33,402 research outputs found
A Compactification of Open Varieties
In this paper we show a general method to compactify certain open varieties
by adding normal crossing divisors. This is done by proving that {\it blowing
up along an arrangement of subvarieties} can be carried out. Important examples
such as Ulyanov's configuration spaces, spaces of holomorphic maps, etc., are
covered. Intersection ring and (non-recursive) Hodge polynomails are computed.
Further general structures arising from the blowup process are described and
studied.Comment: 21 page
Instance and Output Optimal Parallel Algorithms for Acyclic Joins
Massively parallel join algorithms have received much attention in recent
years, while most prior work has focused on worst-optimal algorithms. However,
the worst-case optimality of these join algorithms relies on hard instances
having very large output sizes, which rarely appear in practice. A stronger
notion of optimality is {\em output-optimal}, which requires an algorithm to be
optimal within the class of all instances sharing the same input and output
size. An even stronger optimality is {\em instance-optimal}, i.e., the
algorithm is optimal on every single instance, but this may not always be
achievable.
In the traditional RAM model of computation, the classical Yannakakis
algorithm is instance-optimal on any acyclic join. But in the massively
parallel computation (MPC) model, the situation becomes much more complicated.
We first show that for the class of r-hierarchical joins, instance-optimality
can still be achieved in the MPC model. Then, we give a new MPC algorithm for
an arbitrary acyclic join with load O ({\IN \over p} + {\sqrt{\IN \cdot \OUT}
\over p}), where \IN,\OUT are the input and output sizes of the join, and
is the number of servers in the MPC model. This improves the MPC version of
the Yannakakis algorithm by an O (\sqrt{\OUT \over \IN} ) factor.
Furthermore, we show that this is output-optimal when \OUT = O(p \cdot \IN),
for every acyclic but non-r-hierarchical join. Finally, we give the first
output-sensitive lower bound for the triangle join in the MPC model, showing
that it is inherently more difficult than acyclic joins
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