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

(I0, I1, ... , In-1) where Ii 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 N1 and N2, create

new root with N1 and N2 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 EN is its entry for N. Adjust the rectangle for EN to tightly enclose N.

3. If NN exists, add entry ENN to P. ENN points to NN and its rectangle tightly encloses NN.

4. If necessary, split P5. Set N=P and go to step 2.

Deletion1. 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 EN be

the entry that points to N. If N has less than m entries, delete EN from P and add N to Q.

4. If N has at least m entries then set the rectangle of EN 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 NodesProblem: 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 Algorithm1. Find pair of entries E1 and E2 that maximizes area(J)

- area(E1) - area(E2) where J is covering rectangle.

2. Put E1 in one group, E2 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 d1 and d2 where di is the minimum area increase in covering rectangle of Group i when E is added.

5. Find E with maximum |d1 - d2| 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 Algorithm1. 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 10th 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.