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Register a new userSorting Networks: to the End and Back Again
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Added by
Peter Schneider-Kamp
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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This paper studies new properties of the front and back ends of a sorting network, and illustrates the utility of these in the search for new bounds on optimal sorting networks. Search focuses first on the outsides of the network and then on the inner part. All previous works focus only on properties of the front end of networks and on how to apply these to break symmetries in the search. The new, out-side-in, properties help shed understanding on how sorting networks sort, and facilitate the computation of new bounds on optimal sorting networks. We present new parallel sorting networks for 17 to 20 inputs. For 17, 19, and 20 inputs these networks are faster than the previously known best networks. For 17 inputs, the new sorting network is shown optimal in the sense that no sorting network using less layers exists.
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This paper shows an application of the theory of sorting networks to facilitate the synthesis of optimized general purpose sorting libraries. Standard sorting libraries are often based on combinations of the classic Quicksort algorithm with insertion sort applied as the base case for small fixed numbers of inputs. Unrolling the code for the base case by ignoring loop conditions eliminates branching and results in code which is equivalent to a sorting network. This enables the application of further program transformations based on sorting network optimizations, and eventually the synthesis of code from sorting networks. We show that if considering the number of comparisons and swaps then theory predicts no real advantage of this approach. However, significant speed-ups are obtained when taking advantage of instruction level parallelism and non-branching conditional assignment instructions, both of which are common in modern CPU architectures. We provide empirical evidence that using code synthesized from efficient sorting networks as the base case for Quicksort libraries results in significant real-world speed-ups.
Previous work identifying depth-optimal $n$-channel sorting networks for $9leq n leq 16$ 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 $n=40$.
Sorting a Permutation by Transpositions (SPbT) is an important problem in Bioinformtics. In this paper, we improve the running time of the best known approximation algorithm for SPbT. We use the permutation tree data structure of Feng and Zhu and improve the running time of the 1.375 Approximation Algorithm for SPbT of Elias and Hartman to $O(nlog n)$. The previous running time of EH algorithm was $O(n^2)$.
We study the commutative algebras $Z_{JK}$ appearing in Brown and Goodearls extension of the $mathcal{H}$-stratification framework, and show that if $A$ is the single parameter quantized coordinate ring of $M_{m,n}$, $GL_n$ or $SL_n$, then the algebras $Z_{JK}$ can always be constructed in terms of centres of localizations. The main purpose of the $Z_{JK}$ is to study the structure of the topological space $spec(A)$, which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real $mtimes n$ matrices in terms of its restricted permutation.
We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (l-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive l-monoids). This contrasts with the fact that, as proved by Repnitskii, there exist inverse-free equations that are valid in all Abelian l-groups but not in all commutative distributive l-monoids, and, as we prove here, there exist inverse-free equations that hold in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM.
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