# Comp.Theory FAQ

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

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

• The Turing Machine
• Other equivalent models
Lambda Calculus, Post Machines.
• 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

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

3. Theoretical Models of More Practical Importance

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

4. Kolmogorov Complexity Motivation

Consider the following sequences of coin tosses:

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

• 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:

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

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

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

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.

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Alex Lopez-Ortiz                            Faculty of Computer Science
Assistant Professor                         University of New Brunswick
alexl-o-theory-faq@interlab.cs.unb.ca        Fredericton, New Brunswick