Average-case complexity: Difference between revisions

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'''Average-case complexity''' is a subfield of [[computational complexity theory]] that studies the complexity of [[algorithms]] onover randominputs drawn randomly from a particular [[probability distribution]]. It is frequently contrasted with [[worst-case complexity]] which considers the maximal complexity of the algorithm over all possible inputs.

There are two primary motivations for studying average-case complexity.<ref name="gol07">O. Goldreich and S. Vadhan, Special issue on worst-case versus average-case complexity, Comput. Complex. 16, 325-330, 2007.</ref> First, although some problems may be intractable in the worst-case, the inputs which elicit this behavior may rarely occur in practice, so the average-case complexity may be a more accurate measure of an algorithm's performance. Second, average-case complexity analysis provides tools and techniques to generate hard instances of problems which can be utilized in areas such as [[cryptography]] and [[derandomization]].

The study of average-case complexity has applications in the theory of [[cryptography]].

==History and Background==

[[Leonid Levin]] presented the motivation for studying average-case complexity as follows:<ref>[http://www.cs.bu.edu/fac/lnd/research/hard.htm "Intractability Concepts for Concrete Problems", [[Leonid Levin]]]</ref>

:"Many combinatorial problems (called search or NP problems) have easy methods of checking solutions for correctness. Examples: finding factors of a long integer, or proofs of math theorems or short fast programs generating a given string. Such problems can be stated as a task to invert a given, easy to compute, function (multiplication or extraction of a theorem from its proof). In 1971 I noticed that many such problems can be proven to be as hard as the Tiling problem (which, I knew for a while, was universal, i.e. at least as hard as any search problem)...

The average-case performance of algorithms has been studied since modern notions of computational efficiency were developed in the 1950s. Much of this initial work focused on problems for which worst-case polynomial time algorithms were already known.<ref name="bog06">A. Bogdanov and L. Trevisan, "Average-Case Complexity," Foundations and Trends in Theoretical Computer Science, Vol. 2, No 1 (2006) 1-106.</ref> In 1973, [[Donald Knuth]]<ref name="knu73">D. Knuth, The Art of Computer Programming. Vol. 3, Addison-Wesley, 1973.</ref> published Volume 3 of the [[Art of Computer Programming]] which extensively surveys average-case performance of algorithms for problems solvable in worst-case polynomial time, such as sorting and median-finding.

:"A common misinterpretation of these results was that all NP-complete problems are hard, no chance for good algorithms. On this basis some such problems generated much hope in cryptography: the adversary would be helpless. Karp and others noticed that this was naive. While worst instances of NP-complete problems defeat our algorithms, such instances may be extremely rare. In fact, fast on average algorithms were found for a great many NP-complete problems. If all NP problems are easy on average, the P=?NP question becomes quite academic. Even if exponentially hard instances exist, those we could ever find might all be easy. Some problems (like factoring) seem hard for typical instances, but nothing is proven at all to support this (crucial, e.g., for cryptography) belief. These issues turned out to be subtle and it was not clear how a theory could distinguish intrinsically hard on average problems. [Levin 86], [Venkatesan, Levin STOC-88], [Impagliazzo, Levin, FOCS-90] proposed such a theory with first average case intractability results. Random (under uniform distribution) instances of some problems are now known to be as hard as random instances of any NP-problem under any samplable distribution."

An efficient algorithm for [[NP-complete]] problems in generally characterized as one which runs in polynomial time for all inputs; this is equivalent to requiring efficient worst-case complexity. However, an algorithm which is inefficient on a "small" number of inputs may still be efficient for "most" inputs that occur in practice. Thus, it is desirable to study the properties of these algorithms where the average-case complexity may differ from the worst-case complexity and find methods to relate the two.

The fundamental notions of average-case complexity were developed by [[Leonid Levin]] in 1986 when he published a one-page paper<ref name="levin86">L. Levin, "Average case complete problems," SIAM Journal on Computing, vol. 15, no. 1, pp. 285-286, 1986.</ref> defining average-case complexity and completeness while giving an example of a complete problem for distNP, the average-case analogue of [[NP (complexity)|NP]].

==Definitions==

===Efficient Average-Case Complexity===

The first task is to precisely define what is meant by an algorithm which is efficient "on average". An initial attempt might define an efficient average-case algorithm as one which runs in expected polynomial time over all possible inputs. Such a definition has various shortcomings; in particular, it is not robust to changes in the computational model. For example, suppose algorithm A runs in time t_A(x) on input x and algorithm B runs in time t_A(x)² on input x; that is, B is quadratically slower than A. Intuitively, any definition of average-case efficiency should capture the idea that A is efficient-on-average if and only if B is efficient on-average. Suppose, however, that the inputs are drawn randomly from the uniform distribution of strings with length n, and that A runs in time n² on all inputs except the string 1ⁿ for which A takes time 2ⁿ. Then it can be easily checked that the expected running time of A is polynomial but the expected running time of B is exponential.<ref name="bog06" />

