15,616 research outputs found
Parallel Algorithms for Geometric Graph Problems
We give algorithms for geometric graph problems in the modern parallel models
inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem
over a set of points in the two-dimensional space, our algorithm computes a
-approximate MST. Our algorithms work in a constant number of
rounds of communication, while using total space and communication proportional
to the size of the data (linear space and near linear time algorithms). In
contrast, for general graphs, achieving the same result for MST (or even
connectivity) remains a challenging open problem, despite drawing significant
attention in recent years.
We develop a general algorithmic framework that, besides MST, also applies to
Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic
framework has implications beyond the MapReduce model. For example it yields a
new algorithm for computing EMD cost in the plane in near-linear time,
. We note that while recently Sharathkumar and Agarwal
developed a near-linear time algorithm for -approximating EMD,
our algorithm is fundamentally different, and, for example, also solves the
transportation (cost) problem, raised as an open question in their work.
Furthermore, our algorithm immediately gives a -approximation
algorithm with space in the streaming-with-sorting model with
passes. As such, it is tempting to conjecture that the
parallel models may also constitute a concrete playground in the quest for
efficient algorithms for EMD (and other similar problems) in the vanilla
streaming model, a well-known open problem
Deterministic Sampling and Range Counting in Geometric Data Streams
We present memory-efficient deterministic algorithms for constructing
epsilon-nets and epsilon-approximations of streams of geometric data. Unlike
probabilistic approaches, these deterministic samples provide guaranteed bounds
on their approximation factors. We show how our deterministic samples can be
used to answer approximate online iceberg geometric queries on data streams. We
use these techniques to approximate several robust statistics of geometric data
streams, including Tukey depth, simplicial depth, regression depth, the
Thiel-Sen estimator, and the least median of squares. Our algorithms use only a
polylogarithmic amount of memory, provided the desired approximation factors
are inverse-polylogarithmic. We also include a lower bound for non-iceberg
geometric queries.Comment: 12 pages, 1 figur
Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms
We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques
Towards Tight Bounds for the Streaming Set Cover Problem
We consider the classic Set Cover problem in the data stream model. For
elements and sets () we give a -pass algorithm with a
strongly sub-linear space and logarithmic
approximation factor. This yields a significant improvement over the earlier
algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of
passes. We complement this result by showing that the tradeoff between the
number of passes and space exhibited by our algorithm is tight, at least when
the approximation factor is equal to . Specifically, we show that any
algorithm that computes set cover exactly using passes
must use space in the regime of .
Furthermore, we consider the problem in the geometric setting where the
elements are points in and sets are either discs, axis-parallel
rectangles, or fat triangles in the plane, and show that our algorithm (with a
slight modification) uses the optimal space to find a
logarithmic approximation in passes.
Finally, we show that any randomized one-pass algorithm that distinguishes
between covers of size 2 and 3 must use a linear (i.e., ) amount of
space. This is the first result showing that a randomized, approximate
algorithm cannot achieve a space bound that is sublinear in the input size.
This indicates that using multiple passes might be necessary in order to
achieve sub-linear space bounds for this problem while guaranteeing small
approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201
- …