132,428 research outputs found
Majority problems of large query size
We study two models of the Majority problem. We are given n balls and an
unknown coloring of them with two colors. We can ask sets of balls of size k as
queries, and in the so-called General Model the answer to a query shows if all
the balls in the set are of the same color or not. In the so-called Counting
Model the answer to a query gives the difference between the cardinalities of
the color classes in the query. Our goal is to show a ball of the larger color
class, or prove that the color classes are of the same size, using as few
queries as possible. In this paper we improve the bounds given by De Marco and
Kranakis for the number of queries needed.Comment: We cut the non-adaptive results from the first version to publish
separatel
On non-adaptive majority problems of large query size
We are given balls and an unknown coloring of them with two colors. Our
goal is to find a ball that belongs to the larger color class, or show that the
color classes have the same size. We can ask sets of balls as queries, and
the problem has different variants, according to what the answers to the
queries can be. These questions has attracted several researchers, but the
focus of most research was the adaptive version, where queries are decided
sequentially, after learning the answer to the previous query. Here we study
the non-adaptive version, where all the queries have to be asked at the same
time.Comment: 12 page
Finding a non-minority ball with majority answers
Suppose we are given a set of balls each colored
either red or blue in some way unknown to us. To find out some information
about the colors, we can query any triple of balls
. As an answer to such a query we obtain (the
index of) a {\em majority ball}, that is, a ball whose color is the same as the
color of another ball from the triple. Our goal is to find a {\em non-minority
ball}, that is, a ball whose color occurs at least times among the
balls. We show that the minimum number of queries needed to solve this
problem is in the adaptive case and in the
non-adaptive case. We also consider some related problems
Pseudo-random graphs and bit probe schemes with one-sided error
We study probabilistic bit-probe schemes for the membership problem. Given a
set A of at most n elements from the universe of size m we organize such a
structure that queries of type "Is x in A?" can be answered very quickly.
H.Buhrman, P.B.Miltersen, J.Radhakrishnan, and S.Venkatesh proposed a bit-probe
scheme based on expanders. Their scheme needs space of bits, and
requires to read only one randomly chosen bit from the memory to answer a
query. The answer is correct with high probability with two-sided errors. In
this paper we show that for the same problem there exists a bit-probe scheme
with one-sided error that needs space of O(n\log^2 m+\poly(\log m)) bits. The
difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh
is that we consider a bit-probe scheme with an auxiliary word. This means that
in our scheme the memory is split into two parts of different size: the main
storage of bits and a short word of bits that is
pre-computed once for the stored set A and `cached'. To answer a query "Is x in
A?" we allow to read the whole cached word and only one bit from the main
storage. For some reasonable values of parameters our space bound is better
than what can be achieved by any scheme without cached data.Comment: 19 page
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