Average-case complexity

About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.

Experimental study on time complexity of GOSCL algorithm for sparse data tables

Abstract: In this paper we provide experimental study on time complexity of GOSCL algorithm according to the sparseness of the input data table. GOSCL is incremental algorithm for the creation of Generalized One-Sided Concept Lattices, which is related to well-known Formal Concept Analysis area, but with the possibility to work with different types of attributes and to produce one-sided concept lattice from the generalized one-sided formal context. Generally, FCA-based algorithms are exponential. However, in practice there are many inputs for which the complexity is reduced. One of the special cases is related to the high number of "zeros" (bottom elements) in data table for so-called sparse data matrices, which is characteristic for some inputs like document-term matrix in text-mining analysis. We describe experimentally the influence of sparseness of data tables on time complexity of GOSCL with different distributions of zeros generated artificially randomly or according to the standard text-mining datasets.

On the complement of one complexity class in another

Abstract: We investigate the diagonalization theorems in [7] (Theorem 23), [3] (Theorem 6) and [10] (main Theorem) and show that they can be strengthened so as to be applicable to most complexity classes, not just to those closed under polynomial-time reducibility. Thus the applications in [7], [3] and [10] (e.g. P ≠ P ⇒ NP\P is not recursively presentable) are not peculiar to P, NP, PSPACE etc.; rather, they are examples of properties common to almost all "reasonable" complexity classes.

How robust is quicksort average complexity

Abstract: The paper questions the robustness of average case time complexity of the fast and popular quicksort algorithm. Among the six standard probability distributions examined in the paper, only continuous uniform, exponential and standard normal are supporting it whereas the others are supporting the worst case complexity measure. To the question -why are we getting the worst case complexity measure each time the average case measure is discredited? -- one logical answer is average case complexity under the universal distribution equals worst case complexity. This answer, which is hard to challenge, however gives no idea as to which of the standard probability distributions come under the umbrella of universality. The morale is that average case complexity measures, in cases where they are different from those in worst case, should be deemed as robust provided only they get the support from at least the standard probability distributions, both discrete and continuous. Regretfully, this is not the case with quicksort.

Properties of NP-Complete Sets

Abstract: We study several properties of sets that are complete for NP. We prove that if L is an NP-complete set and S /spl nsupe/ L is a p-selective sparse set, then L -S is /spl les//sub m//sup p/-hard for NP. We demonstrate existence of a sparse set S /spl isin/ DTIME(2/sup 2n/) such that for every L /spl isin/ NP - P, L - S is not /spl les//sub m//sup p/-hard for NP. Moreover, we prove for every L /spl isin/ NP - P, that there exists a sparse S /spl isin/ EXP such that L - S is not /spl les//sub m//sup p/-hard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure one-way permutations to derive consequences for long-standing open questions about whether NP-complete sets are immune. For example, assuming that pseudorandom generators and secure one-way permutations exist, it follows easily that NP-complete sets are not p-immune. Assuming only that secure one-way permutations exist, we prove that no NP-complete set is DTIME(2/sup ne/)-immune. Also, using these hypotheses we show that no NP-complete set is quasipolynomial-close to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turing-complete sets for NP are not closed under union. Our hypothesis asserts existence of a UP-machine M that accepts 0* such that for some 0 < /spl epsi/ < 1, no 2/sup ne/ time-bounded machine can correctly compute infinitely many accepting computations of M, We show that if UP /spl cap/ coUP contains DTIME(2/sup ne/)-bi-immune sets, then this hypothesis is true.

Erratum for: on basing one-way functions on NP-hardness

Abstract: This is an errata for our STOC'06 paper, "On Basing One-Way Functions on NP-Hardness".There is a gap in the proof of our results regarding adaptive reductions, and we currently do not know whether Theorem 3 (as stated in Section 2) holds.