67 research outputs found
Load-Balancing for Parallel Delaunay Triangulations
Computing the Delaunay triangulation (DT) of a given point set in
is one of the fundamental operations in computational geometry.
Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm
that merges two partial triangulations by re-triangulating a small subset of
their vertices - the border vertices - and combining the three triangulations
efficiently via parallel hash table lookups. The input point division should
therefore yield roughly equal-sized partitions for good load-balancing and also
result in a small number of border vertices for fast merging. In this paper, we
present a novel divide-step based on partitioning the triangulation of a small
sample of the input points. In experiments on synthetic and real-world data
sets, we achieve nearly perfectly balanced partitions and small border
triangulations. This almost cuts running time in half compared to
non-data-sensitive division schemes on inputs exhibiting an exploitable
underlying structure.Comment: Short version submitted to EuroPar 201
Spectral Clustering with Imbalanced Data
Spectral clustering is sensitive to how graphs are constructed from data
particularly when proximal and imbalanced clusters are present. We show that
Ratio-Cut (RCut) or normalized cut (NCut) objectives are not tailored to
imbalanced data since they tend to emphasize cut sizes over cut values. We
propose a graph partitioning problem that seeks minimum cut partitions under
minimum size constraints on partitions to deal with imbalanced data. Our
approach parameterizes a family of graphs, by adaptively modulating node
degrees on a fixed node set, to yield a set of parameter dependent cuts
reflecting varying levels of imbalance. The solution to our problem is then
obtained by optimizing over these parameters. We present rigorous limit cut
analysis results to justify our approach. We demonstrate the superiority of our
method through unsupervised and semi-supervised experiments on synthetic and
real data sets.Comment: 24 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1302.513
Graph-based Semi-Supervised & Active Learning for Edge Flows
We present a graph-based semi-supervised learning (SSL) method for learning
edge flows defined on a graph. Specifically, given flow measurements on a
subset of edges, we want to predict the flows on the remaining edges. To this
end, we develop a computational framework that imposes certain constraints on
the overall flows, such as (approximate) flow conservation. These constraints
render our approach different from classical graph-based SSL for vertex labels,
which posits that tightly connected nodes share similar labels and leverages
the graph structure accordingly to extrapolate from a few vertex labels to the
unlabeled vertices. We derive bounds for our method's reconstruction error and
demonstrate its strong performance on synthetic and real-world flow networks
from transportation, physical infrastructure, and the Web. Furthermore, we
provide two active learning algorithms for selecting informative edges on which
to measure flow, which has applications for optimal sensor deployment. The
first strategy selects edges to minimize the reconstruction error bound and
works well on flows that are approximately divergence-free. The second approach
clusters the graph and selects bottleneck edges that cross cluster-boundaries,
which works well on flows with global trends
Dynamic Balanced Graph Partitioning
This paper initiates the study of the classic balanced graph partitioning
problem from an online perspective: Given an arbitrary sequence of pairwise
communication requests between nodes, with patterns that may change over
time, the objective is to service these requests efficiently by partitioning
the nodes into clusters, each of size , such that frequently
communicating nodes are located in the same cluster. The partitioning can be
updated dynamically by migrating nodes between clusters. The goal is to devise
online algorithms which jointly minimize the amount of inter-cluster
communication and migration cost.
The problem features interesting connections to other well-known online
problems. For example, scenarios with generalize online paging, and
scenarios with constitute a novel online variant of maximum matching. We
present several lower bounds and algorithms for settings both with and without
cluster-size augmentation. In particular, we prove that any deterministic
online algorithm has a competitive ratio of at least , even with significant
augmentation. Our main algorithmic contributions are an -competitive deterministic algorithm for the general setting with
constant augmentation, and a constant competitive algorithm for the maximum
matching variant
Clustering and Community Detection with Imbalanced Clusters
Spectral clustering methods which are frequently used in clustering and
community detection applications are sensitive to the specific graph
constructions particularly when imbalanced clusters are present. We show that
ratio cut (RCut) or normalized cut (NCut) objectives are not tailored to
imbalanced cluster sizes since they tend to emphasize cut sizes over cut
values. We propose a graph partitioning problem that seeks minimum cut
partitions under minimum size constraints on partitions to deal with imbalanced
cluster sizes. Our approach parameterizes a family of graphs by adaptively
modulating node degrees on a fixed node set, yielding a set of parameter
dependent cuts reflecting varying levels of imbalance. The solution to our
problem is then obtained by optimizing over these parameters. We present
rigorous limit cut analysis results to justify our approach and demonstrate the
superiority of our method through experiments on synthetic and real datasets
for data clustering, semi-supervised learning and community detection.Comment: Extended version of arXiv:1309.2303 with new applications. Accepted
to IEEE TSIP
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