of P and NP is to misremember that NP
     stands for "non-polynomial"; avoid this trap, unless you want to prove
     it :-)

     An example of a decision problem in NP is:

     Decision Problem. Composite Number
     Instance. Binary encoding of a positive integer n.
     Language. All instances for which n is composite, i.e., not a prime
     number.

     We can look at this as a language L by simply coding n in log n bits
     as a binary number, so every binary composite number is in L, and
     nothing else. We can show this problem is in NP by providing a
     polynomial time f(x,c) (also known as a "polynomial time proof system"
     for L). In this case, c can be the binary encoding of a non-trivial
     factor of n. Since c can be no bigger than n, the size of c is
     polynomial (at most linear) in the size of n. The function f simply
     checks to see whether c divides n evenly; if it does, then n is proved to
     be composite and f returns True. Since division can be done in time
     polynomial in the size of the operands, Composite Number is in NP.

     (b) What is NP-hard?

     An NP-hard problem is at least as hard as or harder than any problem
     in NP. Given a method for solving an NP-hard problem, we can solve any
     problem in NP with only polynomially more work.

     Here's some more terminology. A language L' is polynomial time
     reducible to a language L if there exists a polynomial time function
     f(x) from strings to strings such that x is in L' if f(x) is in L.
     This means that if we can test strings for membership in L in time t,
     we can use f to test strings for membership in L' in a time polynomial
     in t. (hint) An example of this would be the relationship between
     Composite Number and Boolean Circuit Satisfiability.

     Decision Problem. Boolean Circuit Satisfiability
     Instance. A Boolean circuit with n inputs and one output. (Note: in
     this and the following descriptions of decision problems, it is
     assumed that the actual instance is a reasonable string encoding of
     the given instance, so we can still talk about languages of strings.)
     Language. All instances for which there is an assignment to the inputs
     that causes the output to become True.

     Composite Number is polynomial time reducible to Boolean Circuit
     Satisfiability by the following reduction: To decide whether an
     instance x is in Composite Number, construct a circuit that multiplies
     two integers given in binary on its inputs and compares the result to
     x, giving True as the output if and only if the result of the
     multiplication is x and neither of the input integers is one. The
     multiplier can be constructed and checked in polynomial time and
     space, and the comparison can be done in linear time and space.

     Polynomial time reducibility formalizes the notion of one problem
     being harder than another. If L can be used to solve instances of L',
     then L is at least as hard as or harder than L'.

     Definition A decision problem L is NP-hard if, for every language L'
     in NP, L' is polynomially reducible to L.

     So a solution to an NP-hard problem running in time t can be used to
     solve any problem in NP in a time polynomial in t (possibly different
     polynomials for different problems). NP-hard problems are at least as
     hard as or harder than any problem in NP. Boolean Circuit
     Satisfiability is an example of an NP-hard problem. A related problem,
     Boolean Formula Satisfiability (commonly called SAT), is also NP-hard;
     see Garey and Johnson for a proof of Cook's Theorem, which was the
     first proof to show that a problem (satisfiability) is NP-hard.

     An example of an NP-hard problem that isn't known to be in NP is
     Maximum Satisfiability:

     Decision Problem. Maximum Satisfiability (MAXSAT)
     Instance. A Boolean formula F and an integer k.
     Language. All instances for which F has at least k satisfying
     assignments.

     This problem is harder than SAT because of this reduction: Suppose we
     want to decide whether a formula F is in SAT. We can simply choose k
     to be one and see if (F, k) is in MAXSAT. If so, then there is at
     least one satisfying assignment and the formula is in SAT.

     (c) What is NP-complete?

     Definition A decision problem L is NP-complete if it is both NP-hard
     and in NP.

     So NP-complete problems are the hardest problems in NP. Since Cook's
     Theorem was proved in 196?, thousands of problems have been proved to
     be NP-complete. Probably the most famous example is the Travelling
     Salesman Problem:

     Decision Problem. Travelling Salesman Problem (TSP)
     Instance. A set S of cities, a function f:S x S -> N giving the
     distances between the cities, and an integer k.
     Language. The travelling salesman needs from a starting city, go
     through each city exactly once, and return to the start. The language
     is all instances for which there exists such a tour through the cities
     of S of length less than or equal to k.

     (b) NP complete list

     Pierluigi Crescenzi and Viggo Kann mantain a good list of NP
     optimization problems

     (d) Other complete problems (PSPACE, P).

  8. Complexity Theory

     (a) Lower Bounds

     (b) YACC (Yet Another Complexity Class)

  9. Logic in Computer Science

 10. Algorithm Libraries

     Stony Brook Algorithms Repository

     Library of Efficient Datatypes and Algorithms (LEDA)

 11. Open Problems.

     The are several important open problems within theoretical computer
     science. Among them

     P =? NP

     AC != P

     Find RAM problem with time complexity T(n) = \omega(n log n). T(n) =
     O(n^k).

     Show that sorting is n log n on a RAM with constant word size.

     Find exact time complexity of prime decomposition.

