309 research outputs found
Parallel algorithms and concentration bounds for the Lovasz Local Lemma via witness DAGs
The Lov\'{a}sz Local Lemma (LLL) is a cornerstone principle in the
probabilistic method of combinatorics, and a seminal algorithm of Moser &
Tardos (2010) provides an efficient randomized algorithm to implement it. This
can be parallelized to give an algorithm that uses polynomially many processors
and runs in time on an EREW PRAM, stemming from
adaptive computations of a maximal independent set (MIS). Chung et al. (2014)
developed faster local and parallel algorithms, potentially running in time
, but these algorithms require more stringent conditions than the
LLL.
We give a new parallel algorithm that works under essentially the same
conditions as the original algorithm of Moser & Tardos but uses only a single
MIS computation, thus running in time on an EREW PRAM. This can
be derandomized to give an NC algorithm running in time as well,
speeding up a previous NC LLL algorithm of Chandrasekaran et al. (2013).
We also provide improved and tighter bounds on the run-times of the
sequential and parallel resampling-based algorithms originally developed by
Moser & Tardos. These apply to any problem instance in which the tighter
Shearer LLL criterion is satisfied
A new unifying heuristic algorithm for the undirected minimum cut problems using minimum range cut algorithms
AbstractGiven a connected undirected multigraph with n vertices and m edges, we first propose a new unifying heuristic approach to approximately solving the minimum cut and the s-t minimum cut problems by using efficient algorithms for the corresponding minimum range cut problems. Our method is based on the association of the range value of a cut and its cut value when each edge weight is chosen uniformly randomly from the fixed interval. Our computational experiments demonstrate that this approach produces very good approximate solutions. We shall also propose an O(log2 n) time parallel algorithm using O(n2) processors on an arbitrary CRCW PRAM model for the minimum range cut problems, by which we can efficiently obtain approximate minimum cuts in poly-log time using a polynomial number of processors
Optimal parallel string algorithms: sorting, merching and computing the minimum
We study fundamental comparison problems on strings of characters, equipped with the usual lexicographical ordering. For each problem studied, we give a parallel algorithm that is optimal with respect to at least one criterion for which no optimal algorithm was previously known. Specifically, our main results are: % \begin{itemize} \item Two sorted sequences of strings, containing altogether ~characters, can be merged in time using operations on an EREW PRAM. This is optimal as regards both the running time and the number of operations. \item A sequence of strings, containing altogether ~characters represented by integers of size polynomial in~, can be sorted in time using operations on a CRCW PRAM. The running time is optimal for any polynomial number of processors. \item The minimum string in a sequence of strings containing altogether characters can be found using (expected) operations in constant expected time on a randomized CRCW PRAM, in time on a deterministic CRCW PRAM with a program depending on~, in time on a deterministic CRCW PRAM with a program not depending on~, in expected time on a randomized EREW PRAM, and in time on a deterministic EREW PRAM. The number of operations is optimal, and the running time is optimal for the randomized algorithms and, if the number of processors is limited to~, for the nonuniform deterministic CRCW PRAM algorithm as we
Deterministic parallel algorithms for bilinear objective functions
Many randomized algorithms can be derandomized efficiently using either the
method of conditional expectations or probability spaces with low independence.
A series of papers, beginning with work by Luby (1988), showed that in many
cases these techniques can be combined to give deterministic parallel (NC)
algorithms for a variety of combinatorial optimization problems, with low time-
and processor-complexity.
We extend and generalize a technique of Luby for efficiently handling
bilinear objective functions. One noteworthy application is an NC algorithm for
maximal independent set. On a graph with edges and vertices, this
takes time and processors, nearly
matching the best randomized parallel algorithms. Other applications include
reduced processor counts for algorithms of Berger (1997) for maximum acyclic
subgraph and Gale-Berlekamp switching games.
This bilinear factorization also gives better algorithms for problems
involving discrepancy. An important application of this is to automata-fooling
probability spaces, which are the basis of a notable derandomization technique
of Sivakumar (2002). Our method leads to large reduction in processor
complexity for a number of derandomization algorithms based on
automata-fooling, including set discrepancy and the Johnson-Lindenstrauss
Lemma
- …