Elsevier

Information Systems

Volume 35, Issue 8, December 2010, Pages 911-932
Information Systems

Optimized algorithms for predictive range and KNN queries on moving objects

https://doi.org/10.1016/j.is.2010.05.004Get rights and content

Abstract

There have been many studies on management of moving objects recently. Most of them try to optimize the performance of predictive window queries. However, not much attention is paid to two other important query types: the predictive range query and the predictive k nearest neighbor query. In this article, we focus on these two types of queries. The novelty of our work mainly lies in the introduction of the Transformed Minkowski Sum, which can be used to determine whether a moving bounding rectangle intersects a moving circular query region. This enables us to use the traditional tree traversal algorithms to perform range and kNN searches. We theoretically show that our algorithms based on the Transformed Minkowski Sum are optimal in terms of the number of tree node accesses. We also experimentally verify the effectiveness of our technique and show that our algorithms outperform alternative approaches.

Introduction

As telecommunication technologies such as GPS and mobile devices become widely used, we are able to track cars or mobile phone users and provide location-based services to them. Commonly, these services request information about moving objects (cars, mobile phone users, etc.) for a period of time in the future, which are called predictive queries. Several structures have been proposed to support efficient processing of predictive queries [19], [24], [13], [16]. All these studies focus on one query type, the predictive window query, which is defined as follows. Given a rectangular region WQ and a future time interval QT, retrieve the set of objects that will intersect WQ at any timestamp tQT. For a real-life example, a police officer can issue the query: “report the registration numbers of all the cars that will pass through the central business district in the next 10 min” through a traffic monitoring system. A predictive window query is suitable here since the central business district is probably best described by a rectangle. However, in many other cases, a query region is best described by a circle. For example, a tourist can issue the query: “find all the vacant taxis within 200 m from me in the next 10 min” through a mobile phone. The region of this query is more accurately represented by a circle than by a rectangle. In this regard, we introduce the predictive range query to capture predictive queries with circular regions. A formal definition of the query will be given in Section 2.1.

Consider further the above example. Sometimes, the tourist may not know a proper distance to specify the query range. Suppose there is no taxi that will be within 200 m from the tourist in the next 10 min, then the tourist will get a null answer to the query, and not get any taxi. In this case, another type of query, the predictive nearest neighbor query suits the purpose better. The tourist would now ask: “find the nearest vacant taxi to me in the next 10 min”. Generally, the tourist can request k nearest taxis for consideration, which corresponds to a predictive k nearest neighbor (kNN) query. A formal definition of the query will be given in Section 2.2.

While the predictive range and kNN queries have wide ranges of applications, few studies have specifically focused on optimizing algorithms to process these queries. In this article, we study how to process these two types of queries efficiently and make the following contributions:

  • We introduce an analysis tool, the Transformed Minkowski Sum (TMS), for moving object databases. TMS can be used to determine the intersection of two moving objects of arbitrary shapes. In particular, we apply this technique to determine the intersection between a moving rectangle and a moving circle, which enables us to process queries with circular search regions on moving objects, specifically, predictive range and kNN queries.

  • Exploiting the TMS, we derive an equivalent condition that is easy to evaluate for identifying objects in a moving circular query range. Based on this equivalent condition, we can adopt a tree traversal algorithm for processing the predictive range query. The algorithm can return exactly the objects that intersect the circular range, which is not achieved by any existing algorithm. We prove that this algorithm is optimal in terms of the number of tree node accesses.

  • We also provide tree traversal algorithms for the predictive kNN query, again, enabled by the TMS concept. We prove that our kNN algorithm is optimal in terms of the number of tree node accesses. In addition, we can determine the timestamps when the closest distances between the query point and the k nearest neighbors (NNs) happen.

  • Based on the TMS, we develop a cost model to estimate the number of node accesses for predictive range queries. We also show how it can help estimate the cost of predictive kNN queries.

  • We perform an extensive experimental study. The results verify the effectiveness of our algorithms and the accuracy of our cost model.

The rest of the article is organized as follows: Section 2 gives the formal definitions of the predictive range and kNN queries and discusses our problem setting. Section 3 reviews related work and provides preliminaries. In Section 4, we introduce the Transformed Minkowski Sum. 5 Processing predictive range queries, 6 Processing predictive KNN queries present our algorithms for the predictive range and kNN queries, respectively. Section 8 reports the results of our experimental study and Section 9 concludes the article.

Section snippets

Problem formulation

In this section, we first give the formal definitions of the predictive range query and the predictive kNN query. Then we discuss our problem setting and optimization goals.

Representation of moving objects

Traditionally, a moving point's movement is represented by sampled positions on its trajectory. This approach requires frequent position updates, which impose heavy workload on the system. Sistla et al. [22] propose to model a moving point as a linear function of time t: P(t)=P(tref)+V (ttref), where P(tref) is the point's position at a reference time tref, and V is the point's velocity. This representation allows prediction of the point's future positions. When an object changes its speed,

The Transformed Minkowski Sum

In this section, we propose a method to determine whether a TPBR, that is, a moving rectangle intersects a moving circle, which is essential to processing range and kNN queries. Our method is based on a new concept called the Transformed Minkowski Sum. The Transformed Minkowski Sum is obtained by performing: (i) a coordinate transform according to the movement of the query, and then (ii) the Minkowski enlargement in the transformed coordinate system. This method can be further generalized to

Processing predictive range queries

In this section, we give our algorithm for processing predictive range queries based on the Transformed Minkowski Sum. There are three kinds of predictive range queries as discussed in Section 2.1. We only present the algorithm for the most general version, that is, the moving range query (Definition 3). Given a moving range query Q, we need to find all the objects intersecting Q in QT. According to Theorem 1, the equivalent condition of an object O intersecting Q during QT=[t,t] is SR(O,Q,t,

Processing predictive KNN queries

In this section, we give our algorithm for processing predictive kNN queries. There are three kinds of predictive kNN queries as discussed in Section 2.2. We only present the algorithm for the most general version, that is, the moving kNN query (Definition 4). Given a moving kNN query Q, we need to find the k objects that will be closest to the query point during the querying period QT. Moreover, we would also like to know when the k objects are closest to Q. For example, when a working robot

Cost model

In this section, we develop a cost model to estimate the cost of predictive range queries, which again makes use of the Transformed Minkowski Sum and its sweeping region. We also discuss how the cost model for predictive range queries can help estimate the cost of predictive kNN queries.

Experiments

This section reports our experimental results. We present the results on our predictive range search algorithm in Section 8.1 and the results on our predictive kNN search algorithm in Section 8.2. Finally we evaluate our cost model in Section 8.3.

We have generated datasets and queries using the benchmark data generator proposed by Chen et al. [6]. The parameters of the datasets are described as follows. The moving objects are populated in a 2-dimensional 100,000 ×100,000 coordinate space and

Conclusions and discussions

In this article, we introduced the Transformed Minkowski Sum (TMS) technique for determining the intersection of two moving objects of arbitrary shapes. We have exemplified the usage of this technique by applying it to two typical types of queries on moving objects, that is, the predictive range and kNN queries. Based on the TMS, we proposed an I/O optimal algorithm for the predictive range query, which can return exactly the objects that intersect a moving circular region; this has not been

Acknowledgments

We would like to thank the authors of [4] for providing us the source code of their work, which we used in our performance study. This work is supported under the Australian Research Councils’ Discovery funding schemes (Project number DP0880215 and DP0880250).

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