1,785 research outputs found
Tribes Is Hard in the Message Passing Model
We consider the point-to-point message passing model of communication in
which there are processors with individual private inputs, each -bit
long. Each processor is located at the node of an underlying undirected graph
and has access to private random coins. An edge of the graph is a private
channel of communication between its endpoints. The processors have to compute
a given function of all their inputs by communicating along these channels.
While this model has been widely used in distributed computing, strong lower
bounds on the amount of communication needed to compute simple functions have
just begun to appear. In this work, we prove a tight lower bound of
on the communication needed for computing the Tribes function,
when the underlying graph is a star of nodes that has leaves with
inputs and a center with no input. Lower bound on this topology easily implies
comparable bounds for others. Our lower bounds are obtained by building upon
the recent information theoretic techniques of Braverman et.al (FOCS'13) and
combining it with the earlier work of Jayram, Kumar and Sivakumar (STOC'03).
This approach yields information complexity bounds that is of independent
interest
Logic Column 17: A Rendezvous of Logic, Complexity, and Algebra
This article surveys recent advances in applying algebraic techniques to
constraint satisfaction problems.Comment: 30 page
Online unit clustering in higher dimensions
We revisit the online Unit Clustering and Unit Covering problems in higher
dimensions: Given a set of points in a metric space, that arrive one by
one, Unit Clustering asks to partition the points into the minimum number of
clusters (subsets) of diameter at most one; while Unit Covering asks to cover
all points by the minimum number of balls of unit radius. In this paper, we
work in using the norm.
We show that the competitive ratio of any online algorithm (deterministic or
randomized) for Unit Clustering must depend on the dimension . We also give
a randomized online algorithm with competitive ratio for Unit
Clustering}of integer points (i.e., points in , , under norm). We show that the competitive ratio of
any deterministic online algorithm for Unit Covering is at least . This
ratio is the best possible, as it can be attained by a simple deterministic
algorithm that assigns points to a predefined set of unit cubes. We complement
these results with some additional lower bounds for related problems in higher
dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the
Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA
2017
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