Few Sequence Pairs Suffice: Representing All Rectangle Placements

TL;DR

This paper establishes new bounds on the minimal size of complete representations of rectangle placements using spatial relations, improving upon existing methods and enabling faster VLSI placement algorithms.

Contribution

It introduces tighter bounds on the number of sequence pairs needed for complete rectangle placement representations, leveraging pattern-avoiding permutations.

Findings

New upper bound of O((n! / n^6) * ((11+5√5)/2)^n)

New lower bound of Ω((n! / n^4) * (4 + 2√2)^n)

Improved theoretical bounds enable faster VLSI placement algorithms.

Abstract

We consider representations of general non-overlapping 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.