126,695 research outputs found
Optimal parallel solution of sparse triangular systems
A method for the parallel solution of triangular sets of equations is described that is appropriate when there are many right-handed sides. By preprocessing, the method can reduce the number of parallel steps required to solve Lx = b compared to parallel forward or backsolve. Applications are to iterative solvers with triangular preconditioners, to structural analysis, or to power systems applications, where there may be many right-handed sides (not all available a priori). The inverse of L is represented as a product of sparse triangular factors. The problem is to find a factored representation of this inverse of L with the smallest number of factors (or partitions), subject to the requirement that no new nonzero elements be created in the formation of these inverse factors. A method from an earlier reference is shown to solve this problem. This method is improved upon by constructing a permutation of the rows and columns of L that preserves triangularity and allow for the best possible such partition. A number of practical examples and algorithmic details are presented. The parallelism attainable is illustrated by means of elimination trees and clique trees
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
Constructing BFS Trees Using Tokens To Balance Speed and Network Traffic
Constructing BFS trees rooted at each node of a network helps solve many problems. Reliable communication to other nodes is easily managed and metrics such as the network diameter, shortest path between any two nodes, the center, the radius, and others can be easily computed. A traditional way to form a BFS tree from each node is for all nodes to construct their trees in parallel. While this is the fastest way to accomplish this task, it also requires a large amount of network traffic. In this thesis, we present a way to use a token passing algorithm to form a BFS tree from each node in the network within a desired network traffic limit. We will analyze how the algorithm works on several network topologies and determine the amount of tokens necessary to form BFS trees from each node as quickly as possible without stressing the network more than a desirable limit
Polynomial-Time Space-Optimal Silent Self-Stabilizing Minimum-Degree Spanning Tree Construction
Motivated by applications to sensor networks, as well as to many other areas,
this paper studies the construction of minimum-degree spanning trees. We
consider the classical node-register state model, with a weakly fair scheduler,
and we present a space-optimal \emph{silent} self-stabilizing construction of
minimum-degree spanning trees in this model. Computing a spanning tree with
minimum degree is NP-hard. Therefore, we actually focus on constructing a
spanning tree whose degree is within one from the optimal. Our algorithm uses
registers on bits, converges in a polynomial number of rounds, and
performs polynomial-time computation at each node. Specifically, the algorithm
constructs and stabilizes on a special class of spanning trees, with degree at
most . Indeed, we prove that, unless NP coNP, there are no
proof-labeling schemes involving polynomial-time computation at each node for
the whole family of spanning trees with degree at most . Up to our
knowledge, this is the first example of the design of a compact silent
self-stabilizing algorithm constructing, and stabilizing on a subset of optimal
solutions to a natural problem for which there are no time-efficient
proof-labeling schemes. On our way to design our algorithm, we establish a set
of independent results that may have interest on their own. In particular, we
describe a new space-optimal silent self-stabilizing spanning tree
construction, stabilizing on \emph{any} spanning tree, in rounds, and
using just \emph{one} additional bit compared to the size of the labels used to
certify trees. We also design a silent loop-free self-stabilizing algorithm for
transforming a tree into another tree. Last but not least, we provide a silent
self-stabilizing algorithm for computing and certifying the labels of a
NCA-labeling scheme
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