391,692 research outputs found
Simple parallel and distributed algorithms for spectral graph sparsification
We describe a simple algorithm for spectral graph sparsification, based on
iterative computations of weighted spanners and uniform sampling. Leveraging
the algorithms of Baswana and Sen for computing spanners, we obtain the first
distributed spectral sparsification algorithm. We also obtain a parallel
algorithm with improved work and time guarantees. Combining this algorithm with
the parallel framework of Peng and Spielman for solving symmetric diagonally
dominant linear systems, we get a parallel solver which is much closer to being
practical and significantly more efficient in terms of the total work.Comment: replaces "A simple parallel and distributed algorithm for spectral
sparsification". Minor change
Parallel and Distributed Algorithms for the Housing Allocation Problem
We give parallel and distributed algorithms for the housing allocation
problem. In this problem, there is a set of agents and a set of houses. Each
agent has a strict preference list for a subset of houses. We need to find a
matching such that some criterion is optimized. One such criterion is Pareto
Optimality. A matching is Pareto optimal if no coalition of agents can be
strictly better off by exchanging houses among themselves. We also study the
housing market problem, a variant of the housing allocation problem, where each
agent initially owns a house. In addition to Pareto optimality, we are also
interested in finding the core of a housing market. A matching is in the core
if there is no coalition of agents that can be better off by breaking away from
other agents and switching houses only among themselves.
In the first part of this work, we show that computing a Pareto optimal
matching of a house allocation is in {\bf CC} and computing the core of a
housing market is {\bf CC}-hard. Given a matching, we also show that verifying
whether it is in the core can be done in {\bf NC}. We then give an algorithm to
show that computing a maximum Pareto optimal matching for the housing
allocation problem is in {\bf RNC}^2 and quasi-{\bf NC}^2. In the second part
of this work, we present a distributed version of the top trading cycle
algorithm for finding the core of a housing market. To that end, we first
present two algorithms for finding all the disjoint cycles in a functional
graph: a Las Vegas algorithm which terminates in rounds with high
probability, where is the length of the longest cycle, and a deterministic
algorithm which terminates in rounds, where is the
number of nodes in the graph. Both algorithms work in the synchronous
distributed model and use messages of size
Strong Scaling of Matrix Multiplication Algorithms and Memory-Independent Communication Lower Bounds
A parallel algorithm has perfect strong scaling if its running time on P
processors is linear in 1/P, including all communication costs.
Distributed-memory parallel algorithms for matrix multiplication with perfect
strong scaling have only recently been found. One is based on classical matrix
multiplication (Solomonik and Demmel, 2011), and one is based on Strassen's
fast matrix multiplication (Ballard, Demmel, Holtz, Lipshitz, and Schwartz,
2012). Both algorithms scale perfectly, but only up to some number of
processors where the inter-processor communication no longer scales.
We obtain a memory-independent communication cost lower bound on classical
and Strassen-based distributed-memory matrix multiplication algorithms. These
bounds imply that no classical or Strassen-based parallel matrix multiplication
algorithm can strongly scale perfectly beyond the ranges already attained by
the two parallel algorithms mentioned above. The memory-independent bounds and
the strong scaling bounds generalize to other algorithms.Comment: 4 pages, 1 figur
Non-Local Probes Do Not Help with Graph Problems
This work bridges the gap between distributed and centralised models of
computing in the context of sublinear-time graph algorithms. A priori, typical
centralised models of computing (e.g., parallel decision trees or centralised
local algorithms) seem to be much more powerful than distributed
message-passing algorithms: centralised algorithms can directly probe any part
of the input, while in distributed algorithms nodes can only communicate with
their immediate neighbours. We show that for a large class of graph problems,
this extra freedom does not help centralised algorithms at all: for example,
efficient stateless deterministic centralised local algorithms can be simulated
with efficient distributed message-passing algorithms. In particular, this
enables us to transfer existing lower bound results from distributed algorithms
to centralised local algorithms
Parallel and distributed Gr\"obner bases computation in JAS
This paper considers parallel Gr\"obner bases algorithms on distributed
memory parallel computers with multi-core compute nodes. We summarize three
different Gr\"obner bases implementations: shared memory parallel, pure
distributed memory parallel and distributed memory combined with shared memory
parallelism. The last algorithm, called distributed hybrid, uses only one
control communication channel between the master node and the worker nodes and
keeps polynomials in shared memory on a node. The polynomials are transported
asynchronous to the control-flow of the algorithm in a separate distributed
data structure. The implementation is generic and works for all implemented
(exact) fields. We present new performance measurements and discuss the
performance of the algorithms.Comment: 14 pages, 8 tables, 13 figure
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