5,264 research outputs found
Monadic second-order definable graph orderings
We study the question of whether, for a given class of finite graphs, one can
define, for each graph of the class, a linear ordering in monadic second-order
logic, possibly with the help of monadic parameters. We consider two variants
of monadic second-order logic: one where we can only quantify over sets of
vertices and one where we can also quantify over sets of edges. For several
special cases, we present combinatorial characterisations of when such a linear
ordering is definable. In some cases, for instance for graph classes that omit
a fixed graph as a minor, the presented conditions are necessary and
sufficient; in other cases, they are only necessary. Other graph classes we
consider include complete bipartite graphs, split graphs, chordal graphs, and
cographs. We prove that orderability is decidable for the so called
HR-equational classes of graphs, which are described by equation systems and
generalize the context-free languages
Scheduling Monotone Moldable Jobs in Linear Time
A moldable job is a job that can be executed on an arbitrary number of
processors, and whose processing time depends on the number of processors
allotted to it. A moldable job is monotone if its work doesn't decrease for an
increasing number of allotted processors. We consider the problem of scheduling
monotone moldable jobs to minimize the makespan.
We argue that for certain compact input encodings a polynomial algorithm has
a running time polynomial in n and log(m), where n is the number of jobs and m
is the number of machines. We describe how monotony of jobs can be used to
counteract the increased problem complexity that arises from compact encodings,
and give tight bounds on the approximability of the problem with compact
encoding: it is NP-hard to solve optimally, but admits a PTAS.
The main focus of this work are efficient approximation algorithms. We
describe different techniques to exploit the monotony of the jobs for better
running times, and present a (3/2+{\epsilon})-approximate algorithm whose
running time is polynomial in log(m) and 1/{\epsilon}, and only linear in the
number n of jobs
Unifying mesh- and tree-based programmable interconnect
We examine the traditional, symmetric, Manhattan mesh design for field-programmable gate-array (FPGA) routing along with tree-of-meshes (ToM) and mesh-of-trees (MoT) based designs. All three networks can provide general routing for limited bisection designs (Rent's rule with p<1) and allow locality exploitation. They differ in their detailed topology and use of hierarchy. We show that all three have the same asymptotic wiring requirements. We bound this tightly by providing constructive mappings between routes in one network and routes in another. For example, we show that a (c,p) MoT design can be mapped to a (2c,p) linear population ToM and introduce a corner turn scheme which will make it possible to perform the reverse mapping from any (c,p) linear population ToM to a (2c,p) MoT augmented with a particular set of corner turn switches. One consequence of this latter mapping is a multilayer layout strategy for N-node, linear population ToM designs that requires only /spl Theta/(N) two-dimensional area for any p when given sufficient wiring layers. We further show upper and lower bounds for global mesh routes based on recursive bisection width and show these are within a constant factor of each other and within a constant factor of MoT and ToM layout area. In the process we identify the parameters and characteristics which make the networks different, making it clear there is a unified design continuum in which these networks are simply particular regions
Computing with and without arbitrary large numbers
In the study of random access machines (RAMs) it has been shown that the
availability of an extra input integer, having no special properties other than
being sufficiently large, is enough to reduce the computational complexity of
some problems. However, this has only been shown so far for specific problems.
We provide a characterization of the power of such extra inputs for general
problems. To do so, we first correct a classical result by Simon and Szegedy
(1992) as well as one by Simon (1981). In the former we show mistakes in the
proof and correct these by an entirely new construction, with no great change
to the results. In the latter, the original proof direction stands with only
minor modifications, but the new results are far stronger than those of Simon
(1981). In both cases, the new constructions provide the theoretical tools
required to characterize the power of arbitrary large numbers.Comment: 12 pages (main text) + 30 pages (appendices), 1 figure. Extended
abstract. The full paper was presented at TAMC 2013. (Reference given is for
the paper version, as it appears in the proceedings.
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