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Few Sequence Pairs Suffice: Representing All Rectangle Placements
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-10-01 , DOI: 10.1137/19m1246456
Jannik Silvanus , Jens Vygen

SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2017-2032, January 2020.
We consider representations of general nonoverlapping placements of rectangles by spatial relations (west, south, east, north) of pairs of rectangles. We call a set of representations complete if it contains a representation of every placement of $n$ rectangles. We prove a new upper bound of $\mathcal{O}(\frac{n!}{n^6} \cdot (\frac{11+5 \sqrt 5}{2})^n)$ and a new lower bound of $\Omega(\frac{n!}{n^4} \cdot (4 + 2 \sqrt2)^n)$ on the minimum cardinality of complete sets of representations. A key concept in the proofs of these results are pattern-avoiding permutations. The new upper bound directly improves upon the well-known sequence pair representation, which has size $(n!)^2$, by only considering a restricted set of sequence pairs. It implies theoretically faster algorithms for VLSI placement problems.


中文翻译:

几个序列对就足够了:代表所有矩形位置

SIAM离散数学杂志,第34卷,第4期,第2017-2032页,2020年1月。
我们考虑通过成对的矩形的空间关系(西,南,东,北)来表示矩形的一般非重叠放置。如果一组表示包含$ n $矩形的每个放置位置,我们将其称为完整。我们证明了$ \ mathcal {O}(\ frac {n!} {n ^ 6} \ cdot(\ frac {11 + 5 \ sqrt 5} {2})^ n)$的新上限和新的下限完整表示集的最小基数上$ \ Omega(\ frac {n!} {n ^ 4} \ cdot(4 + 2 \ sqrt2)^ n)$的边界。这些结果证明中的关键概念是避免模式排列。通过仅考虑一组受限的序列对,新的上限直接改善了众所周知的序列对表示形式,其大小为$(n!)^ 2 $。从理论上讲,这意味着可以解决VLSI放置问题的算法。
更新日期:2020-10-02
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