Indexing the Present and Future Positions of Moving Objects Simonas Šaltenis Aalborg University Nykredit Center for Database Research Department of Computer Science, Aalborg University

LBS workshop, Aalborg, June 7-8, Motivation – Background Position-aware, online, moving objects are enabled by the following trends. Miniaturization of electronics Advances in positioning systems (e.g., GPS, assisted GPS,...) Advances in wireless communications Examples of position-aware online moving objects WAP-enabled mobile-phones, as well as diverse types of personal digital assistants (online “cameras,” “wrist watches,” etc.)  By 2005, there wil be 500 million users of mobile phones with GPS. Vehicles, including cars, public transportation, recreational vehicles, sea vessels, etc. The coming years will witness very large quantities of these.

LBS workshop, Aalborg, June 7-8, Motivation – Sample E-Services Traffic coordination and management Identification of impending traffic jams and expected fastest routes between positions Location-aware advertising Consumers may receive sales information for locations close to them. Here, the positional data is used together with an accumulated user profile to provide a better service. Integrated tourist services For example, this covers booking (hotels, concerts, ferries, etc.) and payment, and provision of transportation or travel directions. Safety-related services It is possible to monitor tourists traveling in dangerous environments, and then react to emergencies.

LBS workshop, Aalborg, June 7-8, Motivation – Problem Statement We address the problem of indexing the ever-changing current and predicted future positions of point objects moving in one, two, and three-dimensional space. This also includes continuous variables in process monitoring, e.g., temperature, pressure (1D, nD).

LBS workshop, Aalborg, June 7-8, Outline Data and queries The TPR-tree Bounding rectangles and insertion The inner workings of the tree Performance experiments Nothing is eternal......neither is positional information Conclusions

LBS workshop, Aalborg, June 7-8, Spatial Indexing With the R-Tree The R-tree supports updates, but not continuous movement. Example Query R1 R2 R1R2 R3R4R5 p6p7 p5 p1p2 Pointers to data tuples p8 p3p4 p9p10 p11 p12p13 R6R7 R3 R4 R5 R6 R7 p1 p7 p6 p8 p2 p3 p4 p5 p9 p10 p11 p12 p13

LBS workshop, Aalborg, June 7-8, Modeling Continuous Movement In conventional databases, data is assumed constant unless explicitly modified. With continuous movement, this is problematic. Too frequent updates Outdated, inacurate data

LBS workshop, Aalborg, June 7-8, Modeling Continuous Movement In conventional databases, data is assumed constant unless explicitly modified. With continuous movement, this is problematic. Too frequent updates Outdated, inacurate data Instead of storing position values, we store positions as functions of time, yielding time-parameterized positions. We use linear functions to capture the present and future positions. Updates are necessary only when the parameters of the functions change. For example, given, the current and anticiapted, future position of a two- dimensional point can be described by four parameters.

LBS workshop, Aalborg, June 7-8, Modeling Continuous Movement Three ways to think about continuously moving points in d-dimensional space: Lines in (d+1)-dimensional space  d spatial dimensions and 1 time dimension Points in 2d-dimensional space  d spatial and d velocity dimensions (function parameters: ) Time-parameterized points in d-dimensional space – our approach x t o1o o2o2 o3o3 v x(t 0 ) -0.5 o1o1 o2o2 o3o3

LBS workshop, Aalborg, June 7-8, Queries Type 1: objects that intersect a given rectangle at Type 2: objects that intersect a given rectangle sometime from to Type 3: objects that intersect a given moving rectangle sometime between and x t o1o1 o1o1 o2o2 o3o3 o4o4

LBS workshop, Aalborg, June 7-8, Time-Parameterized Rectangles The TPR-tree is based on the R-tree. Moving points are bounded with time-parameterized rectangles. Are bounding from now on. The R-tree allows overlap. Ideally, bounding rectangles should be always minimal. Excessive storage cost The tree employs conservative bounding rectangles. These are ”tightened” during modifications.

