On the complexity of range searching among curves

Abstract

Modern tracking technology has made the collection of large numbers of densely sampled trajectories of moving objects widely available. We consider a fundamental problem encountered when analysing such data: Given n polygonal curves S in R d , preprocess S into a data structure that answers queries with a query curve q and radius ρ for the curves of S that have Fréchet distance at most ρ to q.We initiate a comprehensive analysis of the space/query-time trade-off for this data structuring problem. Our lower bounds imply that any data structure in the pointer model model that achieves Q(n) + O(k) query time, where k is the output size, has to use roughly Ω (n/Q(n)) 2 space in the worst case, even if queries are mere points (for the discrete Fréchet distance) or line segments (for the continuous Fréchet distance). More importantly, we show that more complex queries and input curves lead to additional logarithmic factors in the lower bound. Roughly speaking, the number of logarithmic factors added is linear in the number of edges added to the query and input curve complexity. This means that the space/query time trade-off worsens by an exponential factor of input and query complexity. This behaviour addresses an open question (see [1,9]) in the range searching literature concerning multilevel partition trees which may be of independent interest, namely, whether it is possible to avoid the additional logarithmic factors in the space and query time of a multilevel partition tree. We answer this question negatively.On the positive side, we show we can build data structures for the Fréchet distance by using semialgebraic range searching. The space/query-time trade-off of our data structure for the discrete Fréchet distance is in line with the lower bound, as the number of levels in the data structure is O(t), where t denotes the maximal number of vertices of a curve. For the continuous Fréchet distance, the number of levels increases to O(t 2 ).

FRESH: Fréchet Similarity with Hashing

“…In 2011, [7] revived the topic motivated by the availability of high-resolution trajectories of soccer players in the emerging area of sports analytics. A comprehensive study of the complexity of range searching under the Fréchet distance appeared in [1], that also gives lower bounds on the space-query-time trade-off of range searching under the Fréchet distance. Recently, the annual data competition within the ACM SIGSPATIAL conference on geographic information science has drawn attention to the timeliness of this problem [30].…”

“…The target of this paper is similarity search for time series and trajectories or, more generally, for curves: indeed, time series and trajectories can be envisioned as polygonal curves with vertices from IR d , for a suitable dimension d ≥ 1. 1 . Similarity search of curves frequently arises in several applications, like ridesharing recommendation [27], frequent routes [25], players performance [21], and seismology [26].…”

“…The Fréchet distance 2 between two curves is traditionally explained with this metaphor: a man is walking on a curve and his dog on another curve; the man and dog follow their curves from start to end and can vary their speeds, but they cannot go backward; the minimum length of the leash necessary to connect man and dog during the walk is the continuous Fréchet distance. The Fréchet distance does not require a one-to-one mapping between points of two curves, and it is hence invariant under differences in speed: this allows, for instance, to detect the trajectories of two cars 1 Usually, we have d = 1 for time series and d > 1 for trajectories. 2 If not differently stated, "Fréchet distance" refers to the continuous definition.…”

Probabilistic Embeddings of the Fréchet Distance

“…Such a random projection could be used in combination with probabilistic data structures, e.g. locality-sensitive hashing [20], but also with the multi-level data structures for Fréchet range searching given by Afshani and Driemel [2]. See below for a more in-depth discussion of these data structures.We show that in the worst case and under certain assumptions, the discrete Fréchet distance between two polygonal curves of complexity t in IR d , where d = {2, 3, 4, 5}, degrades by a factor linear in t with constant probability.…”

“…Such a random projection could be used in combination with probabilistic data structures, e.g. locality-sensitive hashing [20], but also with the multi-level data structures for Fréchet range searching given by Afshani and Driemel [2]. See below for a more in-depth discussion of these data structures.…”