License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2020.30
URN: urn:nbn:de:0030-drops-118911
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/11891/
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Funakoshi, Mitsuru ; Pape-Lange, Julian

Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements

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Abstract

The discrete acyclic convolution computes the 2n+1 sums ∑_{i+j=k|(i,j)∈[0,1,2,… ,n]²} a_i b_j in ?(n log n) time. By using suitable offsets and setting some of the variables to zero, this method provides a tool to calculate all non-zero sums ∑_{i+j=k|(i,j)∈ P∩ℤ²} a_i b_j in a rectangle P with perimeter p in ?(p log p) time.
This paper extends this geometric interpretation in order to allow arbitrary convex polygons P with k vertices and perimeter p. Also, this extended algorithm only needs ?(k + p(log p)² log k) time.
Additionally, this paper presents fast algorithms for counting sub-cadences and cadences with 3 elements using this extended method.

BibTeX - Entry

@InProceedings{funakoshi_et_al:LIPIcs:2020:11891,
  author =	{Mitsuru Funakoshi and Julian Pape-Lange},
  title =	{{Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{30:1--30:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11891},
  URN =		{urn:nbn:de:0030-drops-118911},
  doi =		{10.4230/LIPIcs.STACS.2020.30},
  annote =	{Keywords: discrete acyclic convolutions, string-cadences, geometric algorithms, number theoretic transforms}
}

Keywords: discrete acyclic convolutions, string-cadences, geometric algorithms, number theoretic transforms
Collection: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue Date: 2020
Date of publication: 04.03.2020

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