## Fine Separation of Average Time Complexity Classes (1997)

Venue: | SIAM Journal on Computing |

Citations: | 12 - 2 self |

### BibTeX

@INPROCEEDINGS{Cai97fineseparation,

author = {Jin-yi Cai and Alan L. Selman},

title = {Fine Separation of Average Time Complexity Classes},

booktitle = {SIAM Journal on Computing},

year = {1997},

pages = {331--343},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

We extend Levin's definition of average polynomial time to arbitrary time-bounds in accordance with the following general principles: (1) It essentially agrees with Levin's notion when applied to polynomial time-bounds. (2) If a language L belongs to DTIME(T(n)), for some time-bound T(n), then every distributional problem (L;µ) is T on the µ-average. (3) If L does not belong to DTIME(T(n)) almost everywhere, then no distributional problem (L;µ) is T on the µ-average. We present hierarchy theorems for average-case complexity, for arbitrary timebounds, that are as tight as the well-known Hartmanis-Stearns [HS65] hierarchy theorem for deterministic complexity. As a consequence, for every time-bound T(n), there are distributional problems (L;µ) that can be solved using only a slight increase in time but that cannot be solved on the µ-average in time T(n). Keywords: computational complexity, average time complexity classes, hierarchy, Average-P, logarithmico-exponential ACM Computing R...

### Citations

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(Show Context)
Citation Context ...age case complexity. The average complexity of a problem is, in many cases, a more significant measure than its worst case complexity. This has motivated a rich area in algorithms research, but Levin =-=[Lev86]-=- was the first to advocate the general study of average case complexity. An average case complexity class consists of pairs, called distributional problems. Each pair consists of a decision problem an... |

200 |
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(Show Context)
Citation Context ...hen no distributional problem (L;��) is T on the ��-average. We present hierarchy theorems for average-case complexity, for arbitrary timebounds, that are as tight as the well-known Hartmanis-=-=Stearns [HS65] hie-=-rarchy theorem for deterministic complexity. As a consequence, for every time-bound T(n), there are distributional problems (L;��) that can be solved using only a slight increase in time but that ... |

110 | On the theory of average case complexity
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(Show Context)
Citation Context ...-P are the distributional analogues of NP and P, respectively. Many beautiful results have been obtained. Levin, for example, has proved the existence of complete problems in DistNP. Ben-David et al. =-=[BDCGL92]-=- were the first to suggest a general formulation of average case complexity for time-bounds other than polynomials. We will prove a fine hierarchy theorem using their definition for time-bounds that a... |

82 |
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(Show Context)
Citation Context ... of every random-access machine M 0 that decides L exceeds T 0 (n) almost everywhere. (This result is an almost-everywhere version of a hierarchy theorem of Cook and Reckow for random-access machines =-=[CR73].) Let �-=-��� be any polynomial time computable distribution that satisfies Condition W. It is immediate that the running time of M is O(T(n)) on the ��-average. However, it is also immediate, as in the p... |

72 | Average Case Completeness
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(Show Context)
Citation Context ...n's theory of average polynomial time will recall that a naive, intuitive formulation suffers from serious problems. This issue is discussed in detail by previous authors including, notably, Gurevich =-=[Gur91]-=- and Ben-David et al. [BDCGL92]. Similarly, the path to a correct formulation of average case complexity for arbitrary time-bounds is intricate. We will develop our new definition in Section 4. We wil... |

58 | Cook versus Karp-Levin: Separating completeness notions if NP is not small
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(Show Context)
Citation Context ...stributional problem (L;��) belongs to DistNP but does not belong to AVP. The proof follows from Theorem 4. A weaker conclusion is known to follow from a weaker hypothesis. To wit, Lutz and Mayord=-=omo [LM94]-=- proved that if NP contains a languagesL that is bi-immune to P, then E 6= NE. Ben David et al. [BDCGL92] proved that if E 6= NE, then there is a tally language L in NP \Gamma P such that the distribu... |

