License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ICDT.2021.6
URN: urn:nbn:de:0030-drops-137146
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Lu, Shangqi ; Tao, Yufei

Towards Optimal Dynamic Indexes for Approximate (and Exact) Triangle Counting

LIPIcs-ICDT-2021-6.pdf (1.0 MB)


In ICDT'19, Kara, Ngo, Nikolic, Olteanu, and Zhang gave a structure which maintains the number T of triangles in an undirected graph G = (V, E) along with the edge insertions/deletions in G. Using O(m) space (m = |E|), their structure supports an update in O(√m log m) amortized time which is optimal (up to polylog factors) subject to the OMv-conjecture (Henzinger, Krinninger, Nanongkai, and Saranurak, STOC'15). Aiming to improve the update efficiency, we study:
- the optimal tradeoff between update time and approximation quality. We require a structure to provide the (ε, Γ)-guarantee: when queried, it should return an estimate t of T that has relative error at most ε if T ≥ Γ, or an absolute error at most ε ⋅ Γ, otherwise. We prove that, under any ε ≤ 0.49 and subject to the OMv-conjecture, no structure can guarantee O(m^{0.5-δ}/Γ) expected amortized update time and O(m^{2/3-δ}) query time simultaneously for any constant δ > 0; this is true for Γ = m^c of any constant c in [0, 1/2). We match the lower bound with a structure that ensures Õ((1/ε)³ ⋅ √m/Γ) amortized update time with high probability, and O(1) query time.
- (for exact counting) how to achieve arboricity-sensitive update time. For any 1 ≤ Γ ≤ √m, we describe a structure of O(min{α m + m log m, (m/Γ)²}) space that maintains T precisely, and supports an update in Õ(min{α + Γ, √m}) amortized time, where α is the largest arboricity of G in history (and does not need to be known). Our structure reconstructs the aforementioned ICDT'19 result up to polylog factors by setting Γ = √m, but achieves Õ(m^{0.5-δ}) update time as long as α = O(m^{0.5-δ}).

BibTeX - Entry

  author =	{Lu, Shangqi and Tao, Yufei},
  title =	{{Towards Optimal Dynamic Indexes for Approximate (and Exact) Triangle Counting}},
  booktitle =	{24th International Conference on Database Theory (ICDT 2021)},
  pages =	{6:1--6:23},
  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 =		{},
  URN =		{urn:nbn:de:0030-drops-137146},
  doi =		{10.4230/LIPIcs.ICDT.2021.6},
  annote =	{Keywords: Triangle Counting, Data Structures, Lower Bounds, Graph Algorithms}

Keywords: Triangle Counting, Data Structures, Lower Bounds, Graph Algorithms
Collection: 24th International Conference on Database Theory (ICDT 2021)
Issue Date: 2021
Date of publication: 11.03.2021

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