11,734 research outputs found
Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas
We show lower bounds of and on the
randomized and quantum communication complexity, respectively, of all
-variable read-once Boolean formulas. Our results complement the recent
lower bound of by Leonardos and Saks and
by Jayram, Kopparty and Raghavendra for
randomized communication complexity of read-once Boolean formulas with depth
. We obtain our result by "embedding" either the Disjointness problem or its
complement in any given read-once Boolean formula.Comment: 5 page
Finding the Median (Obliviously) with Bounded Space
We prove that any oblivious algorithm using space to find the median of a
list of integers from requires time . This bound also applies to the problem of determining whether the median
is odd or even. It is nearly optimal since Chan, following Munro and Raman, has
shown that there is a (randomized) selection algorithm using only
registers, each of which can store an input value or -bit counter,
that makes only passes over the input. The bound also implies
a size lower bound for read-once branching programs computing the low order bit
of the median and implies the analog of for length oblivious branching programs
Quantum vs. Classical Read-once Branching Programs
The paper presents the first nontrivial upper and lower bounds for
(non-oblivious) quantum read-once branching programs. It is shown that the
computational power of quantum and classical read-once branching programs is
incomparable in the following sense: (i) A simple, explicit boolean function on
2n input bits is presented that is computable by error-free quantum read-once
branching programs of size O(n^3), while each classical randomized read-once
branching program and each quantum OBDD for this function with bounded
two-sided error requires size 2^{\Omega(n)}. (ii) Quantum branching programs
reading each input variable exactly once are shown to require size
2^{\Omega(n)} for computing the set-disjointness function DISJ_n from
communication complexity theory with two-sided error bounded by a constant
smaller than 1/2-2\sqrt{3}/7. This function is trivially computable even by
deterministic OBDDs of linear size. The technically most involved part is the
proof of the lower bound in (ii). For this, a new model of quantum
multi-partition communication protocols is introduced and a suitable extension
of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to
this model is presented.Comment: 35 pages. Lower bound for disjointness: Error in application of info
theory corrected and regularity of quantum read-once BPs (each variable at
least once) added as additional assumption of the theorem. Some more informal
explanations adde
Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity
In this paper, the space complexity of nonuniform quantum computations is
investigated. The model chosen for this are quantum branching programs, which
provide a graphic description of sequential quantum algorithms. In the first
part of the paper, simulations between quantum branching programs and
nonuniform quantum Turing machines are presented which allow to transfer lower
and upper bound results between the two models. In the second part of the
paper, different variants of quantum OBDDs are compared with their
deterministic and randomized counterparts. In the third part, quantum branching
programs are considered where the performed unitary operation may depend on the
result of a previous measurement. For this model a simulation of randomized
OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte
Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers
The Sensitivity Conjecture and the Log-rank Conjecture are among the most
important and challenging problems in concrete complexity. Incidentally, the
Sensitivity Conjecture is known to hold for monotone functions, and so is the
Log-rank Conjecture for and with monotone
functions , where and are bit-wise AND and XOR,
respectively. In this paper, we extend these results to functions which
alternate values for a relatively small number of times on any monotone path
from to . These deepen our understandings of the two conjectures,
and contribute to the recent line of research on functions with small
alternating numbers
On Computational Power of Quantum Read-Once Branching Programs
In this paper we review our current results concerning the computational
power of quantum read-once branching programs. First of all, based on the
circuit presentation of quantum branching programs and our variant of quantum
fingerprinting technique, we show that any Boolean function with linear
polynomial presentation can be computed by a quantum read-once branching
program using a relatively small (usually logarithmic in the size of input)
number of qubits. Then we show that the described class of Boolean functions is
closed under the polynomial projections.Comment: In Proceedings HPC 2010, arXiv:1103.226
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