Clustering and visualizing high-dimensional (HD) data are important tasks in a variety of fields. For example, in bioinformatics, they are crucial for analyses of single-cell data such as mass cytometry (CyTOF) data. Some of the most effective algorithms for clustering HD data are based on representing the data by nodes in a graph, with edges connecting neighbouring nodes according to some measure of similarity or distance. However, users of graph-based algorithms are typically faced with the critical but challenging task of choosing the value of an input parameter that sets the size of neighbourhoods in the graph, e.g. the number of nearest neighbours to which to connect each node or a threshold distance for connecting nodes. The burden on the user could be alleviated by a measure of inter-node similarity that can have value 0 for dissimilar nodes without requiring any user-defined parameters or thresholds. This would determine the neighbourhoods automatically while still yielding a sparse graph. To this end, I propose a new method called ASTRICS to measure similarity between clusters of HD data points based on local dimensionality reduction and triangulation of critical alpha shapes. I show that my ASTRICS similarity measure can facilitate both clustering and visualization of HD data by using it in Stage 2 of a three-stage pipeline: Stage 1 = perform an initial clustering of the data by any method; Stage 2 = let graph nodes represent initial clusters instead of individual data points and use ASTRICS to automatically define edges between nodes; Stage 3 = use the graph for further clustering and visualization. This trades the critical task of choosing a graph neighbourhood size for the easier task of essentially choosing a resolution at which to view the data. The graph and consequently downstream clustering and visualization are then automatically adapted to the chosen resolution.