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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.55

Lifting to Randomized Parity Decision Trees

Authors: Farzan Byramji and Russell Impagliazzo

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

We prove a lifting theorem from randomized decision tree depth to randomized parity decision tree (PDT) size. We use the same property of the gadget, stifling, which was introduced by Chattopadhyay, Mande, Sanyal and Sherif [ITCS 23] to prove a lifting theorem for deterministic PDTs. Moreover, even the milder condition that the gadget has minimum parity certificate complexity at least 2 suffices for lifting to randomized PDT size. To improve the dependence on the gadget g in the lower bounds for composed functions, we consider a related problem g_* whose inputs are certificates of g. It is implicit in the work of Chattopadhyay et al. that for any function f, lower bounds for the *-depth of f_* give lower bounds for the PDT size of f. We make this connection explicit in the deterministic case and show that it also holds for randomized PDTs. We then combine this with composition theorems for *-depth, which follow by adapting known composition theorems for decision trees. As a corollary, we get tight lifting theorems when the gadget is Indexing, Inner Product or Disjointness.

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Farzan Byramji and Russell Impagliazzo. Lifting to Randomized Parity Decision Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{byramji_et_al:LIPIcs.APPROX/RANDOM.2025.55,
  author =	{Byramji, Farzan and Impagliazzo, Russell},
  title =	{{Lifting to Randomized Parity Decision Trees}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{55:1--55:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.55},
  URN =		{urn:nbn:de:0030-drops-244213},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.55},
  annote =	{Keywords: Parity decision trees, composition}
}

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Lifting to Randomized Parity Decision Trees

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