To create a more robust definition of average-case efficiency, it makes sense to allow an algorithm A to run longer than polynomial time on some inputs but the fraction of inputs on which on which A requires larger and larger running time becomes smaller and smaller. This intuition is captured in the following formula for average polynomial running time, which balances the polynomial trade-off between running time and fraction of inputs:

<math>

\Pr_{x \in_R D_n}[t_A(x) \geq t] \leq \frac{p(n)}{t^\epsilon}

</math>

for every n, t, &epsilon; > 0 and polynomial p, where t_A(x) denotes the running time of algorithm A on input x.<ref name="wangsurvey">J. Wang, "Average-case computational complexity theory," Complexity Theory Retrospective II, pp. 295-328, 1997.</ref> Alternatively, this can be written as

<math>

E_{x \in_R D_n}[\frac{t_{A}(x)^{\epsilon}}{n}] \leq C

</math>

for some constant C, where n = |x|.<ref name="ab09">S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press, New York, NY, 2009.</ref> In other words, an algorithm A has good average-case complexity if, after running for t_A(n) steps, A can solve all but a <math>\frac{n^c}{(t_A(n))^{\epsilon}}</math> fraction of inputs of length n, for some &epsilon;, c > 0.<ref name="bog06"/>

===Distributional Problem===

The next step is to define the "average" input to a particular problem. This is achieved by associating the inputs of each problem with a particular probability distribution. That is, an "average-case" problem consists of a language L and an associated probability distribution D which forms the pair (L, D).<ref name="ab09"/> The two most common classes of distributions which are allowed are:

#Polynomial-time computable distributions (P-computable): these are distributions for which it is possible to compute the cumulative density of any given input x. More formally, given a probability distribution &mu; and a string x &isin; {0, 1}ⁿ it is possible to compute the value <math>\mu(x) = \sum\limits_{y \in \{0, 1\}^n : y \leq x} \Pr[y]</math> in polynomial time. This implies that Pr[x] is also computable in polynomial time.

#Polynomial-time samplable distributions (P-samplable): these are distributions from which it is possible to draw random samples in polynomial time.

These two formulations, while similar, are not equivalent. If a distribution is P-computable it is also P-samplable, but the converse is not true if [[P (complexity)|P]] ≠ P^#P.<ref name="ab09"/>

===AvgP and distNP===

A distributional problem (L, D) is in the complexity class AvgP if there is an efficient average-case algorithm for L, as defined above. The class AvgP is occasionally called distP in the literature.<ref name="ab09"/>

A distributional problem (L, D) is in the complexity class distNP if L is in NP and D is P-computable (or P-samplable).<ref name="ab09"/>

Together, AvgP and DistNP define the average-case analogues of P and NP, respectively.<ref name="ab09"/>

==Reductions Between Distributional Problems==

Let (L,D) and (L',D') be two distributional problems. (L, D) average-case reduces to (L', D') (written $(L, D) &le;_AvgP (L', D')) if there is a function f that for every n, on input x can be computed in time polynomial in n and

#(Correctness) x &isin; L if and only if f(x) &isin; L'

#(Domination) There are polynomials p and m such that, for every n and y, <math>\sum\limits_{x: f(x) = y} D_n(x) \leq p(n)D'_{m(n)}(y)</math>

The domination condition enforces the notion that if problem (L, D) is hard on average, then (L', D') is also hard on average. Intuitively, a reduction should provide a way to solve an instance x of problem L by computing f(x) and feeding the output to the algorithm which solves L'. Without the domination condition, this may not be possible since the algorithm which solves L in polynomial time on average may take super-polynomial time on a small number of inputs but f may map these inputs into a much larger set of D' so that algorithm A' no longer runs in polynomial time on average. The domination condition only allows such strings to occur polynomially as often in D'.<ref name="wangsurvey"/>

===DistNP-Complete Problems===

The average-case analogue to NP-completeness is distNP-completeness. A distributional problem (L', D') is distNP-complete if (L', D') is in distNP and for every (L, D) in distNP, (L, D) is average-case reducible to (L', D').\cite{ab09}

An example of a distNP-complete problem is the Bounded Halting Problem, BH, defined as follows:

BH = {(M,x,1^t) : M is a non-deterministic Turing machine that accepts x in &le; t steps.}<ref name="ab09"/>

In his original paper, Levin showed an example of a distributional tiling problem that is average-case NP-complete.<ref name="levin86"/> A survey of known distNP-complete problems is available online.<ref name="wangsurvey"/>

