A region \(\mathcal {R} \) is a dwell region for a moving object O if, given a threshold distance r q and duration τ q , every point of \(\mathcal {R} \) remains within distance r q from O for at least time τ q . Points within \(\mathcal {R} \) are likely to be of interest to O , so identification of dwell regions has applications such as monitoring and surveillance. We first present a logarithmic-time online algorithm to find dwell regions in an incoming stream of object positions. Our method maintains the upper and lower bounds for the radius of the smallest circle enclosing the object positions, thereby greatly reducing the number of trajectory points needed to evaluate the query. It approximates the radius of the smallest circle enclosing a given subtrajectory within an arbitrarily small user-defined factor, and is also able to efficiently answer decision queries asking whether or not a dwell region exists. For the offline version of the dwell region problem, we first extend our online approach to develop the ρ -Index, which indexes subtrajectories using query radius ranges. We then refine this approach to obtain the τ -Index, which indexes subtrajectories using both query radius ranges and dwell durations. Our experiments using both real-world and synthetic datasets show that the online approach can scale up to hundreds of thousands of moving objects. For archived trajectories, our indexing approaches speed up queries by many orders of magnitude.
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Index-based, High-dimensional, Cosine Threshold Querying with Optimality Guarantees
Given a database of vectors, a cosine threshold query returns all vectors in the database having cosine similarity to a query vector above a given threshold {\theta}. These queries arise naturally in many applications, such as document retrieval, image search, and mass spectrometry. The present paper considers the efficient evaluation of such queries, providing novel optimality guarantees and exhibiting good performance on real datasets. We take as a starting point Fagin's well-known Threshold Algorithm (TA), which can be used to answer cosine threshold queries as follows: an inverted index is first built from the database vectors during pre-processing; at query time, the algorithm traverses the index partially to gather a set of candidate vectors to be later verified for {\theta}-similarity. However, directly applying TA in its raw form misses significant optimization opportunities. Indeed, we first show that one can take advantage of the fact that the vectors can be assumed to be normalized, to obtain an improved, tight stopping condition for index traversal and to efficiently compute it incrementally. Then we show that one can take advantage of data skewness to obtain better traversal strategies. In particular, we show a novel traversal strategy that exploits a common data skewness condition which holds in multiple domains including mass spectrometry, documents, and image databases. We show that under the skewness assumption, the new traversal strategy has a strong, near-optimal performance guarantee. The techniques developed in the paper are quite general since they can be applied to a large class of similarity functions beyond cosine.
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- Award ID(s):
- 1759980
- NSF-PAR ID:
- 10109805
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 127
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 11:1-11:20
- 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
- Conference Paper:
- https://doi.org/https://doi.org/10.4230/LIPIcs.ICDT.2019.11
