## On Quasilinear Time Complexity Theory (1994)

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Citations: | 3 - 0 self |

### BibTeX

@MISC{Naik94onquasilinear,

author = {Ashish V. Naik and Kenneth W. Regan and D. Sivakumar},

title = {On Quasilinear Time Complexity Theory},

year = {1994}

}

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### Abstract

This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilinear-time hierarchy.

### Citations

11403 | Computers and Intractability: A Guide to the Theory of NP-Completeness - Garey, Johnson - 1979 |

4091 |
Introduction to Automata Theory, Languages, and Computation
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Citation Context ...(n log n). Let QBF stand for [ k B k . While QBF is in alternating qlin space, it is not known to be in deterministic qlin space. Moreover, the standard reduction from a language A 2 PSPACE to QBF in =-=[HU79]-=- has a quadratic blowup in size (if A is in linear space). These apparent differences from PSPACE are connected to the issue of whether 4 Savitch's simulation of nondeterministic space s(n) =\Omega\Ga... |

2541 |
The Design and Analysis of Computer Algorithms
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Citation Context ...e running time. To multiply two polynomials of degree k \Gamma 1 over GF(2) and reduce them modulo ff in the field GF(2 k ) takes time t 1 = O(k log k loglog k) on standard Turing machine models (see =-=[AHU74]-=- and [Rab80]). The time to compute a i in GF(2 k ) where isn is t 2 = O(log n \Delta 2k log k loglog k) via repeated squaring. Hence the time to evaluate the monomial is at most O(mt 2 + mt 1 ) = O(m ... |

2064 | The Theory of Error-Correcting Codes - MacWilliams, Sloane - 1977 |

1775 | An Introduction to Kolmogorov Complexity and Its Applications - Li, Vitanyi - 1997 |

708 | Universal classes of hash functions - Carter, Wegman - 1977 |

256 | Small-bias probability spaces: Efficient constructions and applications
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Citation Context ...ter by an order of magnitude in (randomness or number of nondeterministic moves) and running time than all of the previous ones. The idea of using error-correcting codes is mentioned by Naor and Naor =-=[NN93]-=- and ascribed to Bruck, referring the reader to [ABN + 92] for details. However, the construction in [ABN + 92] uses a code of Justesen (see [Jus72, MS77] whose implementation in our setting seems to ... |

255 | Checking computations in polylogarithmic time - Babai, Fortnow, et al. - 1990 |

218 | A taxonomy of problems with fast parallel algorithms - CooK - 1985 |

215 |
NP is as easy as detecting unique solutions
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Citation Context ...esults for the polynomial hierarchy and PSPACE hold also for the quasilinear hierarchy (QLH) and QLSPACE. Section 3 shows that the randomized reduction from NP to parity given by Valiant and Vazirani =-=[VV86]-=- and used by Toda [Tod91], previously proved by constructions which run in quadratic time (see [VV86, Tod89, CRS93, Gup93]), can be made to run in time qlin . Our construction also markedly improves t... |

190 |
PP is as hard as the polynomial-time hierarchy
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Citation Context ...l hierarchy and PSPACE hold also for the quasilinear hierarchy (QLH) and QLSPACE. Section 3 shows that the randomized reduction from NP to parity given by Valiant and Vazirani [VV86] and used by Toda =-=[Tod91]-=-, previously proved by constructions which run in quadratic time (see [VV86, Tod89, CRS93, Gup93]), can be made to run in time qlin . Our construction also markedly improves the number of random bits ... |

190 |
The complexity of optimization problems
- Krentel
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(Show Context)
Citation Context ...ness very tightly. Our hypothesis is the last on the following list: (a) SAT has power index 1 [SH86]. (b) SAT is P-superterse [Bei87a]. (c) The search function for SAT does not belong to PF NP[o(n)] =-=[Kre88]-=-. (d) NP does not have p-measure zero in exponential time [Lut93] (e) The search function for SAT requires\Omega\Gamma n 2 ) query bits to compute in polynomial time, with any oracle set (or at least ... |

188 |
Concatenated Codes
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Citation Context ...hat the density of R 0 is at least \Delta=k, because two distinct elements a 1 ; a 2 2 GF(2 k ) might differ in only one out of k places as binary strings. The key idea, called concatenation of codes =-=[For66]-=-, is to apply a second level of coding to these elements. In this case we take the so-called inner code to be the Hadamard code H k . Then whenever a 1 6= a 2 in GF(2 k ), H k (a 1 ) and H k (a 2 ) di... |

