We study a point pattern detection problem on networks, motivated by geographical analysis tasks, such as crime hotspot detection. Given a network ▫$N$▫ (for example, a street, train, or highway ...network) together with a set of sites which are located on the network (for example, accident locations or crime scenes), we want to find a connected subnetwork ▫$F$▫ of ▫$N$▫ of small total length that contains many sites. That is, we are searching for a subnetwork ▫$F$▫ that spans a cluster of sites which are close with respect to the network distance. We consider different variants of this problem where ▫$N$▫ is either a general graph or restricted to a tree, and the subnetwork ▫$F$▫ that we are looking for is either a simple path, a path with self-intersections at vertices, or a tree. Many of these variants are NP-hard, that is, polynomial-time solutions are very unlikely to exist. Hence we focus on exact algorithms for special cases and efficient algorithms for the general case under realistic input assumptions.
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