A point-disjoint path cover of a directed graph is a collection of point-disjoint paths (some paths possibly having zero length) which covers all the points. A path cover which minimizes the number of paths corresponds to an optimal sequence of the steps of a computer program for efficient coding and documentation. The minimization problem for the general directed graph is hard in the sense of being NP-complete. In the case of cycle-free digraphs, however, the problem is polynomial, for it is shown that it can be reduced to the maximum-matching problem. A heuristic given here for finding a near optimal path cover for the general case is based upon applying the maximum-matching algorithm to the subgraphs of an interval decomposition.

Although machine learning (ML) has shown promise across disciplines, out-of-sample generalizability is concerning. This is currently addressed by sharing multi-site data, but such centralization is challenging/infeasible to scale due to various limitations. Federated ML (FL) provides an alternative paradigm for accurate and generalizable ML, by only sharing numerical model updates. Here we present the largest FL study to-date, involving data from 71 sites across 6 continents, to generate an automatic tumor boundary detector for the rare disease of glioblastoma, reporting the largest such dataset in the literature (n = 6, 314). We demonstrate a 33% delineation improvement for the surgically targetable tumor, and 23% for the complete tumor extent, over a publicly trained model. We anticipate our study to: 1) enable more healthcare studies informed by large diverse data, ensuring meaningful results for rare diseases and underrepresented populations, 2) facilitate further analyses for glioblastoma by releasing our consensus model, and 3) demonstrate the FL effectiveness at such scale and task-complexity as a paradigm shift for multi-site collaborations, alleviating the need for data-sharing.

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The notion of a discrete pattern is formalized and certain properties deduced. A pattern is shown to be a generalization of a formal language. Algorithms for implementing the kinds of patterns in SNOBOL4 are given. The general approach is to create, in-so-far as possible, a bottom-up parse from a top-down specification.