Finding the Most Relevant Fragments in Networks

Abstract

We study a point pattern detection problem on networks, motivated by applications in geographical analysis, such as crime hotspot detection. Given a network N (a connected graph with non-negative edge lengths) together with a set of sites, which lie on the edges or vertices of N , we look for a connected subnetwork F of N of small total length that contains many sites. The edges of F can form parts of the edges of N .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 or a tree. We give polynomial-time algorithms, NP-hardness and NP-completeness proofs, approximation algorithms, and also fixed-parameter tractable algorithms.

Lipschitz Unimodal and Isotonic Regression on Paths and Trees

“…We anonymous reviewers for many helpful remarks that helped improve the presentation of this paper and we thank Günter Rote for suggesting the improvement in the analysis of T(F + G), similar to his paper [10].…”

Significant Linear Hotspot Discovery

“…There are a myriad of other techniques that divide data into groups without considering statistical significance. These include DBScan [16], K-Means [17], KMR [18], and Maximum Subgraph Finding [19]. For example, the algorithm from our previous work [18] on summarizing activities using routes may return routes that are not statistically significant.…”

“…In this work, it is assumed that the number of activities on the road network is fixed and does not change over time. We do not consider techniques that do not employ statistical significance (e.g., DBScan [16], K-Means [17], KMR [18], and Maximum Subgraph Finding [19]). This paper only enumerates shortest paths rather than all possible paths.…”