20,180 research outputs found
Equivalence Classes and Conditional Hardness in Massively Parallel Computations
The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., P ? NP), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems.
In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by MPC(o(log N)), and some standard classes concerning space complexity, namely L and NL, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model
Recognizing Partial Cubes in Quadratic Time
We show how to test whether a graph with n vertices and m edges is a partial
cube, and if so how to find a distance-preserving embedding of the graph into a
hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time
solutions.Comment: 25 pages, five figures. This version significantly expands previous
versions, including a new report on an implementation of the algorithm and
experiments with i
Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford
A \emph{metric tree embedding} of expected \emph{stretch~}
maps a weighted -node graph to a weighted tree with such that, for all ,
and
. Such embeddings are highly useful for designing
fast approximation algorithms, as many hard problems are easy to solve on tree
instances. However, to date the best parallel -depth algorithm that achieves an asymptotically optimal expected stretch of
requires
work and a metric as input.
In this paper, we show how to achieve the same guarantees using
depth and
work, where and is an arbitrarily small constant.
Moreover, one may further reduce the work to at the expense of increasing the expected stretch to
.
Our main tool in deriving these parallel algorithms is an algebraic
characterization of a generalization of the classic Moore-Bellman-Ford
algorithm. We consider this framework, which subsumes a variety of previous
"Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss
it in depth. In our tree embedding algorithm, we leverage it for providing
efficient query access to an approximate metric that allows sampling the tree
using depth and work.
We illustrate the generality and versatility of our techniques by various
examples and a number of additional results
Shortest Path and Distance Queries on Road Networks: An Experimental Evaluation
Computing the shortest path between two given locations in a road network is
an important problem that finds applications in various map services and
commercial navigation products. The state-of-the-art solutions for the problem
can be divided into two categories: spatial-coherence-based methods and
vertex-importance-based approaches. The two categories of techniques, however,
have not been compared systematically under the same experimental framework, as
they were developed from two independent lines of research that do not refer to
each other. This renders it difficult for a practitioner to decide which
technique should be adopted for a specific application. Furthermore, the
experimental evaluation of the existing techniques, as presented in previous
work, falls short in several aspects. Some methods were tested only on small
road networks with up to one hundred thousand vertices; some approaches were
evaluated using distance queries (instead of shortest path queries), namely,
queries that ask only for the length of the shortest path; a state-of-the-art
technique was examined based on a faulty implementation that led to incorrect
query results. To address the above issues, this paper presents a comprehensive
comparison of the most advanced spatial-coherence-based and
vertex-importance-based approaches. Using a variety of real road networks with
up to twenty million vertices, we evaluated each technique in terms of its
preprocessing time, space consumption, and query efficiency (for both shortest
path and distance queries). Our experimental results reveal the characteristics
of different techniques, based on which we provide guidelines on selecting
appropriate methods for various scenarios.Comment: VLDB201
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