Abstract The graph edit distance is an intuitive measure to quantify the dissimilarity of graphs, but its computation is $$\mathsf {NP}$$ NP -hard and challenging in practice. We introduce methods for answering nearest neighbor and range queries regarding this distance efficiently for large databases with up to millions of graphs. We build on the filter-verification paradigm, where lower and upper bounds are used to reduce the number of exact computations of the graph edit distance. Highly effective bounds for this involve solving a linear assignment problem for each graph in the database, which is prohibitive in massive datasets. Index-based approaches typically provide only weak bounds leading to high computational costs verification. In this work, we derive novel lower bounds for efficient filtering from restricted assignment problems, where the cost function is a tree metric. This special case allows embedding the costs of optimal assignments isometrically into $$\ell _1$$ ℓ 1 space, rendering efficient indexing possible. We propose several lower bounds of the graph edit distance obtained from tree metrics reflecting the edit costs, which are combined for effective filtering. Our method termed EmbAssi can be integrated into existing filter-verification pipelines as a fast and effective pre-filtering step. Empirically we show that for many real-world graphs our lower bounds are already close to the exact graph edit distance, while our index construction and search scales to very large databases.

Finding an optimal assignment between two sets of objects is a fundamental problem arising in many applications, including the matching of 'bag-of-words' representations in natural language processing and computer vision. Solving the assignment problem typically requires cubic time and its pairwise computation is expensive on large datasets. In this paper, we develop an algorithm which can find an optimal assignment in linear time when the cost function between objects is represented by a tree distance. We employ the method to approximate the edit distance between two graphs by matching their vertices in linear time. To this end, we propose two tree distances, the first of which reflects discrete and structural differences between vertices, and the second of which can be used to compare continuous labels. We verify the effectiveness and efficiency of our methods using synthetic and real-world datasets.