3 Search Results for "Ahle, Thomas"

Document
DOI: 10.4230/LIPIcs.SoCG.2022.1
Tiling with Squares and Packing Dominos in Polynomial Time

Authors: Anders Aamand, Mikkel Abrahamsen, Thomas Ahle, and Peter M. R. Rasmussen

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container P. We give polynomial-time algorithms for deciding if P can be tiled with k× k squares for any fixed k which can be part of the input (that is, deciding if P is the union of a set of non-overlapping k× k squares) and for packing P with a maximum number of non-overlapping and axis-parallel 2× 1 dominos, allowing rotations by 90^∘. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2× 1 dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2× 2 squares is known to be NP-hard [J. Algorithms 1990]. For our three problems there are known pseudo-polynomial-time algorithms, that is, algorithms with running times polynomial in the area or perimeter of P. However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial-time algorithms for the problems. Concretely, we give a simple O(nlog n)-time algorithm for tiling with squares, where n is the number of corners of P. We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O(n³) vertices. This leads to algorithms with running times O(n³(log³ n)/(log²log n)) and O(n³(log² n)/(log log n)), respectively.
Cite as

Anders Aamand, Mikkel Abrahamsen, Thomas Ahle, and Peter M. R. Rasmussen. Tiling with Squares and Packing Dominos in Polynomial Time. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 1:1-1:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aamand_et_al:LIPIcs.SoCG.2022.1,
  author =	{Aamand, Anders and Abrahamsen, Mikkel and Ahle, Thomas and Rasmussen, Peter M. R.},
  title =	{{Tiling with Squares and Packing Dominos in Polynomial Time}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{1:1--1:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.1},
  URN =		{urn:nbn:de:0030-drops-160096},
  doi =		{10.4230/LIPIcs.SoCG.2022.1},
  annote =	{Keywords: packing, tiling, polyominos}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2021.21
Approximate Similarity Search Under Edit Distance Using Locality-Sensitive Hashing

Authors: Samuel McCauley

Published in: LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)

Abstract
Edit distance similarity search, also called approximate pattern matching, is a fundamental problem with widespread database applications. The goal of the problem is to preprocess n strings of length d, to quickly answer queries q of the form: if there is a database string within edit distance r of q, return a database string within edit distance cr of q. Previous approaches to this problem either rely on very large (superconstant) approximation ratios c, or very small search radii r. Outside of a narrow parameter range, these solutions are not competitive with trivially searching through all n strings. In this work we give a simple and easy-to-implement hash function that can quickly answer queries for a wide range of parameters. Specifically, our strategy can answer queries in time Õ(d3^rn^{1/c}). The best known practical results require c ≫ r to achieve any correctness guarantee; meanwhile, the best known theoretical results are very involved and difficult to implement, and require query time that can be loosely bounded below by 24^r. Our results significantly broaden the range of parameters for which there exist nontrivial theoretical bounds, while retaining the practicality of a locality-sensitive hash function.
Cite as

Samuel McCauley. Approximate Similarity Search Under Edit Distance Using Locality-Sensitive Hashing. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 21:1-21:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mccauley:LIPIcs.ICDT.2021.21,
  author =	{McCauley, Samuel},
  title =	{{Approximate Similarity Search Under Edit Distance Using Locality-Sensitive Hashing}},
  booktitle =	{24th International Conference on Database Theory (ICDT 2021)},
  pages =	{21:1--21:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-179-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{186},
  editor =	{Yi, Ke and Wei, Zhewei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.21},
  URN =		{urn:nbn:de:0030-drops-137299},
  doi =		{10.4230/LIPIcs.ICDT.2021.21},
  annote =	{Keywords: edit distance, approximate pattern matching, approximate nearest neighbor, similarity search, locality-sensitive hashing}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2020.22
The Space Complexity of Inner Product Filters

Authors: Rasmus Pagh and Johan Sivertsen

Published in: LIPIcs, Volume 155, 23rd International Conference on Database Theory (ICDT 2020)

Abstract
Motivated by the problem of filtering candidate pairs in inner product similarity joins we study the following inner product estimation problem: Given parameters d∈ℕ, α>β≥0 and unit vectors x,y∈ ℝ^d consider the task of distinguishing between the cases ⟨x,y⟩≤β and ⟨x,y⟩≥α where ⟨x,y⟩ = ∑_{i=1}^d x_i y_i is the inner product of vectors x and y. The goal is to distinguish these cases based on information on each vector encoded independently in a bit string of the shortest length possible. In contrast to much work on compressing vectors using randomized dimensionality reduction, we seek to solve the problem deterministically, with no probability of error. Inner product estimation can be solved in general via estimating ⟨x,y⟩ with an additive error bounded by ε = α - β. We show that d log₂ (√{1-β}/ε) ± Θ(d) bits of information about each vector is necessary and sufficient. Our upper bound is constructive and improves a known upper bound of d log₂(1/ε) + O(d) by up to a factor of 2 when β is close to 1. The lower bound holds even in a stronger model where one of the vectors is known exactly, and an arbitrary estimation function is allowed.
Cite as

Rasmus Pagh and Johan Sivertsen. The Space Complexity of Inner Product Filters. In 23rd International Conference on Database Theory (ICDT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 155, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{pagh_et_al:LIPIcs.ICDT.2020.22,
  author =	{Pagh, Rasmus and Sivertsen, Johan},
  title =	{{The Space Complexity of Inner Product Filters}},
  booktitle =	{23rd International Conference on Database Theory (ICDT 2020)},
  pages =	{22:1--22:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-139-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{155},
  editor =	{Lutz, Carsten and Jung, Jean Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2020.22},
  URN =		{urn:nbn:de:0030-drops-119468},
  doi =		{10.4230/LIPIcs.ICDT.2020.22},
  annote =	{Keywords: Similarity, estimation, dot product, filtering}
}
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