# Quasilinear time complexity theory

## 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 quasilineartime hierarchy. Whereas all previously known versions of the Valiant-Vazirani reduction from NP to parity run in quadratic time, we give a new construction using error-correcting codes that runs in quasilinear time. We show, however, that the important equivalence between search problems and decision problems in polynomial time is unlikely to carry over: if search reduces to decision for *SAT* in quasi-linear time, then all of NP is contained in quasi-polynomial time. Other connections to work by Stearns and Hunt [SH86, SH90, HS90] on “power indices” of NP languages are made.

## Keywords

Polynomial Time Success Probability Turing Machine Power Index Quadratic Time## Preview

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