159 | On isomorphism and density of NP and other complete sets - Berman, Hartmanis - 1977 |

115 | Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs - Alon, Bruck, et al. - 1992 |

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90 | The quantitative structure of exponential time
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Citation Context ...st: (a) SAT has power index 1 [SH86]. (b) SAT is P-superterse [Bei87a]. (c) The search function for SAT does not belong to PF NP[o(n)] [Kre88]. (d) NP does not have p-measure zero in exponential time =-=[Lut93]-=- (e) The search function for SAT requires\Omega\Gamma n 2 ) query bits to compute in polynomial time, with any oracle set (or at least any oracle set in NQL). 19 It would be interesting to seek closer... |

88 | A taxonomy of complexity classes of functions - Selman - 1994 |

85 | Complete sets and the polynomial-time hierarchy - Wrathall - 1977 |

85 |
P-selective sets, tally languages, and the behavior of polynomial time reducibilities
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Citation Context ...uces to decision in quasilinear time for SAT A . This also gives a sense in which the quasi-polynomial upper bound for NP in Theorem 4.2 appears to be optimal. First we observe that a lemma of Selman =-=[Sel79]-=- carries over for quasilinear time reductions. Lemma 5.5 If L 1 and L 2 are such that L 1 j ql m L 2 and search reduces to decision in quasilinear time for L 1 , then search reduces to decision in qua... |

84 | Probabilistic algorithms in finite fields
- Rabin
- 1980
(Show Context)
Citation Context ...me. To multiply two polynomials of degree k \Gamma 1 over GF(2) and reduce them modulo ff in the field GF(2 k ) takes time t 1 = O(k log k loglog k) on standard Turing machine models (see [AHU74] and =-=[Rab80]-=-). The time to compute a i in GF(2 k ) where isn is t 2 = O(log n \Delta 2k log k loglog k) via repeated squaring. Hence the time to evaluate the monomial is at most O(mt 2 + mt 1 ) = O(m log(n)k log ... |

82 |
Reckhow, Time bounded random access machines
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- 1973
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Citation Context ...independence on particular machine models that makes polynomial time such a robust concept. Gurevich and Shelah [GS89] showed that a wide variety of models related to the RAM under log-cost criterion =-=[CR73]-=- accept the same class of languages in quasilinear time---we call this class DNLT. They also showed that nondeterministic qlin 1 time for these machines, namely NNLT, equals NQL. However, currently it... |

81 | Emde Boas, Machine models and simulations - van - 1990 |

75 |
Two remarks on the power of counting
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Citation Context ...) ]. The proof of (a) uses Schnorr's construction and Lemma 2.1, and in fact gives NQL QBF = DQL[QBF ]. Statement (b) holds for the standard oracle B separating NP B from P B in [HU79]. The result of =-=[PZ83]-=- that \Phi P \Phi P = \Phi P also carries over because of the quasilinear bound on the total length of all queries in an oracle computation: \Phi QL \Phi QL = \Phi QL. However, it is unclear whether t... |

73 |
Nondeterminism within P
- Buss, Goldsmith
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(Show Context)
Citation Context ...problems also belong to NQL and are complete for NQL undersql m , so that the question NQL ? = DQL takes on much the same shape as NP ? = P. Related classes within P are studied by Buss and Goldsmith =-=[BG93]-=-. One theoretical difficulty with the concept of quasilinear time is that it appears not to share the degree of independence on particular machine models that makes polynomial time such a robust conce... |

71 | A class of constructive asymptotically good algebraic codes - Justesen - 1972 |

71 | Some connections between bounded query classes and non-uniform complexity - Amir, Beigel, et al. - 1990 |

70 | New algorithms for finding irreducible polynomials over finite fields
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Citation Context ...j) := u \Delta v /*Hadamard code applied here*/ 12. Accept iff R(x; y)sG(y; j) = 1. Steps 1--5 take linear time. It is now possible to do Step 6 deterministically in time that is polynomial in k (see =-=[Sho88]-=-, and since k is approximately log q + log n+ log(1=ffl), which is O(log n) when ffl is fixed, the time for step 6 is negligible. Step 7 takes time about nk= log n, which for fixed ffl is asymptotical... |

67 | Bounded queries to SAT and the Boolean hierarchy - Beigel - 1991 |

66 | Efficient checking of polynomials and proofs and the hardness of approximation problems - Sudan - 1992 |

