R-Trees: A Dynamic Index Structure For Spatial Searching Antonin Guttman

Introduction Range queries in multiple dimensions:  Computer Aided Design (CAD)  Geo-data applications Support spacial data objects (boxes) Index structure is dynamic.

R-Tree Balanced (similar to B+ tree) I is an n-dimensional rectangle of the form (I 0, I 1,..., I n-1 ) where I i is a range [a,b]  [- ,  ] Leaf node index entries: (I, tuple_id) Non-leaf node entry: (I, child_ptr) M is maximum entries per node. m  M/2 is the minimum entries per node.

Invariants 1.Every leaf (non-leaf) has between m and M records (children) except for the root. 2.Root has at least two children unless it is a leaf. 3.For each leaf (non-leaf) entry, I is the smallest rectangle that contains the data objects (children). 4.All leaves appear at the same level.

Example (part 1)

Example (part 2)

Searching Given a search rectangle S... 1.Start at root and locate all child nodes whose rectangle I intersects S (via linear search). 2.Search the subtrees of those child nodes. 3.When you get to the leaves, return entries whose rectangles intersect S. Searches may require inspecting several paths. Worst case running time is not so good...

S = R16

Insertion Insertion is done at the leaves Where to put new index E with rectangle R? 1.Start at root. 2.Go down the tree by choosing child whose rectangle needs the least enlargement to include R. In case of a tie, choose child with smallest area. 3.If there is room in the correct leaf node, insert it. Otherwise split the node (to be continued...) 4.Adjust the tree... 5.If the root was split into nodes N 1 and N 2, create new root with N 1 and N 2 as children.

Adjusting the tree 1.N = leaf node. If there was a split, then NN is the other node. 2.If N is root, stop. Otherwise P = N’s parent and E N is its entry for N. Adjust the rectangle for E N to tightly enclose N. 3.If NN exists, add entry E NN to P. E NN points to NN and its rectangle tightly encloses NN. 4.If necessary, split P 5.Set N=P and go to step 2.

Deletion 1.Find the entry to delete and remove it from the appropriate leaf L. 2.Set N=L and Q = . (Q is set of eliminated nodes) 3.If N is root, go to step 6. Let P be N’s parent and E N be the entry that points to N. If N has less than m entries, delete E N from P and add N to Q. 4.If N has at least m entries then set the rectangle of E N to tightly enclose N. 5.Set N=P and repeat from step 3. 6.*Reinsert entries from eliminated leaves. Insert non- leaf entries higher up so that all leaves are at the same level. 7.If root has 1 child, make the child the new root.

Why Reinsert? Nodes can be merged with sibling whose area will increase the least, or entries can be redistributed. In any case, nodes may need to be split. Reinsertion is easier to implement. Reinsertion refines the spatial structure of the tree. Entries to be reinserted are likely to be in memory because their pages are visited during the search to find the index to delete.

Other Operations To update, delete the appropriate index, modify it, and reinsert. Search for objects completely contained in rectangle R. Search for objects that contain a rectangle. Range deletion.

Splitting Nodes Problem: Divide M+1 entries among two nodes so that it is unlikely that the nodes are needlessly examined during a search. Solution: Minimize total area of the covering rectangles for both nodes. Exponential algorithm. Quadratic algorithm. Linear time algorithm.

Splitting Nodes – Exhaustive Search Try all possible combinations. Optimal results! Bad running time!

Splitting Nodes – Quadratic Algorithm 1.Find pair of entries E 1 and E 2 that maximizes area(J) - area(E 1 ) - area(E 2 ) where J is covering rectangle. 2.Put E 1 in one group, E 2 in the other. 3.If one group has M-m+1 entries, put the remaining entries into the other group and stop. If all entries have been distributed then stop. 4.For each entry E, calculate d 1 and d 2 where d i is the minimum area increase in covering rectangle of Group i when E is added. 5.Find E with maximum |d 1 - d 2 | and add E to the group whose area will increase the least. 6.Repeat starting with step 3.

Greedy continued Algorithm is quadratic in M. Linear in number of dimensions. But not optimal.

Splitting Nodes – Linear Algorithm 1.For each dimension, choose entry with greatest range. 2.Normalize by dividing the range by the width of entire set along that dimension. 3.Put the two entries with largest normalized separation into different groups. 4.Randomly, but evenly divide the rest of the entries between the two groups. Algorithm is linear, almost no attempt at optimality.

Performance Tests CENTRAL circuit cell (1057 rectangles) Measure performance on last 10% inserts. Search used randomly generated rectangles that match about 5% of the data. Delete every 10 th data item.

Performance

With linear-time splitting, inserts spend very little time doing splits. Increasing m reduces splitting (and insertion) cost because when a groups becomes too full, the rest of the entries are assigned to the other group. As expected, most of the space is taken up by the leaves.

Performance

Deletion cost affected by size of m. For large m:  More nodes become underfull.  More reinserts take place.  More possible splits.  Running time is pretty bad for m = M/2. Search is relatively insensitive to splitting algorithm. Smaller values of m reduce average number of entries per node, so less time is spent on search in the node (?).

Space Efficiency Stricter node fill produces smaller index. For very small m, linear algorithm balances nodes. Other algorithms tend to produce unbalanced groups which are likely to split, wasting more space.

Conclusions Linear time splitting algorithm is almost as good as the others. Low node-fill requirement reduces space- utilization but is not siginificantly worse than stricter node-fill requirements. R-tree can be added to relational databases.