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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.
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The German Tank Problem with Multiple Factories
During the Second World War, estimates of the number of tanks deployed by Germany were critically needed. The Allies adopted a successful statistical approach to estimate this information: assuming that the tanks are sequentially numbered starting from 1, if we observe k tanks from an unknown total of N, then the best linear unbiased estimator for N is M(1+1/k)-1 where M is the maximum observed serial number. However, in many situations, the original German Tank Problem is insufficient, since typically there are l > 1 factories, and tanks produced by different factories may have serial numbers in disjoint ranges that are often far separated.Clark, Gonye and Miller presented an unbiased estimator for N when the minimum serial number is unknown. Provided one identifies which samples correspond to which factory, one can then estimate each factory's range and summing the sizes of these ranges yields an estimate for the rival's total productivity. We construct an efficient procedure to estimate the total productivity and prove that it is effective when log l/log k is sufficiently small. In the final section, we show that given information about the gaps, we can make an estimator that performs orders of magnitude better when we have a small number of samples.
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- Award ID(s):
- 2241623
- PAR ID:
- 10580624
- Publisher / Repository:
- PUMP
- Date Published:
- Journal Name:
- The PUMP Journal of Undergraduate Research
- Volume:
- 7
- ISSN:
- 2765-8724
- Page Range / eLocation ID:
- 231 to 257
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.46787/pump.v7i0.4249
