Average-case complexity

In computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource (typically time) used by the algorithm, averaged over all possible inputs. It is frequently contrasted with worst-case complexity which considers the maximal complexity of the algorithm over all possible inputs.

There are three primary motivations for studying average-case complexity.^[1] 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. Third, average-case complexity allows discriminating the most efficient algorithm in practice among algorithms of equivalent best case complexity (for instance Quicksort).

Average-case analysis requires a notion of an "average" input to an algorithm, which leads to the problem of devising a probability distribution over inputs. Alternatively, a randomized algorithm can be used. The analysis of such algorithms leads to the related notion of an expected complexity.^[2]^: 28

History and background

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.^[3] In 1973, Donald Knuth^[4] 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.

An efficient algorithm for $NP$ -complete problems is 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^[5] defining average-case complexity and completeness while giving an example of a complete problem for $distNP$ , the average-case analogue of $NP$ .

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 $n 2$ on all inputs except the string $1 n$ for which $A$ takes time $2 n$ . Then it can be easily checked that the expected running time of $A$ is polynomial but the expected running time of $B$ is exponential.^[3]

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 $A$ 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:

\Pr _{x\in _{R}D_{n}}\left[t_{A}(x)\geq t\right]\leq {\frac {p(n)}{t^{\epsilon }}}

for every $n, t > 0$ and polynomial $p$ , where $t A (x)$ denotes the running time of algorithm $A$ on input $x$ , and $ε$ is a positive constant value.^[6] Alternatively, this can be written as

E_{x\in _{R}D_{n}}\left[{\frac {t_{A}(x)^{\epsilon }}{n}}\right]\leq C

for some constants $C$ and $ε$ , where $n = | x |$ .^[7] 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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠nc/(tA(n))ε⁠$ fraction of inputs of length $n$ , for some $ε, c > 0$ .^[3]

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 $(L, D)$ .^[7] 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 $μ$ and a string $x \in {0, 1} n$ it is possible to compute the value $\mu (x)=\sum \limits _{y\in \{0,1\}^{n}:y\leq x}\Pr[y]$ 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 $P$ ≠ $P #P$ .^[7]

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.^[7]

A distributional problem $(L, D)$ is in the complexity class $distNP$ if $L$ is in $NP$ and $D$ is $P$ -computable. When $L$ is in $NP$ and $D$ is $P$ -samplable, $(L, D)$ belongs to $sampNP$ .^[7]

Together, $AvgP$ and $distNP$ define the average-case analogues of $P$ and $NP$ , respectively.^[7]

Reductions between distributional problems

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

(Correctness) $x \in L$ if and only if $f (x) \in L'$
(Domination) There are polynomials $p$ and $m$ such that, for every $n$ and $y$ , $\sum \limits _{x:f(x)=y}D_{n}(x)\leq p(n)D'_{m(n)}(y)$

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 $D'$ .^[6]

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 $(L', D')$ .^[7]

An example of a $distNP$ -complete problem is the Bounded Halting Problem, $(BH, D)$ (for any $P$ -computable D) defined as follows:

$BH=\{(M,x,1^{t}):M{\text{ is a non-deterministic Turing machine that accepts }}x{\text{ in}}\leq t{\text{ steps}}\}$ ^[7]

In his original paper, Levin showed an example of a distributional tiling problem that is average-case $NP$ -complete.^[5] A survey of known $distNP$ -complete problems is available online.^[6]

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 $EXP$ = $NEXP$ .^[8] (A flat distribution $μ$ is one for which there exists an $ε > 0$ such that for any $x$ , $μ (x) \leq 2 -| x | ε$ .) A result by Livne shows that all natural $NP$ -complete problems have $DistNP$ -complete versions.^[9] However, the goal of finding a natural distributional problem that is $DistNP$ -complete has not yet been achieved.^[10]

Applications

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 $O(n 2)$ , but an average-case running time of $O(n log(n))$ , where $n$ is the length of the input to be sorted.^[2]

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.^[11]

Thus, all secure cryptographic schemes rely on the existence of one-way functions.^[3] Although the existence of one-way functions is still an open problem, many candidate one-way functions are based on 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 not believed to be $NP$ -complete.^[7] 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

Yao's principle, from a 1978 paper by Andrew Yao, shows that for broad classes of computational problems, average-case complexity for a hard input distribution and a deterministic algorithm adapted to that distribution is the same thing as expected complexity for a fast randomized algorithm and its worst-case input.^[12]

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.^[13] Applying this theory to natural distributional problems remains an outstanding open question.^[3]

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.^[14] 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 $NP$ .^[15] In 2003, Bogdanov and Trevisan generalized this result to arbitrary non-adaptive reductions.^[16] These results show that it is unlikely that any association can be made between average-case complexity and worst-case complexity via reductions.^[3]

References

^ O. Goldreich and S. Vadhan, Special issue on worst-case versus average-case complexity, Comput. Complex. 16, 325–330, 2007.
^ ^a ^b 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.
^ ^a ^b ^c ^d ^e ^f A. Bogdanov and L. Trevisan, "Average-Case Complexity," Foundations and Trends in Theoretical Computer Science, Vol. 2, No 1 (2006) 1–106.
^ D. Knuth, The Art of Computer Programming. Vol. 3, Addison-Wesley, 1973.
^ ^a ^b L. Levin, "Average case complete problems," SIAM Journal on Computing, vol. 15, no. 1, pp. 285–286, 1986.
^ ^a ^b ^c J. Wang, "Average-case computational complexity theory," Complexity Theory Retrospective II, pp. 295–328, 1997.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press, New York, NY, 2009.
^ Y. Gurevich, "Complete and incomplete randomized NP problems", Proc. 28th Annual Symp. on Found. of Computer Science, IEEE (1987), pp. 111–117.
^ N. Livne, "All Natural NP-Complete Problems Have Average-Case Complete Versions," Computational Complexity (2010) 19:477. https://doi.org/10.1007/s00037-010-0298-9
^ O. Goldreich, "Notes on Levin's theory of average-case complexity," Technical Report TR97-058, Electronic Colloquium on Computational Complexity, 1997.
^ J. Katz and Y. Lindell, Introduction to Modern Cryptography (Chapman and Hall/Crc Cryptography and Network Security Series), Chapman and Hall/CRC, 2007.
^ Yao, Andrew (1977), "Probabilistic computations: Toward a unified measure of complexity", Proceedings of the 18th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 222–227, doi:10.1109/SFCS.1977.24
^ 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.
^ 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.
^ J. Feigenbaum and L. Fortnow, "Random-self-reducibility of complete sets," SIAM Journal on Computing, vol. 22, pp. 994–1005, 1993.
^ 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.