One area of active research involves finding new distNP-complete problems. However, finding such problems can be complicated due to a result of Gurevich which shows that any distributional problem with a flat distribution cannot be distNP-complete unless [[EXP]] = [[NEXP]].<ref name="gur87">Y. Gurevich, "Complete and incomplete randomized NP problems", Proc. 28th Annual Symp. on Found. of Computer Science, IEEE (1987), pp. 111-117.</ref> (A flat distribution &mu; is one for which there exists an &epsilon; > 0 such that for any x, &mu;(x) &le; 2^{-|x|^&epsilon;}. A result by Livne shows that all natural NP problems have DistNP-complete versions.<ref name="livne06">N. Livne, "All Natural NPC Problems Have Average-Case Complete Versions," Computational Complexity, to appear. Preliminary version in ECCC, TR06-122, 2006.</ref> However, the goal of finding a natural distributional problem that is DistNP-complete has not yet been achieved.<ref name="gol97">O. Goldreich, "Notes on Levin's theory of average-case complexity," Technical Report TR97-058, Electronic Colloquium on Computational Complexity, 1997.</ref>

==Applicatons==

===Sorting Algorithms===

As mentioned above, much early work relating to average-case complexity focused on problems for which polynomial-time algorithms already existed, such as sorting. For example, many sorting algorithms which utilize randomness, such as [[Quicksort]], have a worst-case running time of O(n²), but an average-case running time of O(nlog(n)), where n is the length of the input to be sorted.<ref name="clrs">Cormen, Thomas H.; Leiserson, Charles E., Rivest, Ronald L., Stein, Clifford (2009) [1990]. Introduction to Algorithms (3rd ed.). MIT Press and McGraw-Hill. ISBN 0-262-03384-4.</ref>

===Cryptography===

For most problems, average-case complexity analysis is undertaken to find efficient algorithms for a problem that is considered difficult in the worst-case. In cryptographic applications, however, the opposite is true: the worst-case complexity is irrelevant; we instead want a guarantee that the average-case complexity of every algorithm which "breaks" the cryptographic scheme is inefficient.<ref name="katz07">J. Katz and Y. Lindell, Introduction to Modern Cryptography (Chapman and Hall/Crc Cryptography and Network Security Series), Chapman and Hall/CRC, 2007.</ref>

Thus, all secure cryptographic schemes rely on the existence of [[one-way functions]].<ref name="bog06"/> Although the existence of one-way functions is still an open problem, many candidate one-way functions are based on NP-hard problems such as [[integer factorization]] or computing the [[discrete log]]. Note that it is not desirable for the candidate function to be NP-complete since this would only guarantee that there is likely no efficient algorithm for solving the problem in the worst case; what we actually want is a guarantee that no efficient algorithm can solve the problem over random inputs (i.e. the average case). In fact, both the integer factorization and discrete log problems are in NP &cap; [[coNP]], and are therefore believed to be [[NP-hard]] but not NP-complete.<ref name="ab09"/> The fact that all of cryptography is predicated on the existence of average-case intractable problems in NP is one of the primary motivations for studying average-case complexity.

==Other Results==

In 1990, Impagliazzo and Levin showed that if there is an efficient average-case algorithm for a distNP-complete problem under the uniform distribution, then there is an average-case algorithm for every problem in NP under any polynomial-time samplable distribution.<ref name="imp90">R. Impagliazzo and L. Levin, "No Better Ways to Generate Hard NP Instances than Picking Uniformly at Random," in Proceedings of the 31st IEEE Sympo- sium on Foundations of Computer Science, pp. 812-821, 1990.</ref> Applying this theory to natural distributional problems remains an outstanding open question.<ref name="bog06"/>

In 1992, Ben-David et al., showed that if all languages in distNP have good-on-average decision algorithms, they also have good-on-average search algorithms. Further, they show that this conclusion holds under a weaker assumption: if every language in NP is easy on average for decision algorithms with respect to the uniform distribution, then it is also easy on average for search algorithms with respect to the uniform distribution.<ref name="bd92">S. Ben-David, B. Chor, O. Goldreich, and M. Luby, "On the theory of average case complexity," Journal of Computer and System Sciences, vol. 44, no. 2, pp. 193-219, 1992.</ref> Thus, cryptographic one-way functions can exist only if there are distNP problems over the uniform distribution that are hard on average for decision algorithms.

In 1993, Feigenbaum and Fortnow showed that it is not possible to prove, under non-adaptive random reductions, that the existence of a good-on-average algorithm for a distNP-complete problem under the uniform distribution implies the existence of worst-case efficient algorithms for all problems in NP.<ref name="ff93">J. Feigenbaum and L. Fortnow, "Random-self-reducibility of complete sets," SIAM Journal on Computing, vol. 22, pp. 994-1005, 1993.</ref> In 2003, Bogdanov and Trevisan generalized this result to arbitrary non-adaptive reductions.<ref name="bog03">A. Bogdanov and L. Trevisan, "On worst-case to average-case reductions for NP problems," in Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, pp. 308-317, 2003.</ref> These results show that it is unlikely that any association can be made between average-case complexity and worst-case complexity via reductions.<ref name="bog06"/>

==Related topics==

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[[Category:Probabilistic complexity theory]]

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