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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
780-850 A.D.
Abu Ja'far Mohammed Ben Musa al-Khwarizmi publishes two books,
"Al-Khwarizmi on the Hindu Art of Recknoning" and "Hisab al-jabr
w'al-muqabala". His surname, al-Khowarazmi, is the root from
which the word algorithm is derived. The word algebra is derived
from the title of the second book.
1424 A.D.
Ghiyath al-Din Jamshid Mas'ud al-Kashi computes pi to sixteen
decimal places.
EARLY ANALYSIS
1845
Gabriel Lame showed that Euclid's algorithm takes no more
division steps than 5 times the number of decimal digits of the
smaller number
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 (arithmetic) 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 devices
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 cannot yet be solved
in polynomial time.
References
S. Cook. ``The complexity of theorem-proving 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 departs from a starting city, goes
through each city exactly once, and returns 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.
There 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"
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 Techni cal 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 Server
"http://glimpse.cs.arizona.edu:1994/bib/" Networked Computer
Science "http://www.ncstrl.org/"
+ Survey in combinatorial topology by Dey, Edelsbrunner, and
Guha. (includes descriptions of applications).
"http://www.ics.uci.edu/~eppstein/gina/DeyEdelsbrunnerGuha.ps.Z"
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
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.
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. Godel, Escher 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.
---
Alex Lopez-Ortiz Faculty of Computer Science
Assistant Professor University of New Brunswick
alexl-o-theory-faq@interlab.cs.unb.ca Fredericton, New Brunswick
http://www.cs.unb.ca/~alopez-o Canada, E3B 5A3
User Contributions:
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