The harmonious coloring number of a graph

AbstractHopcroft and Krishnamoorthy (1983) have shown that the harmonious coloring problem is NP-complete, introducing the notion of a harmonious coloring of a graph as being a vertex coloring for which no two edges receive the same color-pair. In this report we construct efficient harmonious colorings of complete binary trees, 2 and 3-dimensional grids, and n-dimensional cubes

Expansion of layouts of complete binary trees into grids

AbstractLet Th be the complete binary tree of height h. Let M be the infinite grid graph with vertex set Z2, where two vertices (x1,y1) and (x2,y2) of M are adjacent if and only if |x1−x2|+|y1−y2|=1. Suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansion ratio n(T)/n(Th)=n(T)/(2h+1−1) is bounded below by 1.122 for h sufficiently large. That is, we give bounds on how many vertices of degree 2 must be inserted along the edges of Th in order that the resulting tree can be laid out in the grid. Concerning the constructive end of VLSI design, suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of the n×n grid graph. Define the expansion ratio of such a layout to be n2/n(Th)=n2/(2h+1−1). We show constructively that the minimum possible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h. That is, we give efficient layouts of complete binary trees into square grids, making improvements upon the previous work of others. We also give bounds for the point expansion and expansion problems for layouts of Th into extended grids, i.e. grids with added diagonals

OpenMx 2.0:Extended Structural Equation and Statistical Modeling

The new software package OpenMx 2.0 for structural equation and other statistical modeling is introduced and its features are described. OpenMx is evolving in a modular direction and now allows a mix-and-match computational approach that separates model expectations from fit functions and optimizers. Major backend architectural improvements include a move to swappable open-source optimizers such as the newly-written CSOLNP. Entire new methodologies such as Item Factor analysis (IRT) and State-space modeling have been implemented. New model expectation functions including support for the expression of models in LISREL syntax and a simplified multigroup expectation function are available. Ease-of-use improvements include helper functions to standardize model parameters and compute their Jacobian-based standard errors, access to model components through standard R $ mechanisms, and improved tab completion from within the R Graphical User Interface

Maintained Individual Data Distributed Likelihood Estimation (MIDDLE)

Maintained Individual Data Distributed Likelihood Estimation (MIDDLE) is a novel paradigm for research in the behavioral, social, and health sciences. The MIDDLE approach is based on the seemingly-impossible idea that data can be privately maintained by participants and never revealed to researchers, while still enabling statistical models to be fit and scientific hypotheses tested. MIDDLE rests on the assumption that participant data should belong to, be controlled by, and remain in the possession of the participants themselves. Distributed likelihood estimation refers to fitting statistical models by sending an objective function and vector of parameters to each participants' personal device (e.g., smartphone, tablet, computer), where the likelihood of that individual's data is calculated locally. Only the likelihood value is returned to the central optimizer. The optimizer aggregates likelihood values from responding participants and chooses new vectors of parameters until the model converges. A MIDDLE study provides significantly greater privacy for participants, automatic management of optin and opt-out consent, lower cost for the researcher and funding institute, and faster determination of results. Furthermore, if a participant opts into several studies simultaneously and opts into data sharing, these studies automatically have access to individual-level longitudinal data linked across all studies