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Abstract Graphs in metric spaces appear in a wide range of data sets, and there is a large body of work focused on comparing, matching, or analyzing collections of graphs in different ambient spaces. In this survey, we provide an overview of a diverse collection of distance measures that can be defined on the set of finite graphs immersed (and in some cases, embedded) in a metric space. For each of the distance measures, we recall their definitions and investigate which of the properties of a metric they satisfy. Furthermore we compare the distance measures based on these properties and discuss their computational complexity. 

Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than 100x on the benchmark datasets while maintaining an error rate below 0.1%. 

Comparing two road maps is a basic operation that arises in a variety of situations. A map comparison method that is commonly used, mainly in the context of comparing reconstructed maps to ground truth maps, is based ongraph sampling. The essential idea is to first compute a set of point samples on each map, and then to match pairs of samples—one from each map—in a one-to-one fashion. For deciding whether two samples can be matched, different criteria, e.g., based on distance or orientation, can be used. The total number of matched pairs gives a measure of how similar the maps are. Since the work of Biagioni and Eriksson [11, 12], graph sampling methods have become widely used. However, there are different ways to implement each of the steps, which can lead to significant differences in the results. This means that conclusions drawn from different studies that seemingly use the same comparison method, cannot necessarily be compared. In this work we present a unified approach to graph sampling for map comparison. We present the method in full generality, discussing the main decisions involved in its implementation. In particular, we point out the importance of the sampling method (GEO vs. TOPO) and that of the matching definition, discussing the main options used, and proposing alternatives for both key steps. We experimentally evaluate the different sampling and matching options considered on map datasets and reconstructed maps. Furthermore, we provide a code base and an interactive visualization tool to set a standard for future evaluations in the field of map construction and map comparison. 

We present a series of nine Computational Geometry Concept Videos, available on Youtube. The videos are aimed at a general audience and introduce concepts ranging from closest and farthest pairs to data structures for range searching and for point location. The video series grew out of the development of an online graduate course on computational geometry, and the beginning portions of the videos are used in the course to motivate the concept and to tie it to a "real" problem in New Orleans. Thus our videos serve a dual purpose of outreach and education.