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We study subtrajectory clustering under the Fréchet distance. Given one or more trajectories, the task is to split the trajectories into several parts, such that the parts have a good clustering structure. We approach this problem via a new set cover formulation, which we think provides a natural formalization of the problem as it is studied in many applications. Given a polygonal curve P with n vertices in fixed dimension, integers k, ℓ ≥ 1, and a real value Δ > 0, the goal is to find k center curves of complexity at most ℓ such that every point on P is covered by a subtrajectory that has small Fréchet distance to one of the k center curves (≤ Δ). In many application scenarios, one is interested in finding clusters of small complexity, which is controlled by the parameter ℓ. Our main result is a bicriterial approximation algorithm: if there exists a solution for given parameters k, ℓ, and Δ, then our algorithm finds a set of k' center curves of complexity at most ℓ with covering radius Δ' with k' in O(kℓ2 log (kℓ)), and Δ' ≤ 19Δ. Moreover, within these approximation bounds, we can minimize k while keeping the other parameters fixed. If ℓ is a constant independent of n, then, the approximation factor for the number of clusters k is O(log k) and the approximation factor for the radius Δ is constant. In this case, the algorithm has expected running time in Õ(km2 + mn) and uses space in O(n + m), where m=⌈L/Δ⌉ and L is the total arclength of the curve P.more » « less

Abe, S ; Abhir, J ; Acciari, V A ; Agudo, I ; Aniello, T ; Ansoldi, S ; Antonelli, L A ; ArbetEngels, A ; Arcaro, C ; Artero, M ; et al ( , Astronomy & Astrophysics)
Aims. We have performed the first broadband study of Mrk 421 from radio to TeV gamma rays with simultaneous measurements of the Xray polarization from IXPE.Methods. The data were collected as part of an extensive multiwavelength campaign carried out between May and June 2022 using MAGIC,Fermi LAT,NuSTAR ,XMMNewton ,Swift , and several optical and radio telescopes to complement IXPE data.Results. During the IXPE exposures, the measured 0.2–1 TeV flux was close to the quiescent state and ranged from 25% to 50% of the Crab Nebula without intranight variability. Throughout the campaign, the very highenergy (VHE) and Xray emission are positively correlated at a 4σ significance level. The IXPE measurements reveal an Xray polarization degree that is a factor of 2–5 higher than in the optical/radio bands; that implies an energystratified jet in which the VHE photons are emitted cospatially with the Xrays, in the vicinity of a shock front. The June 2022 observations exhibit a rotation of the Xray polarization angle. Despite no simultaneous VHE coverage being available during a large fraction of the swing, theSwift XRT monitoring reveals an Xray flux increase with a clear spectral hardening. This suggests that flares in high synchrotron peaked blazars can be accompanied by a polarization angle rotation, as observed in some flat spectrum radio quasars. Finally, during the polarization angle rotation,NuSTAR data reveal two contiguous spectral hysteresis loops in opposite directions (clockwise and counterclockwise), implying important changes in the particle acceleration efficiency on approximately hour timescales.Free, publiclyaccessible full text available April 1, 2025