3 Search Results for "Zeng, Ji"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.59
On Higher Dimensional Point Sets in General Position

Authors: Andrew Suk and Ji Zeng

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
A finite point set in ℝ^d is in general position if no d + 1 points lie on a common hyperplane. Let α_d(N) be the largest integer such that any set of N points in ℝ^d with no d + 2 members on a common hyperplane, contains a subset of size α_d(N) in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that α₂(N) < N^{5/6 + o(1)}. In this paper, we also use the container method to obtain new upper bounds for α_d(N) when d ≥ 3. More precisely, we show that if d is odd, then α_d(N) < N^{1/2 + 1/(2d) + o(1)}, and if d is even, we have α_d(N) < N^{1/2 + 1/(d-1) + o(1)}. We also study the classical problem of determining the maximum number a(d,k,n) of points selected from the grid [n]^d such that no k + 2 members lie on a k-flat. For fixed d and k, we show that a(d,k,n)≤ O(n^{d/{2⌊(k+2)/4⌋}(1- 1/{2⌊(k+2)/4⌋d+1})}), which improves the previously best known bound of O(n^{d/⌊(k + 2)/2⌋}) due to Lefmann when k+2 is congruent to 0 or 1 mod 4.
Cite as

Andrew Suk and Ji Zeng. On Higher Dimensional Point Sets in General Position. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 59:1-59:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{suk_et_al:LIPIcs.SoCG.2023.59,
  author =	{Suk, Andrew and Zeng, Ji},
  title =	{{On Higher Dimensional Point Sets in General Position}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{59:1--59:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.59},
  URN =		{urn:nbn:de:0030-drops-179097},
  doi =		{10.4230/LIPIcs.SoCG.2023.59},
  annote =	{Keywords: independent sets, hypergraph container method, generalised Sidon sets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.62
A Positive Fraction Erdős-Szekeres Theorem and Its Applications

Authors: Andrew Suk and Ji Zeng

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

Abstract
A famous theorem of Erdős and Szekeres states that any sequence of n distinct real numbers contains a monotone subsequence of length at least √n. Here, we prove a positive fraction version of this theorem. For n > (k-1)², any sequence A of n distinct real numbers contains a collection of subsets A_1,…, A_k ⊂ A, appearing sequentially, all of size s = Ω(n/k²), such that every subsequence (a_1,…, a_k), with a_i ∈ A_i, is increasing, or every such subsequence is decreasing. The subsequence S = (A_1,…, A_k) described above is called block-monotone of depth k and block-size s. Our theorem is asymptotically best possible and follows from a more general Ramsey-type result for monotone paths, which we find of independent interest. We also show that for any positive integer k, any finite sequence of distinct real numbers can be partitioned into O(k²log k) block-monotone subsequences of depth at least k, upon deleting at most (k-1)² entries. We apply our results to mutually avoiding planar point sets and biarc diagrams in graph drawing.
Cite as

Andrew Suk and Ji Zeng. A Positive Fraction Erdős-Szekeres Theorem and Its Applications. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{suk_et_al:LIPIcs.SoCG.2022.62,
  author =	{Suk, Andrew and Zeng, Ji},
  title =	{{A Positive Fraction Erd\H{o}s-Szekeres Theorem and Its Applications}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{62:1--62:15},
  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.2022.62},
  URN =		{urn:nbn:de:0030-drops-160703},
  doi =		{10.4230/LIPIcs.SoCG.2022.62},
  annote =	{Keywords: Erd\H{o}s-Szekeres, block-monotone, monotone biarc diagrams, mutually avoiding sets}
}
Document
DOI: 10.4230/LIPIcs.TQC.2013.192
Symmetries of Codeword Stabilized Quantum Codes

Authors: Salman Beigi, Jianxin Chen, Markus Grassl, Zhengfeng Ji, Qiang Wang, and Bei Zeng

Published in: LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

Abstract
Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q. We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.
Cite as

Salman Beigi, Jianxin Chen, Markus Grassl, Zhengfeng Ji, Qiang Wang, and Bei Zeng. Symmetries of Codeword Stabilized Quantum Codes. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 192-206, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{beigi_et_al:LIPIcs.TQC.2013.192,
  author =	{Beigi, Salman and Chen, Jianxin and Grassl, Markus and Ji, Zhengfeng and Wang, Qiang and Zeng, Bei},
  title =	{{Symmetries of Codeword Stabilized Quantum Codes}},
  booktitle =	{8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)},
  pages =	{192--206},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-55-2},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{22},
  editor =	{Severini, Simone and Brandao, Fernando},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.192},
  URN =		{urn:nbn:de:0030-drops-43129},
  doi =		{10.4230/LIPIcs.TQC.2013.192},
  annote =	{Keywords: CWS Codes, Union Stabilizer Codes, Permutation Symmetry, Toric Code}
}
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