Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Structure in Complexity Theory Annual Conference

On the Monte Carlo Boolean decision tree complexity of read-once formulae

Miklós Sántha

Published Dec 10, 2002• 21 citations
Publisher

Abstract

In the Boolean decision tree model there is at least a linear gap between the Monte Carlo and the Las Vegas complexity of a function depending on the error probability. The author proves for a large class of read-once formulae that this trivial speed-up is the best that a Monte Carlo algorithm can achieve. For every formula F belonging to that class it is shown that the Monte Carlo complexity of F with two-sided error p is (1-2p)R(F), and with one-sided error p is (1-p)R(F), where R(F) denotes the Las Vegas complexity of F. The result follows from a general lower bound that is derived on the Monte Carlo complexity of these formulae.>

DOI
10.1109/sct.1991.160259

Related papers

Bounding the randomized decision tree complexity of read-once Boolean functions

Kazuyuki Amano

We investigate the deterministic and the randomized decision tree complexities of Boolean functions, denoted by D(f) and R(f), respectively. A long standing conjecture is that, for every Boolean) function f, R(f) =

Symposium on Discrete Algorithms•Jan 23, 2011•2 citations

Improved Quantum Query Upper Bounds Based on Classical Decision Trees

Arjan Cornelissen, Nikhil S. Mande, Subhasree Patro

Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost $O(\sqrt{s})$ to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size $s$. Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.

arXiv:2203.02968•Leibniz international proceedings in informatics•Jan 1, 2022•2 citations

Improved Quantum Query Upper Bounds Based on Classical Decision Trees

Arjan Cornelissen, Nikhil S. Mande, Subhasree Patro

Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost O(s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s .Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree.Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.

Quantum•Jun 23, 2025•0 citations

Computer-assisted proof of lower bounds on the randomized query complexity of certain read-once threshold functions

Nikos Leonardos

Abstract: In this work we prove lower bounds on the randomized decision tree complexity of several read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, in which every internal node is labeled by a threshold function (with output 1 only when at least out of input bits are 1) and each leaf by a variable. We consider the randomized decision tree complexity of such functions, when the underlying tree is a uniform and balanced tree with all its internal nodes labeled by the same threshold function. We prove lower bounds of the form , where is the depth of the tree, for certain values of and. The value of each parameter depends on the solution of a linear program, which is provided by computational methods. Keywords: Decision trees, query complexity, randomized computation, lower bounds, read-once functions. Title: Computer-assisted proof of lower bounds on the randomized query complexity of certain read-once threshold functions Author: Nikos Leonardos International Journal of Recent Research in Mathematics Computer Science and Information Technology ISSN 2350-1022 Vol. 9, Issue 2, October 2022 - March 2023 Page No: 11-17 Paper Publications Website: www.paperpublications.org Published Date: 10-December-2022 DOI: https://doi.org/10.5281/zenodo.7421269 Paper Download Link (Source) https://www.paperpublications.org/upload/book/Computer-assisted%20proof%20of%20lower-10122022-1.pdf

•Dec 10, 2022•0 citations

Lower bounds on quantum query complexity for read-once decision trees with parity nodes

Hideaki Fukuhara, Eiji Takimoto

We introduce a complexity measure for decision trees called the soft rank, which measures how well-balanced a given tree is. The soft rank is a somehow relaxed variant of the rank. Among all decision trees of depth d, the complete binary decision tree (the most balanced tree) has maximum soft rank d, the decision list (the most unbalanced tree) has minimum soft rank √d, and any other trees have soft rank between √d and d. We show that, for any decision tree T in some class G of decision trees which includes all read-once decision trees, the soft rank of T is a lower bound on the quantum query complexity of the Boolean function that T represents. This implies that for any Boolean function f that is represented by a decision tree in G, the deterministic query complexity of f is only quadratically larger than the quantum query complexity of f.