61 | Relativized circuit complexity - Wilson - 1985 |

57 | Two-tape simulation of multitape Turing machines - Hennie, Stearns - 1966 |

53 | Approximable sets - Beigel, Kummer, et al. - 1995 |

49 | On Truth-Table Reducibility to SAT - Buss, Hay - 1991 |

48 | The complexity and distribution of hard problems
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Citation Context ...lylog n query bits, then NP ` DTIME[2 polylogn ], and n 1+ffl query bits would place MaxClique into DTIME[2 n ffl ]. The closest impact of (e) may be in relation to (d). By results of Juedes and Lutz =-=[JL93]-=-, (d) implies that there exists ffl ? 0 such that SAT does not have power index ffl, hence that search does not reduce to decision for SAT in time O(n 1+ffl ). We believe there should be deeper connec... |

43 | Reductions on NP and p-selective sets - Selman - 1982 |

42 | The polynomial time hierarchy - Stockmeyer - 1977 |

41 | Query-limited reducibilities
- Beigel
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(Show Context)
Citation Context ..., and SAT in particular, not only lie outside P, but also pack their hardness very tightly. Our hypothesis is the last on the following list: (a) SAT has power index 1 [SH86]. (b) SAT is P-superterse =-=[Bei87a]-=-. (c) The search function for SAT does not belong to PF NP[o(n)] [Kre88]. (d) NP does not have p-measure zero in exponential time [Lut93] (e) The search function for SAT requires\Omega\Gamma n 2 ) que... |

39 | P-selective sets and reducing search to decision vs. self-reducibility - Hemaspaandra, Naik, et al. - 1996 |

36 | Satisfiability is quasilinear complete in NQL - Schnorr - 1978 |

29 | Polynomial terse sets - Amir, Gasarch - 1988 |

29 | Terse, superterse, and verbose sets - Beigel, Gasarch, et al. - 1993 |

28 | A structural theorem that depends quantitatively on the complexity of SAT - Beigel - 1987 |

27 |
Some observations on the probabilistic algorithms and NP - hard problems
- Ko
- 1982
(Show Context)
Citation Context ...i P also carries over because of the quasilinear bound on the total length of all queries in an oracle computation: \Phi QL \Phi QL = \Phi QL. However, it is unclear whether the theorem BPP BPP = BPP =-=[Ko82]-=- carries over, because the amplification of success probability to 1\Gamma2 \Gamma polylog obtainable for BQL seems insufficient. For similar reasons we do not know whether Toda's lemma \Phi P[BP[C]] ... |

27 |
On helping by robust oracle machines
- Ko
- 1987
(Show Context)
Citation Context ...ding on the choice of witness predicate R for A. The desire to find a property of decision problems alone that facilitates search led to several notions of helping proposed by Schoning [Sch85] and Ko =-=[Ko87]-=-. We extend their definitions from polynomial time to arbitrary time bounds t(n) under our oracle convention. An oracle TM M is robust if for every oracle B, M with oracle B halts for all inputs, and ... |

25 | Refining nondeterminism in relativized polynomial-time bounded computations - Kintala, Fisher - 1980 |

24 |
Randomnessoptimal unique element isolation with applications to perfect matching and related problems
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- 1995
(Show Context)
Citation Context ... using small families H = f H k g of 5 universal 2 hash functions [CW79, BCGL89] h k : f 0; 1 g q ! f 0; 1 g k (1sksq + 1) cuts the number r(n) of random bits used to 2q(n). A related construction of =-=[CRS93]-=- achieves the same effect, still with quadratic runtime when q(n) = n. Gupta [Gup93] gives a randomized reduction to parity which achieves constant success probability 3/16 with only (n) = q(n) nondet... |

22 | Classes of bounded nondeterminism - Díaz, Torán - 1990 |

21 |
The network complexity and the Turing machine complexity of finite functions
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- 1975
(Show Context)
Citation Context ...W. Regan SUNY Buffalo regan@cs.buffalo.edu D. Sivakumar SUNY Buffalo sivak-d@cs.buffalo.edu August 20, 1993 Abstract This paper furthers the study of quasi-linear time complexity initiated by Schnorr =-=[Sch76]-=- and Gurevich and Shelah [GS89]. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilinear-time hierarchy. Whereas all previously known versions of the Val... |

21 | Rudimentary predicates and relative computation - WRATHALL - 1978 |