A complete set of filters @math for the optimal-depth @math -input sorting network problem is such that if there exists an @math -input sorting network of depth @math then there exists one of the form @math for some @math . Previous work on the topic presents a method for finding complete set of filters @math and @math that consists only of networks of depths one and two respectively, whose outputs are minimal and representative up to permutation and reflection. We present a novel practical approach for finding a complete set of filters @math containing only networks of depth three whose outputs are minimal and representative up to permutation and reflection. In previous work, we have developed a highly efficient algorithm for finding extremal sets ( i.e. outputs of comparator networks; itemsets; ) up to permutation. In this paper we present a modification to this algorithm that identifies the representative itemsets up to permutation and reflection. Hence, the here presented practical approach is the successful combination of known theory and practice that we apply to the domain of sorting networks. For all @math , we empirically compute the complete set of filters @math of the representative minimal up to permutation and reflection @math -input networks of depth three.
Parberry @cite_1 presented a computer assisted proof for the minimal depth of a nine and ten-input sorting networks. He significantly reduced network level candidates for the first two levels, in comparison to the naive approach, by exploiting symmetries of the networks (referred to as first and second normal form @cite_1 ).
@cite_1
1726079515 2067081749 2155720305 1990538552
We prove depth optimality of sorting networks from "The Art of Computer Programming".Sorting networks posses symmetry that can be used to generate a few representatives.These representatives can be efficiently encoded using regular expressions.We construct SAT formulas whose unsatisfiability is sufficient to show optimality.Resulting algorithm is orders of magnitude faster than prior work on small instances. We solve a 40-year-old open problem on depth optimality of sorting networks. In 1973, Donald E. Knuth detailed sorting networks of the smallest depth known for n ź 16 inputs, quoting optimality for n ź 8 (Volume 3 of "The Art of Computer Programming"). In 1989, Parberry proved optimality of networks with 9 ź n ź 10 inputs. We present a general technique for obtaining such results, proving optimality of the remaining open cases of 11 ź n ź 16 inputs. Exploiting symmetry, we construct a small set R n of two-layer networks such that: if there is a depth-k sorting network on n inputs, then there is one whose first layers are in R n . For each network in R n , we construct a propositional formula whose satisfiability is necessary for the existence of a depth-k sorting network. Using an off-the-shelf SAT solver we prove optimality of the sorting networks listed by Knuth. For n ź 10 inputs, our algorithm is orders of magnitude faster than prior ones. It is demonstrated that there is no nine-input sorting network of depth six. The proof was obtained by executing on a supercomputer a branch-and-bound algorithm which constructs and tests a critical subset of all possible candidates. Such proofs can be classified as experimental science, rather than mathematics. In keeping with the paradigms of experimental science, a high-level description of the experiment and analysis of the result are given. Previous work identifying depth-optimal n-channel sorting networks for 9 a#x2264; n a#x2264; 16 is based on exploiting symmetries of the first two layers. However, the naive generate-and-test approach typically applied does not scale. This paper revisits the problem of generating two-layer prefixes modulo symmetries. An improved notion of symmetry is provided and a novel technique based on regular languages and graph isomorphism is shown to generate the set of non-symmetric representations. An empirical evaluation demonstrates that the new method outperforms the generate-and-test approach by orders of magnitude and easily scales until n = 40. We consider the problem of approximating a Boolean functionf∶ 0,1 n → 0,1 by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP A is properly contained in PSPACE A . We are also able to prove the old AC0 exponential-size lower bounds in a new way. This allows us to prove the new result that an AC0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.