Computing: The Australasian Theory Symposium•Jan 1, 2009•0 citations

Towards Better Separation between Deterministic and Randomized Query Complexity

Sagnik Mukhopadhyay, Swagato Sanyal

We show that there exists a Boolean function F which gives the following separations among deterministic query complexity (D(F)), randomized zero error query complexity (R_0(F)) and randomized one-sided error query complexity (R_1(F)): R_1(F) = ~O(sqrt{D(F)) and R_0(F)=~O(D(F))^3/4. This refutes the conjecture made by Saks and Wigderson that for any Boolean function f, R_0(f)=Omega(D(f))^0.753.. . This also shows widest separation between R_1(f) and D(f) for any Boolean function. The function F was defined by Göös, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function F certify optimal (quadratic) separation between D(f) and R_0(f), and polynomial separation between R_0(f) and R_1(f). Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considered in the work of Ambainis et al are different variants of F, in this work we show that the original function F itself is sufficient to refute the Saks-Wigderson conjecture and obtain widest possible separation between the deterministic and one-sided error randomized query complexity.

arXiv:1506.06399•Leibniz international proceedings in informatics•Jan 1, 2015•4 citations
In the Boolean decision tree model there is at least a linear gap between the Monte Carlo and the Las Vegas complexity of a function depending on the error probability. The author proves for a large class of read-once formulae that this trivial speed-up is the best that a Monte Carlo algorithm can achieve. For every formula F belonging to that class it is shown that the Monte Carlo complexity of F with two-sided error p is (1-2p)R(F), and with one-sided error p is (1-p)R(F), where R(F) denotes the Las Vegas complexity of F. The result follows from a general lower bound that is derived on the Monte Carlo complexity of these formulae.>
Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost O(s)O(\sqrt{s})O(s​) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size sss. Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.
Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost O(s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s .Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree.Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.
Abstract: In this work we prove lower bounds on the randomized decision tree complexity of several read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, in which every internal node is labeled by a threshold function (with output 1 only when at least out of input bits are 1) and each leaf by a variable. We consider the randomized decision tree complexity of such functions, when the underlying tree is a uniform and balanced tree with all its internal nodes labeled by the same threshold function. We prove lower bounds of the form , where is the depth of the tree, for certain values of and. The value of each parameter depends on the solution of a linear program, which is provided by computational methods. Keywords: Decision trees, query complexity, randomized computation, lower bounds, read-once functions. Title: Computer-assisted proof of lower bounds on the randomized query complexity of certain read-once threshold functions Author: Nikos Leonardos International Journal of Recent Research in Mathematics Computer Science and Information Technology ISSN 2350-1022 Vol. 9, Issue 2, October 2022 - March 2023 Page No: 11-17 Paper Publications Website: www.paperpublications.org Published Date: 10-December-2022 DOI: https://doi.org/10.5281/zenodo.7421269 Paper Download Link (Source) https://www.paperpublications.org/upload/book/Computer-assisted%20proof%20of%20lower-10122022-1.pdf
We introduce a complexity measure for decision trees called the soft rank, which measures how well-balanced a given tree is. The soft rank is a somehow relaxed variant of the rank. Among all decision trees of depth d, the complete binary decision tree (the most balanced tree) has maximum soft rank d, the decision list (the most unbalanced tree) has minimum soft rank √d, and any other trees have soft rank between √d and d. We show that, for any decision tree T in some class G of decision trees which includes all read-once decision trees, the soft rank of T is a lower bound on the quantum query complexity of the Boolean function that T represents. This implies that for any Boolean function f that is represented by a decision tree in G, the deterministic query complexity of f is only quadratically larger than the quantum query complexity of f.
We show that there exists a Boolean function F which gives the following separations among deterministic query complexity (D(F)), randomized zero error query complexity (R_0(F)) and randomized one-sided error query complexity (R_1(F)): R_1(F) = ~O(sqrt{D(F)) and R_0(F)=~O(D(F))^3/4. This refutes the conjecture made by Saks and Wigderson that for any Boolean function f, R_0(f)=Omega(D(f))^0.753.. . This also shows widest separation between R_1(f) and D(f) for any Boolean function. The function F was defined by Göös, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function F certify optimal (quadratic) separation between D(f) and R_0(f), and polynomial separation between R_0(f) and R_1(f). Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considered in the work of Ambainis et al are different variants of F, in this work we show that the original function F itself is sufficient to refute the Saks-Wigderson conjecture and obtain widest possible separation between the deterministic and one-sided error randomized query complexity.