2 Search Results for "Roth, Tal"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.95
The Communication Complexity of Set Intersection Under Product Distributions

Authors: Rotem Oshman and Tal Roth

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
We consider a multiparty setting where k parties have private inputs X_1,…,X_k ⊆ [n] and wish to compute the intersection ⋂_{𝓁 =1}^k X_𝓁 of their sets, using as little communication as possible. This task generalizes the well-known problem of set disjointness, where the parties are required only to determine whether the intersection is empty or not. In the worst-case, it is known that the communication complexity of finding the intersection is the same as that of solving set disjointness, regardless of the size of the intersection: the cost of both problems is Ω(n log k + k) bits in the shared blackboard model, and Ω (nk) bits in the coordinator model. In this work we consider a realistic setting where the parties' inputs are independent of one another, that is, the input is drawn from a product distribution. We show that this makes finding the intersection significantly easier than in the worst-case: only Θ̃((n^{1-1/k} (H(S) + 1)^{1/k}) + k) bits of communication are required, where {H}(S) is the Shannon entropy of the intersection S. We also show that the parties do not need to know the exact underlying input distribution; if we are given in advance O(n^{1/k}) samples from the underlying distribution μ, we can learn enough about μ to allow us to compute the intersection of an input drawn from μ using expected communication Θ̃((n^{1-1/k}𝔼[|S|]^{1/k}) + k), where |S| is the size of the intersection.
Cite as

Rotem Oshman and Tal Roth. The Communication Complexity of Set Intersection Under Product Distributions. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 95:1-95:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{oshman_et_al:LIPIcs.ICALP.2023.95,
  author =	{Oshman, Rotem and Roth, Tal},
  title =	{{The Communication Complexity of Set Intersection Under Product Distributions}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{95:1--95:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.95},
  URN =		{urn:nbn:de:0030-drops-181472},
  doi =		{10.4230/LIPIcs.ICALP.2023.95},
  annote =	{Keywords: Communication complexity, intersection, set disjointness}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.54
Candidate Tree Codes via Pascal Determinant Cubes

Authors: Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
Tree codes are combinatorial structures introduced by Schulman [Schulman, 1993] as key ingredients in interactive coding schemes. Asymptotically-good tree codes are long known to exist, yet their explicit construction remains a notoriously hard open problem. Even proposing a plausible construction, without the burden of proof, is difficult and the defining tree code property requires structure that remains elusive. To the best of our knowledge, only one candidate appears in the literature, due to Moore and Schulman [Moore and Schulman, 2014]. We put forth a new candidate for an explicit asymptotically-good tree code. Our construction is an extension of the vanishing rate tree code by Cohen-Haeupler-Schulman [Cohen et al., 2018], and its correctness relies on a conjecture that we introduce on certain Pascal determinants indexed by the points of the Boolean hypercube. Furthermore, using the vanishing distance tree code by Gelles et al. [Gelles et al., 2016] enables us to present a construction that relies on an even weaker assumption. We furnish evidence supporting our conjecture through numerical computation, combinatorial arguments from planar path graphs and based on well-studied heuristics from arithmetic geometry.
Cite as

Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan. Candidate Tree Codes via Pascal Determinant Cubes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 54:1-54:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2021.54,
  author =	{Ben Yaacov, Inbar and Cohen, Gil and Narayanan, Anand Kumar},
  title =	{{Candidate Tree Codes via Pascal Determinant Cubes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{54:1--54:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.54},
  URN =		{urn:nbn:de:0030-drops-147474},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.54},
  annote =	{Keywords: Tree codes, Sparse polynomials, Explicit constructions}
}
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