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DOI: 10.4230/LIPIcs.CCC.2022.19

The Composition Complexity of Majority

Authors: Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

We study the complexity of computing majority as a composition of local functions: Maj_n = h(g_1,…,g_m), where each g_j: {0,1}ⁿ → {0,1} is an arbitrary function that queries only k ≪ n variables and h: {0,1}^m → {0,1} is an arbitrary combining function. We prove an optimal lower bound of m ≥ Ω(n/k log k) on the number of functions needed, which is a factor Ω(log k) larger than the ideal m = n/k. We call this factor the composition overhead; previously, no superconstant lower bounds on it were known for majority. Our lower bound recovers, as a corollary and via an entirely different proof, the best known lower bound for bounded-width branching programs for majority (Alon and Maass '86, Babai et al. '90). It is also the first step in a plan that we propose for breaking a longstanding barrier in lower bounds for small-depth boolean circuits. Novel aspects of our proof include sharp bounds on the information lost as computation flows through the inner functions g_j, and the bootstrapping of lower bounds for a multi-output function (Hamming weight) into lower bounds for a single-output one (majority).

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Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan. The Composition Complexity of Majority. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 19:1-19:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lecomte_et_al:LIPIcs.CCC.2022.19,
  author =	{Lecomte, Victor and Ramakrishnan, Prasanna and Tan, Li-Yang},
  title =	{{The Composition Complexity of Majority}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{19:1--19:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.19},
  URN =		{urn:nbn:de:0030-drops-165818},
  doi =		{10.4230/LIPIcs.CCC.2022.19},
  annote =	{Keywords: computational complexity, circuit lower bounds}
}

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DOI: 10.4230/LIPIcs.ESA.2020.68

Settling the Relationship Between Wilber’s Bounds for Dynamic Optimality

Authors: Victor Lecomte and Omri Weinstein

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract

In FOCS 1986, Wilber proposed two combinatorial lower bounds on the operational cost of any binary search tree (BST) for a given access sequence X ∈ [n]^m. Both bounds play a central role in the ongoing pursuit of the dynamic optimality conjecture (Sleator and Tarjan, 1985), but their relationship remained unknown for more than three decades. We show that Wilber’s Funnel bound dominates his Alternation bound for all X, and give a tight Θ(lg lg n) separation for some X, answering Wilber’s conjecture and an open problem of Iacono, Demaine et. al. The main ingredient of the proof is a new symmetric characterization of Wilber’s Funnel bound, which proves that it is invariant under rotations of X. We use this characterization to provide initial indication that the Funnel bound matches the Independent Rectangle bound (Demaine et al., 2009), by proving that when the Funnel bound is constant, IRB_upRect is linear. To the best of our knowledge, our results provide the first progress on Wilber’s conjecture that the Funnel bound is dynamically optimal (1986).

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Victor Lecomte and Omri Weinstein. Settling the Relationship Between Wilber’s Bounds for Dynamic Optimality. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 68:1-68:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lecomte_et_al:LIPIcs.ESA.2020.68,
  author =	{Lecomte, Victor and Weinstein, Omri},
  title =	{{Settling the Relationship Between Wilber’s Bounds for Dynamic Optimality}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{68:1--68:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.68},
  URN =		{urn:nbn:de:0030-drops-129342},
  doi =		{10.4230/LIPIcs.ESA.2020.68},
  annote =	{Keywords: data structures, binary search trees, dynamic optimality, lower bounds}
}

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Documents authored by Lecomte, Victor

The Composition Complexity of Majority

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Settling the Relationship Between Wilber’s Bounds for Dynamic Optimality

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