LBS workshop, Aalborg, June 7-8, Insertion: Grouping Points How to group moving points? The R-tree’s algorithms minimize characteristics of MBRs such as area, overlap, and margin. How does that work for moving points?

LBS workshop, Aalborg, June 7-8, Insertion in the TPR-Tree The bounding rectangle characteristics (area, overlap, and margin) are functions of time. The goal is to minimize these for all time points from now to now+H. Minimizing the characteristics for time now + H/2 does not work (e.g., the area of a conservative bounding rectangle is not linear). where A(t) is, e.g., the area of an MBR We use the regular R*-tree algorithms, but all bounding rectangle characteristics are replaced by their integrals. What H to use? H depends on the update rate, and on how far queries may reach into the future (W).

LBS workshop, Aalborg, June 7-8, Example I We illustrate the working of the TPR-tree by means of an example. The subsequent figures are generated automatically, by the index code used for performance experiments. Data 20 one-dimensional points are used. Index Parameters Page size = 64 (5 entries in leaf nodes and 3 in non-leaf nodes). H = 8.

LBS workshop, Aalborg, June 7-8, Example II CT = 0 At CT=1, the point at x = 20, v = 0 is updated to have x = 18.5, v = -0.5.

LBS workshop, Aalborg, June 7-8, Example III CT = 1 Inserting a moving point at position 14 with v = –0.5.

LBS workshop, Aalborg, June 7-8, Example IV After insertion

LBS workshop, Aalborg, June 7-8, Performance Experiments I Simulation-based performance study A GIST-based implementation of the TPR-tree is used. Trees are initially bulk loaded, then subjected to workloads intermixing modifications and queries for 600 minutes x 1000 km space and from 100,000 to 900,000 point objects. The speeds of the objects range from 0 to 180 km/h. On average, each object sends an update each 60 minutes (yielding a total of from 1,666 to 15,000 updates per minute). 2D and 3D uniform data is used, as well as 2D data generated according to a scenario, where objects move on a fully connected graph of two-way roads. Indices compared Straigthforward R-tree TPR-tree TPR-tree with load-time bounding rectangles

LBS workshop, Aalborg, June 7-8, Performance Experiments III Search performance for 2D data with varying skew (the number of destinations). The more objects move similarly, the easier it is to index them. Tightening of bounding rectangles on updates results in a significant decrease in the search I/O. Not possible with the dual transformation.

LBS workshop, Aalborg, June 7-8, Performance Experiments IV Degradation of search performance across time for 2D data, with W = 40. Due to the constant influx of updates, the performance of the TPR-tree does not degrade after reaching a certain level.

LBS workshop, Aalborg, June 7-8, Performance Experiments V Search performance for 2D data with varying numbers of moving objects (average # of returned objects per query is not changed). As expected, the TPR-tree shows almost no decrease in performance (as long as the number of tree levels does not change).

LBS workshop, Aalborg, June 7-8, Improving the TPR-tree Coping with out-dated data In the highly dynamic environment of moving objects data becomes inaccurate and obsolete very fast. Unreliable communication channels lead to no guaranties of objects updating or deleting themselves. Solution – expiration time t exp associated with each object Expiration times depend on (or can be derived from):  Desired uncertainty threshold and speeds of objects  Underlying infrastructure (e.g., road network) restricting movement Expiration times may be a good idea even if objects always update and delete themselves on time. Indexing segments of future trajectories instead of infinite lines should be easier.

LBS workshop, Aalborg, June 7-8, Expiration Times Two issues related to expiration times Purging expired entries from the index Filtering expired entries from query answers and taking advantage of having finite line segments instead of infinite lines x t o1o o2o2 o3o3

LBS workshop, Aalborg, June 7-8, Purging Expired Entries I Purging done in a lazy fashion – expired entries are ignored in all index algorithms and they are physically removed when a node is written to the disk. Insertion and deletion algorithms are modified to account for under-full nodes at any stage of the algorithm.

