Using a CAS/DGS to Analyze Computationally the Configuration of Planar Bar Linkage Mechanisms Based on Partial Latin Squares

Ganfornina, Raúl Manuel Falcón

doi:10.1007/s11786-019-00428-1

Cited by 3 publications

(1 citation statement)

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“…Autoparatopisms and symmetry for partial Latin squares were studied in [63,3] and several constructions of partial Latin rectangles with trivial autotopism groups for various autoparatopism groups was given in [36]. Computational methods for determining autotopism groups of partial Latin rectangles were compared in [17,18,69,29]. For Latin squares of order n 17, identifying when #PLR((Θ, π)) = 0 (throughout this paper # denotes the cardinality of a set) was done for isotopisms in [64] and paratopisms in [56], with prior work in [26,32].…”

Section: Introductionmentioning

confidence: 99%

Enumerating Partial Latin Rectangles

Ganfornina

Stones

2020

Electron. J. Combin.

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This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

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Section: Introductionmentioning

confidence: 99%

Enumerating Partial Latin Rectangles

Ganfornina

Stones

2020

Electron. J. Combin.

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Computing Autotopism Groups of Partial Latin Rectangles

Stones

Ganfornina

Kotlar

et al. 2020

ACM J. Exp. Algorithmics

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Computing the autotopism group of a partial Latin rectangle (PLR) can be performed in multiple ways. This study has two aims: comparing some of these methods experimentally to identify those that are competitive; and identifying design goals for developing practical software. We compare six families of algorithms (two backtracking and four graph-theoretic methods), with and without using entry invariants (EIs), in a range of settings. Two EIs are considered: frequencies of row, column, and symbol representatives; and 2 × 2 submatrices. The best approach to computing autotopism groups varies. When PLRs have many autotopisms (such as having very few entries or being a group table), the McKay, Meynert, and Myrvold (MMM) method computes generators for the autotopism group efficiently. (The MMM method is the standard way to compute autotopisms.) Otherwise, PLRs ordinarily have trivial or small autotopism groups, and the task is to verify this. The so-called PLR graph method is slightly more efficient in this setting than the MMM method (in some circumstances, around twice as fast). With an intermediate number of entries, the quick-to-compute strong EIs are effective at reducing the need for computation without introducing significant overhead. With a full or almost-full PLR, a more sophisticated EI is needed to reduce down-the-line computation. These results suggest a hybrid approach to computing autotopism groups: The software decides on suitable EIs based on the input; and the user chooses between the MMM or the PLR graph methods, depending on their dataset. This article expands the authors’ previous article Computing autotopism groups of PLRs: a pilot study .

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