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Chemical tagging of stellar debris from disrupted open clusters and associations underpins the science cases for next-generation multi-object spectroscopic surveys. As part of the Galactic Archaeology project TraCD (Tracking Cluster Debris), a preliminary attempt at reconstructing the birth clouds of now phase-mixed thin disk debris is undertaken using a parametric minimum spanning tree (MST) approach. Empirically-motivated chemical abundance pattern uncertainties (for a 10-dimensional chemistry-space) are applied to NBODY6-realised stellar associations dissolved into a background sea of field stars, all evolving in a Milky Way potential. We demonstrate that significant population reconstruction degeneracies appear when the abundance uncertainties approach 0.1 dex and the parameterised MST approach is employed; more sophisticated methodologies will be required to ameliorate these degeneracies.
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The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper, we present two new distributed approximation algorithms for the MDST problem. Our first result is a randomized distributed algorithm that constructs a spanning tree of maximum degree $hat d = O(dlog{n})$. It requires $O((D + sqrt{n}) log^2 n)$ rounds (w.h.p.), where $D$ is the graph diameter, which matches (within log factors) the optimal round complexity for the related minimum spanning tree problem. Our second result refines this approximation factor by constructing a tree with maximum degree $hat d = O(d + log{n})$, though at the cost of additional polylogarithmic factors in the round complexity. Although efficient approximation algorithms for the MDST problem have been known in the sequential setting since the 1990s, our results are first efficient distributed solutions for this problem.
The minimum spanning tree (MST), a graph constructed from a distribution of points, draws lines between pairs of points so that all points are linked in a single skeletal structure that contains no loops and has minimal total edge length. The MST has been used in a broad range of scientific fields such as particle physics (to distinguish classes of events in collider collisions), in astronomy (to detect mass segregation in star clusters) and cosmology (to search for filaments in the cosmic web). Its success in these fields has been driven by its sensitivity to the spatial distribution of points and the patterns within. MiSTree, a public Python package, allows a user to construct the MST in a variety of coordinates systems, including Celestial coordinates used in astronomy. The package enables the MST to be constructed quickly by initially using a k-nearest neighbour graph (kNN, rather than a matrix of pairwise distances) which is then fed to Kruskals algorithm to construct the MST. MiSTree enables a user to measure the statistics of the MST and provides classes for binning the MST statistics (into histograms) and plotting the distributions. Applying the MST will enable the inclusion of high-order statistics information from the cosmic web which can provide additional information to improve cosmological parameter constraints. This information has not been fully exploited due to the computational cost of calculating N-point statistics. MiSTree was designed to be used in cosmology but could be used in any field which requires extracting non-Gaussian information from point distributions. The source code for MiSTree is available on GitHub at https://github.com/knaidoo29/mistree
A distributed proof (also known as local certification, or proof-labeling scheme) is a mechanism to certify that the solution to a graph problem is correct. It takes the form of an assignment of labels to the nodes, that can be checked locally. There exists such a proof for the minimum spanning tree problem, using $O(log n log W)$ bit labels (where $n$ is the number of nodes in the graph, and $W$ is the largest weight of an edge). This is due to Korman and Kutten who describe it in concise and formal manner in [Korman and Kutten 07]. In this note, we propose a more intuitive description of the result, as well as a gentle introduction to the problem.
A complete understanding of real networks requires us to understand the consequences of the uneven interaction strengths between a systems components. Here we use the minimum spanning tree (MST) to explore the effect of weight assignment and network topology on the organization of complex networks. We find that if the weight distribution is correlated with the network topology, the MSTs are either scale-free or exponential. In contrast, when the correlations between weights and topology are absent, the MST degree distribution is a power-law and independent of the weight distribution. These results offer a systematic way to explore the impact of weak links on the structure and integrity of complex networks.
I present an overview of the science goals and achievements of ongoing spectroscopic surveys of individual stars in the nearby Universe. I include a brief discussion of the development of the field of Galactic Archaeology - using the fossil record in old stars nearby to infer how our Galaxy evolved and place the Milky Way in cosmological context.
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