Abstract

We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t, report all K points of S that lie inside R at time t. We first present an indexing structure that, for any given constant >0, uses O(N/B) disk blocks and answers a query in O((N/B)^1/2++K/B) I/Os, where B is the block size. It can also report all the points of S that lie inside R during a given time interval. A point can be inserted or deleted, or the trajectory of a point can be changed, in O(log_B² N) I/Os. Next, we present a general approach that improves the query time if the queries arrive in chronological order, by allowing the index to evolve over time. We obtain a tradeoff between the query time and the number of times the index needs to be updated as the points move. We also describe an indexing scheme in which the number of I/Os required to answer a query depends monotonically on the difference between the query time stamp t and the current time. Finally, we develop an efficient indexing scheme to answer approximate nearest-neighbor queries among moving points.

Article Outline

1. Introduction

1.1. Problem statement

1.2. Previous results

1.3. Our results

2. Geometric preliminaries

2.1. Duality

2.2. External partition trees

2.3. Simplified partition trees for random points

3. Time-oblivious indexing

3.1. Multilevel partition trees

3.2. Multilevel grid trees

3.3. Further extensions

4. One-dimensional chronological queries

4.1. Kinetic B-trees

4.2. Query/update tradeoffs

4.3. Tradeoffs for grid trees

5. Two-dimensional chronological queries

5.1. External range trees

5.2. Kinetic external range trees

5.3. Query/update tradeoffs

6. Time-responsive indexing

6.1. Recent-past and near-future queries

6.2. Answering distant-future queries

7. Approximate nearest-neighbour searching

8. Conclusion

References

Corresponding author

*1 An extended abstract of this paper appeared in the Proceedings of the 19th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems [2].

¹ Supported in part by National Science Foundation Grants EIA-9870734, EIA-9972879, and CCR-9732787, by Army Research Office MURI Grant DAAH04-96-1-0013, by a Sloan fellowship, and by a grant from the US–Israeli Binational Science Foundation.

² Supported in part by National Science Foundation grants EIA-9870734, EIA-9972879, and CR-9984099.

³ Supported in part by National Science Foundation grant DMS-9627683, by US Army Research Office MURI grant DAAH04-96-1-0013, and by a Sloan Fellowship.