73 research outputs found
Parallelism with limited nondeterminism
Computational complexity theory studies which computational problems can be solved with limited access to resources. The past fifty years have seen a focus on the relationship between intractable problems and efficient algorithms. However, the relationship between inherently sequential problems and highly parallel algorithms has not been as well studied. Are there efficient but inherently sequential problems that admit some relaxed form of highly parallel algorithm? In this dissertation, we develop the theory of structural complexity around this relationship for three common types of computational problems.
Specifically, we show tradeoffs between time, nondeterminism, and parallelizability. By clearly defining the notions and complexity classes that capture our intuition for parallelizable and sequential problems, we create a comprehensive framework for rigorously proving parallelizability and non-parallelizability of computational problems. This framework provides the means to prove whether otherwise tractable problems can be effectively parallelized, a need highlighted by the current growth of multiprocessor systems. The views adopted by this dissertation—alternate approaches to solving sequential problems using approximation, limited nondeterminism, and parameterization—can be applied practically throughout computer science
Space-Round Tradeoffs for MapReduce Computations
This work explores fundamental modeling and algorithmic issues arising in the
well-established MapReduce framework. First, we formally specify a
computational model for MapReduce which captures the functional flavor of the
paradigm by allowing for a flexible use of parallelism. Indeed, the model
diverges from a traditional processor-centric view by featuring parameters
which embody only global and local memory constraints, thus favoring a more
data-centric view. Second, we apply the model to the fundamental computation
task of matrix multiplication presenting upper and lower bounds for both dense
and sparse matrix multiplication, which highlight interesting tradeoffs between
space and round complexity. Finally, building on the matrix multiplication
results, we derive further space-round tradeoffs on matrix inversion and
matching
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