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Electronic Colloquium on Computational Complexity

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REPORTS > KEYWORD > DECISION TREES:
Reports tagged with decision trees:
TR95-046 | 4th August 1995
Vince Grolmusz

On the Power of Circuits with Gates of Low L_1 Norms

We examine the power of Boolean functions with low L_1 norms in several
settings. In large part of the recent literature, the degree of a polynomial
which represents a Boolean function in some way was chosen to be the measure of the complexity of the Boolean function.
However, some functions ... more >>>


TR96-008 | 22nd January 1996
F. Bergadano, N.H. Bshouty, Stefano Varricchio

Learning Multivariate Polynomials from Substitution and Equivalence Queries

It has been shown in previous recent work that
multiplicity automata are predictable from multiplicity
and equivalence queries. In this paper we generalize
related notions in a matrix representation
and obtain a basis for the solution
of a number of open problems in learnability theory.
Membership queries are generalized ... more >>>


TR01-039 | 18th May 2001
Stasys Jukna, Stanislav Zak

On Uncertainty versus Size in Branching Programs

Revisions: 1

We propose an information-theoretic approach to proving lower
bounds on the size of branching programs. The argument is based on
Kraft-McMillan type inequalities for the average amount of
uncertainty about (or entropy of) a given input during the various
stages of computation. The uncertainty is measured by the average
more >>>


TR02-009 | 17th January 2002
Petr Savicky

On determinism versus unambiquous nondeterminism for decision trees

Let $f$ be a Boolean function. Let $N(f)=\dnf(f)+\dnf(\neg f)$ be the
sum of the minimum number of monomials in a disjunctive normal form
for $f$ and $\neg f$. Let $p(f)$ be the minimum size of a partition
of the Boolean cube into disjoint subcubes such that $f$ is constant on
more >>>


TR09-110 | 5th November 2009
Scott Aaronson, Andris Ambainis

The Need for Structure in Quantum Speedups

Revisions: 1

Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate.

First, we show that for any problem that ... more >>>


TR13-049 | 1st April 2013
Amir Shpilka, Ben Lee Volk, Avishay Tal

On the Structure of Boolean Functions with Small Spectral Norm

Revisions: 1

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of $f$ is $\|\hat{f}\|_1=\sum_{\alpha}|\hat{f}(\alpha)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n\to \{0,1\}$ with $\|\hat{f}\|_1=A$.

1. There is a subspace $V$ of co-dimension at most $A^2$ such that $f|_V$ is constant.

2. ... more >>>


TR13-164 | 28th November 2013
Scott Aaronson, Andris Ambainis, Kaspars Balodis, Mohammad Bavarian

Weak Parity

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that ... more >>>


TR18-153 | 22nd August 2018
Krishnamoorthy Dinesh, Samir Otiv, Jayalal Sarma

New Bounds for Energy Complexity of Boolean Functions

Revisions: 1

For a Boolean function $f:\{0,1\}^n \to \{0,1\}$ computed by a circuit $C$ over a finite basis $\cal{B}$, the energy complexity of $C$ (denoted by $\mathbf{EC}_{{\cal B}}(C)$) is the maximum over all inputs $\{0,1\}^n$ the numbers of gates of the circuit $C$ (excluding the inputs) that output a one. Energy Complexity ... more >>>


TR19-166 | 20th November 2019
Guy Blanc, Jane Lange, Li-Yang Tan

Top-down induction of decision trees: rigorous guarantees and inherent limitations

Consider the following heuristic for building a decision tree for a function $f : \{0,1\}^n \to \{\pm 1\}$. Place the most influential variable $x_i$ of $f$ at the root, and recurse on the subfunctions $f_{x_i=0}$ and $f_{x_i=1}$ on the left and right subtrees respectively; terminate once the tree is an ... more >>>


TR20-096 | 22nd June 2020
Igor Sergeev

On the asymptotic complexity of sorting

We investigate the number of pairwise comparisons sufficient to sort $n$ elements chosen from a linearly ordered set. This number is shown to be $\log_2(n!) + o(n)$ thus improving over the previously known upper bounds of the form $\log_2(n!) + \Theta(n)$. The new bound is achieved by the proposed group ... more >>>


TR21-065 | 5th May 2021
Nikhil Mande, Swagato Sanyal

One-way communication complexity and non-adaptive decision trees

Revisions: 1

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let $IP$ denote Inner Product on ... more >>>


TR22-173 | 3rd December 2022
Paul Beame, Sajin Koroth

On Disperser/Lifting Properties of the Index and Inner-Product Functions

Revisions: 1

Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated `lifted' function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. Several important complexity ... more >>>


TR22-186 | 31st December 2022
Prashanth Amireddy, Sai Jayasurya, Jayalal Sarma

Power of Decision Trees with Monotone Queries

In this paper, we initiate study of the computational power of adaptive and non-adaptive monotone decision trees – decision trees where each query is a monotone function on the input bits. In the most general setting, the monotone decision tree height (or size) can be viewed as a measure of ... more >>>


TR24-037 | 26th February 2024
Yaroslav Alekseev, Yuval Filmus, Alexander Smal

Lifting dichotomies

Revisions: 2

Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure $A$ for some function $f$, we compose $f$ with a carefully chosen gadget function $g$ and get essentially the same lower bound on a complexity measure $B$ for the ... more >>>




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