31 |
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(Show Context)
Citation Context ...n-zero constant c, such that lim x! t(x)=s(x) = c. Let f (`) denote the function that iterates ` applications of f . That is, f (1) (x) = f (x) and f (`+1) (x) = f ( f (`) (x)), for `s1. Hardy proved =-=[Har11]-=- that for every function t 2 L, if lim x! t(x) = , then there is some constant ` so that log (`) (x) = o(t(x)), as well as t(x) = o(exp (`) (x)) ---informally, a logarithmico-exponential function that... |

24 | Average case complexity under the universal distribution equals worst-case complexity - Li, Vitányi - 1992 |

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Citation Context ...o the customary convention that T(n)sn + 1 (S(x)sjxj + 1), for any Turing machine running time T (S, respectively). The following proposition is one of the main theorems of Geske, Huynh, and Seiferas =-=[GHS91]-=-. (See also, the paper by Geske, Huynh, and Selman [GHS87].) Proposition 1 If S(x) is fully time-constructible, then there is a language L 2 DTIME(O(S(x))) such that for every function S 0 , if S 0 (x... |

20 |
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Citation Context ...obability of x in fx j jxj = ng. That is, �� 0 n (x) = �� 0 (x)=u n , if u n ? 0, and �� 0 n (x) = 0 for x 2 fx j jxj = ng, if u n = 0. A function �� from S to [0; 1] is computable in =-=polynomial time [Ko83] if ther-=-e is a polynomial time-bounded transducer T such that for every string x and every positive integer n, j��(x) \Gamma T(x;1 n )j ! 1 2 n . We restrict our attention to distributions �� that are... |

17 |
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(Show Context)
Citation Context ...e established definition. (We will discuss this point further in Section 4.) 2.3 Hardy's class of logarithmico-exponential functions We will need the notion of a class of functions L defined by Hardy =-=[Har24]-=-, called the logarithmico-exponential functions. Every function in L is a real-valued function of one variable that is defined on all sufficiently large real numbers. The class L is defined to be the ... |

14 | Structural properties of complete problems for exponential time
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- 1997
(Show Context)
Citation Context ...s and constructions that are not possible with smaller time bounds. For this reason, much more is known about complete problems for exponential time than is currently known about NP complete problems =-=[Hom97]-=-. We can anticipate the same situation for average-case complexity. With regard to applications of average-case complexity theory, here too, average-case exponential time is worth exploring. For examp... |

10 |
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(Show Context)
Citation Context ... for any Turing machine running time T (S, respectively). The following proposition is one of the main theorems of Geske, Huynh, and Seiferas [GHS91]. (See also, the paper by Geske, Huynh, and Selman =-=[GHS87]-=-.) Proposition 1 If S(x) is fully time-constructible, then there is a language L 2 DTIME(O(S(x))) such that for every function S 0 , if S 0 (x)logS 0 (x) = o(S(x)), then every Turing machine M that ac... |

6 |
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(Show Context)
Citation Context ...elhauer [RS93] introduce a concept of average-time complexity according to "rankable" distributions that possesses a tight hierarchy. Their concept is different than Levin's notion. Belanger=-= and Wang [BW95]-=- contains a discussion. In addition, Belanger and Wang [BW95] obtain a weak hierarchy theorem for the notion of average time given by Definition 3.1. We believe that it is interesting and useful to ha... |

5 |
azar and U. Sch oning. Bi-immune sets for complexity classes
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(Show Context)
Citation Context ... ae AVE and AVE ae AVEXP. A language L is bi-immune to a complexity class C if L is infinite, no infinite subset of L belongs to C , and no infinite subset of L belongs to C . Balc azar and Sch oning =-=[BS85]-=- proved that for every time-constructible function T , L does not belong to DTIME(T(n)) almost everywhere if and only if L is bi-immune to DTIME(T(n)). Recall that DistNP is the class of distributiona... |

3 | Reductions do not preserve fast convergence rates in average time
- Belanger, Pavan, et al.
- 1999
(Show Context)
Citation Context ... appropriate reductions, provides a hardness notion. In this paper we have focused on extending the classification to arbitrary time-bounds. We refer the reader to papers by Belanger, Pavan, and Wang =-=[BPW97]-=- and Pavan and Selman [PS97] for research on issues concerning reductions and complete problems as they relate to the new definitions we have given here. 5 Random-access machines In order to illustrat... |