Isoperimetric Graph Partitioning for Data Clustering and Image Segmentation

Spectral methods of graph partitioning have been shown to provide a powerful approach to the image segmentation problem. In this paper, we adopt a different approach, based on estimating the isoperimetric constant of an image graph. Our algorithm produces the high quality segmentations and data clustering of spectral methods, but with improved speed and stability.

Isoperimetric Partitioning: A New Algorithm for Graph Partitioning

Temporal structure is skilled, fluent action exists at several nested levels. At the largest scale considered here, short sequences of actions that are planned collectively in prefronatal cortex appear to be queued for performance by a cyclic competitive process that operates in concert with a parallel analog representation that implicitly specifies the relative priority of elements of the sequence. At an intermediate scale, single acts, like reaching to grasp, depend on coordinated scaling of the rates at which many muscles shorten or lengthen in parallel. To ensure success of acts such as catching an approaching ball, such parallel rate scaling, which appears to be one function of the basal ganglia, must be coupled to perceptual variables such as time-to-contact. At a finer scale, within each act, desired rate scaling can be realized only if precisely timed muscle activations first accelerate and then decelerate the limbs, to ensure that muscle length changes do not under- or over- shoot the amounts needed for precise acts. Each context of action may require a different timed muscle activation pattern than similar contexts. Because context differences that require different treatment cannot be known in advance, a formidable adaptive engine-the cerebellum-is needed to amplify differences within, and continuosly search, a vast parallel signal flow, in order to discover contextual "leading indicators" of when to generate distinctive patterns of analog signals. From some parts of the cerebellum, such signals control muscles. But a recent model shows how the lateral cerebellum may serve the competitive queuing system (frontal cortex) as a repository of quickly accessed long-term sequence memories. Thus different parts of the cerebellum may use the same adaptive engine design to serve the lowest and highest of the three levels of temporal structure treated. If so, no one-to-one mapping exists between leveels of temporal structure and major parts of the brain. Finally, recent data cast doubt on network-delay models of cerebellar adaptive timing.

The Graph Analysis Toolbox: Image Processing on Arbitrary Graphs

Office of Naval Research (N00014-01-1-0624)

Excitation of highly conjugated (porphinato)palladium(II) and (porphinato)platinum(II) oligomers produces long-lived, triplet states at unit quantum yield that absorb strongly over broad spectral domains of the NIR.

Transient dynamical studies of bis[(5,5'-10,20-bis(2,6-bis(3,3-dimethylbutoxy)phenyl)porphinato)palladium(II)]ethyne (PPd(2)), 5,15-bis{[(5'-10,20-bis(2,6-bis(3,3-dimethylbutoxy)phenyl)porphinato)palladium(II)]ethynyl}(10,20-bis(2,6-bis(3,3-dimethylbutoxy)phenyl)porphinato)palladium(II) (PPd(3)), bis[(5,5'-10,20-bis(2,6-bis(3,3-dimethylbutoxy)phenyl)porphinato)platinum(II)]ethyne (PPt(2)), and 5,15-bis{[(5'-10,20-bis(2,6-bis(3,3-dimethylbutoxy)phenyl)porphinato)platinum(II)]ethynyl}(10,20-bis(2,6-bis(3,3-dimethylbutoxy)phenyl)porphinato)platinum(II) (PPt(3)) show that the electronically excited triplet states of these highly conjugated supermolecular chromophores can be produced at unit quantum yield via fast S(1) → T(1) intersystem crossing dynamics (τ(isc): 5.2-49.4 ps). These species manifest high oscillator strength T(1) → T(n) transitions over broad NIR spectral windows. The facts that (i) the electronically excited triplet lifetimes of these PPd(n) and PPt(n) chromophores are long, ranging from 5 to 50 μs, and (ii) the ground and electronically excited absorptive manifolds of these multipigment ensembles can be extensively modulated over broad spectral domains indicate that these structures define a new precedent for conjugated materials featuring low-lying π-π* electronically excited states for NIR optical limiting and related long-wavelength nonlinear optical (NLO) applications.