 12. Electronic Resources
        o ACM Computing Research Repository "http://www.acm.org/repository"
        o SIGACT Home Page "http://sigact.acm.org/sigact/"
        o Fundamentals of Computing "http://www.cs.bu.edu/fac/lnd/toc/" By
          Leonid Levin.
        o Analysis of Algorithms Home Page
          "http://pauillac.inria.fr/algo/AofA/index.html"
        o Computer Science Bibliography Collection
          "http://liinwww.ira.uka.de/bibliography/index.html"
        o Average-Case Complexity Forum "http://www.uncg.edu/mat/acc-forum"
        o The Steiner Tree Web Page "http://ganley.org/steiner/"
        o Search Trees with Relaxed Balance Home Page
          "http://www.imada.ou.dk/~kslarsen/RelBal/"
        o The Alonzo Church Archive "http://www.alonzo.org"
        o Theoretical Computer Science On The Web
          "http://robotics.stanford.edu/~suresh/theory/"
        o Electronic Colloqium on Computational Complexity
          http://www.eccc.uni-trier.de/eccc/index.html
        o by Eitan Gurari, Ohio State University Computer Science Press,
          1989, ISBN 0-7167-8182-4
          "http://www.cis.ohio-state.edu/~gurari/theory-bk/theory-bk.html"
        o Online search systems for CS publications:
             + Bibnet "http://www.netlib.no/netlib/bibnet/faq.html"
             + Technical Report Research Service
               "http://www.hensa.ac.uk/search/techreps/"
             + Unified Computer Science Index
               "http://www.cs.indiana.edu:800/cstr/search" Technical
               Reports Library
             + The Hypertext Bibliography Project
               "http://theory.lcs.mit.edu/~dmjones/hbp/"
             + On-line CS Techreports
               "http://www.cs.cmu.edu/afs/cs.cmu.edu/user/jblythe/Mosaic/cs-reports.html"
             + The New Zealand Digital Library
               "http://www.cs.waikato.ac.nz/~nzdl/"
             + Computer Science Technical Reports
               "http://www.rdt.monash.edu.au/tr/siteslist.html" Archive
               Sites
             + Networked Computer Science "http://www.ncstrl.org/"
               Technical Reports Library
             + Computer Science Bibliography Glimpse
               "http://glimpse.cs.arizona.edu:1994/bib/" Server Networked
               Computer Science "http://www.ncstrl.org/"
             + "http://www.ics.uci.edu/~eppstein/gina/DeyEdelsbrunnerGuha.ps.Z"
               Survey in combinatorial topology by Dey, Edelsbrunner, and
               Guha. (includes descriptions of applications).

 13. Bibliography

     Among the truly few FAQs in this newsgroup are recommendations for a
     Data Structures book and a Complexity Theory book. Here are some of
     titles. In brackets I've added the number of e-mail recommendations
     that I get plus any comments.

     Data Structures and Analysis of Algorithms

     Textbooks

     Baase, Sara. Computer algorithms : introduction to design and analysis
     of algorithms. 2nd ed. Addison-Wesley Pub. Co., c1988.

     Flajolet, Philippe and Sedgewick, Robert. An introduction to the
     analysis of algorithms. Addison-Wesley, c1996.

     Lewis, H and Denenberg, L. Data Structures and their Algorithms.
     Harper-Collins, 1991.

     Cormen, Thomas; Leiserson, Charles; Rivest, Ronald. Introduction to
     algorithms. MIT Press, 1989.

     Goodman and Hedetniemi. Introduction to the Design and Analysis of
     Algorithms. McGraw-Hill.

     Hopcroft and Ullman. Introduction to Authomata Theory, Languages and
     Computation. Addison-Wesley.

     All this books has excelent topics on computation and computational
     complexity.

     Complexity Theory

     Papadimitriou, Christos H. Computational complexity. Addison-Wesley,
     c1994.

     D.Bovet and P. Crescenzi. Introduction to the Theory of Complexity.
     Prentice Hall.

     C.-K. Yap: "Theory of Complexity Classes". Via FTP. General Interest

     Hofstadter, D. Escher, Godel and Bach: An Eternal Golden Braid,
     Penguin Books.

     Lewis and Papadimitrou, C. The Efficiency of Algoritms. Scientific
     American 238 1 (1978).

     Homer, S. and Selman A. Complexity Theory Algorithms

     The Algorithm Design Manual. Steve Skiena. Springer-Verlag, 1997.

     References

     Knuth, D. The Art of Computer Programming. Addison Wesley.

---
Dr. Alex Lopez-Ortiz                        Faculty of Computer Science
Assistant Professor                         University of New Brunswick
e-mail: alexl-o-theory-faq@unb.ca            Fredericton, New Brunswick
http://www.cs.unb.ca/~alopez-o                          Canada, E3B 5A3
Phone: (506)-447-3336                               Fax: (506)-453-3566

-- 
Alex Lopez-Ortiz                                         alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o                       Assistant Professor	
Faculty of Computer Science                  University of New Brunswick

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