LBS workshop, Aalborg, June 7-8, Purging Expired Entries II

LBS workshop, Aalborg, June 7-8, Improving Bounding Rectangles The obvious and simple solution (reduce velocity extents of BRs as much as possible) does not work! x t o1o o2o2 o3o3 CT = 0 x t o1o o2o2 o3o3 CT = 1.5

LBS workshop, Aalborg, June 7-8, Optimal BRs (1-dimensional) We need to find a time-parameterized bounding interval that minimizes the integral of its length from CT to CT+H. Equivalently – find a minimum area (up to CT+H) trapezoid enclosing a set of given intervals. Such a trapezoid has the upper and the lower sides that support two edges of the convex hull of intervals. The edges are those that cross the vertical line CT + H/2. x t H/2H = 6 Such a minimum trapezoid can be found by computing the convex hull (O(n log h), where h – the number of c.h. vertices). We need only two vertices. This can be done in worst-case linear time or with simplified algorithm in ~ 3.2n expected time (O(n 2 ) worst-case).

LBS workshop, Aalborg, June 7-8, Conclusions The TPR-tree indexes the current and predicted future positions of moving objects. The TPR-tree is based on the proven, widely used R-tree technology The tree extends the R*-tree by introducing conservative, time- parameterized bounding rectangles, which are tightened regularly. The tree’s algorithms use integrals of area, overlap, etc. The tree can be tuned to take advantage of a specific update rate and querying window length. Out-dated data can be automatically purged from the index Other types of queries that are supported by the R-tree can be supported by the TPR-tree, e.g., nearest-neighbor queries.

LBS workshop, Aalborg, June 7-8, Related Work The current and future positions (mostly 1D) Tayeb et al. use PMR Quadtrees to index lines in (x,t)-space.  Index has to be periodically rebuilt  Storage overhead Kollios et al. use a duality transformation.  Several schemes are proposed to index the resulting static points. Agarwal et al. apply the techniques of kinetic data structures and persistance to external range trees and partition trees.  Show good worst-case bounds. Indexing of past positions Pfoser et al. The representation of moving points has been explored. E.g., Wolfson et al., Moreira et al., Pfoser et al. Capture uncertainty

LBS workshop, Aalborg, June 7-8, Performance Experiments II Search performance for 2D data, average update interval 120, and varying H. Update interval 120 implies that, on average, each 120 time units, a leaf-node will have new entries and a new BR. The best choice of H correlates strongly with the ”life time” of tree-nodes, which, in turn, is related to the update rate.

LBS workshop, Aalborg, June 7-8, TPR-Tree Bulk Loading I Regular R-tree bulk loading aims at dividing the space into roughly equal, square MBRs (e.g., STR). It is a straightforward approach to use STR, interpreting the d-dimensional moving points as 2d-dimensional points. The ratio between the spatial and velocity extents of BRs is important. We introduce a parameter that expresses how many times the velocity extents of BRs are longer than their spatial extents. Given a fixed number of BRs to produce, we can  make them large in the spatial dimensions, but small in the velocity dimensions (i.e., growing slowly). This corresponds to a small.  make them small in the spatial dimensions, but large in the velocity dimensions (i.e., fast growing). This corresponds to a large.

LBS workshop, Aalborg, June 7-8, TPR-Tree Bulk Loading II Example: Using either initially large, but slow-growing, or initially small, but fast-growing, time-parameterized BRs. S V s v x(t 0 ) v x(t 0 ) t x 3456 t x 3456 (good for a large H) (good for a small H) Goal: find an ”optimal” as a function of H.

LBS workshop, Aalborg, June 7-8, TPR-Tree Bulk Loading III One-dimensional uniform data Let k be the number of nodes and the space-time ratio For 2D data,, and for 3D data, gets smaller with increasing dimensionality Intuitive as hyper-volumes grow faster in more dimensions

LBS workshop, Aalborg, June 7-8, TPR-Tree Bulk Loading IV A modified STR is used that achieves the desired -ratio. Example: 2000 points, 20 points per leaf-node.