6,952 research outputs found
A New Lower Bound for Semigroup Orthogonal Range Searching
We report the first improvement in the space-time trade-off of lower bounds
for the orthogonal range searching problem in the semigroup model, since
Chazelle's result from 1990. This is one of the very fundamental problems in
range searching with a long history. Previously, Andrew Yao's influential
result had shown that the problem is already non-trivial in one
dimension~\cite{Yao-1Dlb}: using units of space, the query time must
be where is the
inverse Ackermann's function, a very slowly growing function.
In dimensions, Bernard Chazelle~\cite{Chazelle.LB.II} proved that the
query time must be where .
Chazelle's lower bound is known to be tight for when space consumption is
`high' i.e., . We have two main results.
The first is a lower bound that shows Chazelle's lower bound was not tight for
`low space': we prove that we must have . Our lower bound does not close the gap to the existing data
structures, however, our second result is that our analysis is tight. Thus, we
believe the gap is in fact natural since lower bounds are proven for idempotent
semigroups while the data structures are built for general semigroups and thus
they cannot assume (and use) the properties of an idempotent semigroup. As a
result, we believe to close the gap one must study lower bounds for
non-idempotent semigroups or building data structures for idempotent
semigroups. We develope significantly new ideas for both of our results that
could be useful in pursuing either of these directions
Non abelian tensor square of non abelian prime power groups
For every -group of order with the derived subgroup of order ,
Rocco in \cite{roc} has shown that the order of tensor square of is at most
. In the present paper not only we improve his bound for
non-abelian -groups but also we describe the structure of all non-abelian
-groups when the bound is attained for a special case. Moreover, our results
give as well an upper bound for the order of .Comment: enriched with contributions of F.G. Russ
Eisenstein's Irreducibility Criterion for Polynomials over Semirings
In this short note, we generalize Eisenstein's irreducibility criterion for
semirings.Comment: A small typo correcte
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