1,844 research outputs found
Optimizing Constrained Subtrees of Trees
Given a tree G = (V, E) and a weight function defined on subsets of its nodes, we consider two associated problems. The first, called the "rooted subtree problem", is to find a maximum weight subtree, with a specified root, from a given set of subtrees. The second problem, called "the subtree packing problem", is to find a maximum weight packing of node disjoint subtrees chosen from a given set of subtrees, where the value of each subtree may depend on its root. We show that the complexity status of both problems is related, and that the subtree packing problem is polynomial if and only if each rooted subtree problem is polynomial. In addition we show that the convex hulls of the feasible solutions to both problems are related: the convex hull of solutions to the packing problem is given by "pasting together" the convex hulls of the rooted subtree problems. We examine in detail the case where the set of feasible subtrees rooted at node i consists of all subtrees with at most k nodes. For this case we derive valid inequalities, and specify the convex hull when k < 4
Asymmetries arising from the space-filling nature of vascular networks
Cardiovascular networks span the body by branching across many generations of
vessels. The resulting structure delivers blood over long distances to supply
all cells with oxygen via the relatively short-range process of diffusion at
the capillary level. The structural features of the network that accomplish
this density and ubiquity of capillaries are often called space-filling. There
are multiple strategies to fill a space, but some strategies do not lead to
biologically adaptive structures by requiring too much construction material or
space, delivering resources too slowly, or using too much power to move blood
through the system. We empirically measure the structure of real networks (18
humans and 1 mouse) and compare these observations with predictions of model
networks that are space-filling and constrained by a few guiding biological
principles. We devise a numerical method that enables the investigation of
space-filling strategies and determination of which biological principles
influence network structure. Optimization for only a single principle creates
unrealistic networks that represent an extreme limit of the possible structures
that could be observed in nature. We first study these extreme limits for two
competing principles, minimal total material and minimal path lengths. We
combine these two principles and enforce various thresholds for balance in the
network hierarchy, which provides a novel approach that highlights the
trade-offs faced by biological networks and yields predictions that better
match our empirical data.Comment: 17 pages, 15 figure
Lossless and near-lossless source coding for multiple access networks
A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences {X-i}(i=1)(infinity), and {Y-i}(i=1)(infinity) is drawn independent and identically distributed (i.i.d.) according to joint probability mass function (p.m.f.) p(x, y); the encoder for each source operates without knowledge of the other source; the decoder jointly decodes the encoded bit streams from both sources. The work of Slepian and Wolf describes all rates achievable by MASCs of infinite coding dimension (n --> infinity) and asymptotically negligible error probabilities (P-e((n)) --> 0). In this paper, we consider the properties of optimal instantaneous MASCs with finite coding dimension (n 0) performance. The interest in near-lossless codes is inspired by the discontinuity in the limiting rate region at P-e((n)) = 0 and the resulting performance benefits achievable by using near-lossless MASCs as entropy codes within lossy MASCs. Our central results include generalizations of Huffman and arithmetic codes to the MASC framework for arbitrary p(x, y), n, and P-e((n)) and polynomial-time design algorithms that approximate these optimal solutions
A Constrained Sequential-Lamination Algorithm for the Simulation of Sub-Grid Microstructure in Martensitic Materials
We present a practical algorithm for partially relaxing multiwell energy
densities such as pertain to materials undergoing martensitic phase
transitions. The algorithm is based on sequential lamination, but the evolution
of the microstructure during a deformation process is required to satisfy a
continuity constraint, in the sense that the new microstructure should be
reachable from the preceding one by a combination of branching and pruning
operations. All microstructures generated by the algorithm are in static and
configurational equilibrium. Owing to the continuity constrained imposed upon
the microstructural evolution, the predicted material behavior may be
path-dependent and exhibit hysteresis. In cases in which there is a strict
separation of micro and macrostructural lengthscales, the proposed relaxation
algorithm may effectively be integrated into macroscopic finite-element
calculations at the subgrid level. We demonstrate this aspect of the algorithm
by means of a numerical example concerned with the indentation of an Cu-Al-Ni
shape memory alloy by a spherical indenter.Comment: 27 pages with 9 figures. To appear in: Computer Methods in Applied
Mechanics and Engineering. New version incorporates minor revisions from
revie
SHADHO: Massively Scalable Hardware-Aware Distributed Hyperparameter Optimization
Computer vision is experiencing an AI renaissance, in which machine learning
models are expediting important breakthroughs in academic research and
commercial applications. Effectively training these models, however, is not
trivial due in part to hyperparameters: user-configured values that control a
model's ability to learn from data. Existing hyperparameter optimization
methods are highly parallel but make no effort to balance the search across
heterogeneous hardware or to prioritize searching high-impact spaces. In this
paper, we introduce a framework for massively Scalable Hardware-Aware
Distributed Hyperparameter Optimization (SHADHO). Our framework calculates the
relative complexity of each search space and monitors performance on the
learning task over all trials. These metrics are then used as heuristics to
assign hyperparameters to distributed workers based on their hardware. We first
demonstrate that our framework achieves double the throughput of a standard
distributed hyperparameter optimization framework by optimizing SVM for MNIST
using 150 distributed workers. We then conduct model search with SHADHO over
the course of one week using 74 GPUs across two compute clusters to optimize
U-Net for a cell segmentation task, discovering 515 models that achieve a lower
validation loss than standard U-Net.Comment: 10 pages, 6 figure
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