11 research outputs found
An Efficient Construction of Yao-Graph in Data-Distributed Settings
A sparse graph that preserves an approximation of the shortest paths between
all pairs of points in a plane is called a geometric spanner. Using range trees
of sublinear size, we design an algorithm in massively parallel computation
(MPC) model for constructing a geometric spanner known as Yao-graph. This
improves the total time and the total memory of existing algorithms for
geometric spanners from subquadratic to near-linear
A 2-Approximation Algorithm for Data-Distributed Metric k-Center
In a metric space, a set of point sets of roughly the same size and an
integer are given as the input and the goal of data-distributed
-center is to find a subset of size of the input points as the set of
centers to minimize the maximum distance from the input points to their closest
centers. Metric -center is known to be NP-hard which carries to the
data-distributed setting.
We give a -approximation algorithm of -center for sublinear in the
data-distributed setting, which is tight. This algorithm works in several
models, including the massively parallel computation model (MPC)
Massively-Parallel Heat Map Sorting and Applications To Explainable Clustering
Given a set of points labeled with labels, we introduce the heat map
sorting problem as reordering and merging the points and dimensions while
preserving the clusters (labels). A cluster is preserved if it remains
connected, i.e., if it is not split into several clusters and no two clusters
are merged.
We prove the problem is NP-hard and we give a fixed-parameter algorithm with
a constant number of rounds in the massively parallel computation model, where
each machine has a sublinear memory and the total memory of the machines is
linear. We give an approximation algorithm for a NP-hard special case of the
problem. We empirically compare our algorithm with k-means and density-based
clustering (DBSCAN) using a dimensionality reduction via locality-sensitive
hashing on several directed and undirected graphs of email and computer
networks
A Massively Parallel Dynamic Programming for Approximate Rectangle Escape Problem
Sublinear time complexity is required by the massively parallel computation
(MPC) model. Breaking dynamic programs into a set of sparse dynamic programs
that can be divided, solved, and merged in sublinear time.
The rectangle escape problem (REP) is defined as follows: For
axis-aligned rectangles inside an axis-aligned bounding box , extend each
rectangle in only one of the four directions: up, down, left, or right until it
reaches and the density is minimized, where is the maximum number
of extensions of rectangles to the boundary that pass through a point inside
bounding box . REP is NP-hard for . If the rectangles are points of a
grid (or unit squares of a grid), the problem is called the square escape
problem (SEP) and it is still NP-hard.
We give a -approximation algorithm for SEP with with time
complexity . This improves the time complexity of existing
algorithms which are at least quadratic. Also, the approximation ratio of our
algorithm for is which is tight. We also give a
-approximation algorithm for REP with time complexity and
give a MPC version of this algorithm for which is the first parallel
algorithm for this problem