2,403 research outputs found
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
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
Separations in Query Complexity Based on Pointer Functions
In 1986, Saks and Wigderson conjectured that the largest separation between
deterministic and zero-error randomized query complexity for a total boolean
function is given by the function on bits defined by a complete
binary tree of NAND gates of depth , which achieves . We show this is false by giving an example of a total
boolean function on bits whose deterministic query complexity is
while its zero-error randomized query complexity is . We further show that the quantum query complexity of the same
function is , giving the first example of a total function
with a super-quadratic gap between its quantum and deterministic query
complexities.
We also construct a total boolean function on variables that has
zero-error randomized query complexity and bounded-error
randomized query complexity . This is the first
super-linear separation between these two complexity measures. The exact
quantum query complexity of the same function is .
These two functions show that the relations and are optimal, up to poly-logarithmic factors. Further
variations of these functions give additional separations between other query
complexity measures: a cubic separation between and , a -power
separation between and , and a 4th power separation between
approximate degree and bounded-error randomized query complexity.
All of these examples are variants of a function recently introduced by
\goos, Pitassi, and Watson which they used to separate the unambiguous
1-certificate complexity from deterministic query complexity and to resolve the
famous Clique versus Independent Set problem in communication complexity.Comment: 25 pages, 6 figures. Version 3 improves separation between Q_E and
R_0 and updates reference
The zero-error randomized query complexity of the pointer function
The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson
\cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to
prove separation results among various measures of complexity such as
deterministic, randomized and quantum query complexities, exact and approximate
polynomial degrees, etc. In particular, the widest possible (quadratic)
separations between deterministic and zero-error randomized query complexity,
as well as between bounded-error and zero-error randomized query complexity,
have been obtained by considering {\em
variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function.
However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the
precise zero-error complexity of the original pointer function was not known.
We show a lower bound of on the zero-error
randomized query complexity of the pointer function on bits;
since an upper bound is already known
\cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a
factor of \polylog\, n
Towards Better Separation between Deterministic and Randomized Query Complexity
We show that there exists a Boolean function which observes the following
separations among deterministic query complexity , randomized zero
error query complexity and randomized one-sided error query
complexity : and
. This refutes the conjecture made by Saks
and Wigderson that for any Boolean function ,
. This also shows widest separation between
and for any Boolean function. The function was defined by
G{\"{o}}{\"{o}}s, 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 certify optimal (quadratic)
separation between and , and polynomial separation between
and . Viewed as separation results, our results are subsumed
by those of Ambainis et al. However, while the functions considerd in the work
of Ambainis et al are different variants of , we work with the original
function itself.Comment: Reference adde
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