Indexing and Range Queries in Spatio-Temporal Databases Danzhou Liu, Wei Cui, Yun Fan School of Computer Science University of Central Florida

Outline Introduction The R*-tree The TPR-tree The TPR*-tree Experiments Conclusions

Introduction Spatio-temporal databases record moving objects’ geographical locations (sometimes also shapes) at various timestamps. support queries that explore their historical and future (predictive) behaviors. Applications. applications: flight control systems, weather forecast and mobile computing The database stores the motion functions of moving objects. For each object o, its motion function gives its location o(t) at any future time t. A predictive window query specifies a query region qR and a future time interval qT retrieves the set of all objects that will fall in qR during qT. our goal: index moving objects so that a predictive window query can be answered with as few disk I/Os as possible. Examples Find all airplanes that will be over Florida in the next 10 minutes. Report all vessels that will enter the United States in the next hour.

Motion Function We consider linear motion. For each object, the database stores Its minimum bounding rectangle (MBR) at the reference time 0 Its current velocity bounding rectangle (VBR) Examples: MBR(a)={2,4,3,4}, VBR(a)={1,1,1,1}; MBR(c)={8,9,3,4}, VBR(c)={-2,0,0,2}; An update is necessary only when an object’s VBR changes.

R*-tree The R*-tree aims at minimizing: the area The perimeter of each MBR The overlap between two MBRs (e.g., N1, N2) in the same node The distance between the centroid of an MBR and that of the node containing it

R*-tree Insertion

The Time Parameterized R-Tree (TPR-Tree) Extends the R-tree by introducing the velocity bounding rectangle (VBR) in all entries. Queries are compared with conservative MBRs of non-leaf entries. N1v={-2,1,-2,1} and N2v={-2,0,-1,2}

TPR*-Tree Our goal index moving objects so that a predictive window query can be answered with as few disk I/Os as possible. A mathematical model that estimates the cost of answering a predictive window query using TPR-like structures. Number of node accesses. Application of the model to derive the optimal performance. The TPR-tree is much worse than the optimal structure. Exam the algorithms of the TPR-tree, identify their deficiencies, and propose new ones. The TPR*-tree.

TPR deficiency 1: Choosing sub-tree to insert To insert an entry, the TPR-tree picks the sub-tree incurring the minimum penalty (smallest MBR/VBR enlargement). May result in inserting an entry into a bad sub-tree; this problem is increasingly serious as time evolves.

TPR* solution: Choose path Aims at finding the best insertion path globally, namely, among all possible paths. Observation: We can find this path by accessing only a few more nodes (than the TPR-tree algorithm). Maintain a heap: [(g),0], [(h),0], [(i),20] the path expanded so far the accumulated penalty so far

TPR* solution: Choose path Aims at finding the best insertion path globally, namely, among all possible paths. Observation: We can find this path by accessing only a few more nodes (than the TPR-tree algorithm). Visit node g: [(h),0], [(a,g),3], [(i),20], [(b,g),32] complete paths already although nodes a and b are not visited

TPR* solution: Choose path Aims at finding the best insertion path globally, namely, among all possible paths. Observation: We can find this path by accessing only a few more nodes (than the TPR-tree algorithm). Visit node h: [(a,g),3], [(d,h),9], [(c,h),17], [(i),20], [(b,g),32] The algorithm stops now.

TPR deficiency 2: Which entries to re-insert When a node overflows, some of its entries are re-inserted to defer node split (the ones that diverge most from the node centroid). The entries chosen by the TPR-tree are very likely to be re-inserted back to the same node, so that a node split is still necessary.

TPR* solution: Pick worst Aims at selecting entries that can most effectively “shrink” the MBR or VBR of the node for re-insertion. The first step picks an appropriate dimension (either spatial or velocity) based purely on estimation using our cost model (see the paper for details). The second step performs sorting on this dimension and decides the entries to be removed . Example: If the axis chosen in the first step is the x-axis, then the sorting list is {b,d,a,c}. Either b or c is removed.

TPR deficiency 3: Tightening MBR in deletion Entry deletion requires first finding the entry, which accesses many nodes of the tree. The TPR-tree uses this fact to tighten the MBR of non-leaf entries. Assume nodes h and i are accessed before e is found; then the TPR-tree will tighten the MBR of i only (enclosing g and f).

TPR deficiency 3: Tightening MBR in deletion Entry deletion requires first finding the entry, which accesses many nodes of the tree. The TPR-tree uses this fact to tighten the MBR of non-leaf entries. Assume nodes h and i are accessed before e is found; then the TPR-tree will tighten the MBR of i only (enclosing g and f).

TPR* solution: Active tightening Tightening more entries for free. Assume nodes h and i are accessed before e is found; then the TPR*-tree will tighten the MBR of both h and i.

TPR* solution: Active tightening Tightening more entries for free. Assume nodes h and i are accessed before e is found; then the TPR*-tree will tighten the MBR of both h and i.

TPR* solution: Active tightening (Cont.) Another example: Assume the shaded nodes are accessed to find e. The active tightening can tighten the MBR of n5, n6, n3, and n4. But not n1 and n2.

Challenge of Migration 3 Operating Systems: Microsoft Windows Sun Solaris Redhat Fedora Core 1 2 Compilers: CL, GCC (2.9.5, 3.3.2) Difference of Code Conversion How close the compilers to the standard? Compatibility of Library

Experiments: Settings (query and tree) Dataset 50,000 sampled objects’ MBRs are taken from a real spatial dataset NJ [Tiger] each object is associated with a VBR such that on each dimension The velocity extent is zero (i.e., the object does not change spatial extents during its movement) the velocity value distribution is randomed in range [0,8] the velocity can be positive or negative with equal probability. We compare TPR*- with TPR-trees. Disk page size=1k bytes (node capacity=27 for both trees). For each object update, perform a deletion followed by an insertion on each tree. Each predictive query is a moving rectangle, and has these parameters: qRlen: The length of the query’s MBR qVlen: The length of the query’s VBR qTlen: The number of timestamps covered.

TPR-tree

TPR*-tree

Conclusions The TPR-tree combines the idea of conservative MBR directly with the tree construction algorithms of R*-trees. The TPR*-tree improves it by designing algorithms that take into account the special features for moving objects. Cost model for performance analysis The optimal performance of a “hypothetically best structure” Reduce disk I/Os for predictive queries

Q&A

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