El tronco de dipirámide regular octagonal afín recta

A partir de tres vectores linealmente independientes en R3 , y bajo otras condiciones, se construye una norma ' sobre R3 cuyas esferas de centro G y radio r > 0, son troncos de dipirámide regular octagonal afín recta de centro G. También, dado un poliedro F de este tipo, se establece que F, es también un cuerpo normado, respecto a esa norma ' construida a partir de F. La representación unificada de ' permite el estudio riguroso y versátil de la estructura geométrica de F, asistida por la noción de homotecia.

Parallel dynamic graph-partitioning for unstructured meshes

A parallel method for the dynamic partitioning of unstructured meshes is described. The method introduces a new iterative optimization technique known as relative gain optimization which both balances the workload and attempts to minimize the interprocessor communications overhead. Experiments on a series of adaptively refined meshes indicate that the algorithm provides partitions of an equivalent or higher quality to static partitioners (which do not reuse the existing partition) and much more rapidly. Perhaps more importantly, the algorithm results in only a small fraction of the amount of data migration compared to the static partitioners.

Multilevel mesh partitioning for optimizing domain shape

Multilevel algorithms are a successful class of optimization techniques that address the mesh partitioning problem for mapping meshes onto parallel computers. They usually combine a graph contraction algorithm together with a local optimization method that refines the partition at each graph level. To date, these algorithms have been used almost exclusively to minimize the cut-edge weight in the graph with the aim of minimizing the parallel communication overhead. However, it has been shown that for certain classes of problems, the convergence of the underlying solution algorithm is strongly influenced by the shape or aspect ratio of the subdomains. Therefore, in this paper, the authors modify the multilevel algorithms to optimize a cost function based on the aspect ratio. Several variants of the algorithms are tested and shown to provide excellent results.

Exact colorations of graphs and digraphs

A coloration is an exact regular coloration if whenever two vertices are colored the same they have identically colored neighborhoods. For example, if one of the two vertices that are colored the same is connected to three yellow vertices, two white and red, then the other vertex is as well. Exact regular colorations have been discussed informally in the social network literature. However they have been part of the mathematical literature for some time, though in a different format. We explore this concept in terms of social networks and illustrate some important results taken from the mathematical literature. In addition we show how the concept can be extended to ecological and perfect colorations, and discuss how the CATREGE algorithm can be extended to find the maximal exact regular coloration of a graph.

Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Multilevel mesh partitioning for optimising aspect ratio

Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. To date these algorithms have been used almost exclusively to minimise the cut-edge weight, however it has been shown that for certain classes of solution algorithm, the convergence of the solver is strongly influenced by the subdomain aspect ratio. In this paper therefore, we modify the multilevel algorithms in order to optimise a cost function based on aspect ratio. Several variants of the algorithms are tested and shown to provide excellent results.

A multilevel algorithm for force-directed graph drawing

We describe a heuristic method for drawing graphs which uses a multilevel technique combined with a force-directed placement algorithm. The multilevel process groups vertices to form clusters, uses the clusters to define a new graph and is repeated until the graph size falls below some threshold. The coarsest graph is then given an initial layout and the layout is successively refined on all the graphs starting with the coarsest and ending with the original. In this way the multilevel algorithm both accelerates and gives a more global quality to the force- directed placement. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on a number of examples ranging from 500 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 30 seconds for a 10,000 vertex graph to around 10 minutes for the largest graph. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

Computing regular equivalence: practical and theoretical issues

Social network analysts have tried to capture the idea of social role explicitly by proposing a framework that precisely gives conditions under which group actors are playing equivalent roles. They term these methods positional analysis techniques. The most general definition is regular equivalence which captures the idea that equivalent actors are related in a similar way to equivalent alters. Regular equivalence gives rise to a whole class of partitions on a network. Given a network we have two different computational problems. The first is how to find a particular regular equivalence. An algorithm exists to find the largest regular partition but there are not efficient algorithms to test whether there is a regular k-partition. That is a partition in k groups that is regular. In addition, when dealing with real data, it is unlikely that any regular partitions exist. To overcome this problem relaxations of regular equivalence have been proposed along with optimisation techniques to find nearly regular partitions. In this paper we review the algorithms that have developed to find particular regular equivalences and look at some of the recent theoretical results which give an insight into the complexity of finding regular partitions.