In-memory immutable 2D R-tree implementation in java using RxJava Observables for reactive processing of search results.

Status: released to Maven Central, ensure you use at least 0.4 to get fix for #7

An R-tree is a commonly used spatial index.

This was fun to make, has an elegant concise algorithm, is thread-safe and fast.

The algorithm to achieve immutability is cute. For insertion/deletion it involves recursion down to the required leaf node then recursion back up to replace the parent nodes up to the root. The guts of it is in Leaf.java and NonLeaf.java.

Backpressure support required some complexity because effectively a bookmark needed to be kept for a position in the tree and returned to later to continue traversal. An immutable stack containing the node and child index of the path nodes came to the rescue here and recursion was abandoned in favour of looping to prevent stack overflow (unfortunately java doesn't support tail recursion!).

Maven site reports are here including javadoc.

• immutable R-tree suitable for concurrency
• Guttman's heuristics (Quadratic splitter) (paper)
• R*-tree heuristics (paper)
• Customizable splitter and selector
• search returns Observable
• search is cancelled by unsubscription
• search is O(log(n)) on average
• insert, delete are O(n) worst case
• all search methods return lazy-evaluated streams offering efficiency and flexibility of functional style including functional composition and concurrency
• balanced delete
• uses structural sharing
• supports backpressure
• JMH benchmarks
• visualizer included
• 100% unit test code coverage
• R*-tree performs 530,000 searches/second returning 22 entries from a tree of 38,377 Greek earthquake locations on [email protected] (maxChildren=4, minChildren=4). Insert at 135,000 entries per second.
• requires java 1.6 or later

Number of points = 1000, max children per node 8:

Quadratic split	R*-tree split

Notice that there is little overlap in the R*-tree split compared to the Quadratic split. This should provide better search performance (and in general benchmarks show this).

Add this maven dependency to your pom.xml:

<dependency>
  <groupId>com.github.davidmoten</groupId>
  <artifactId>rtree</artifactId>
  <version>0.6.1</version>
</dependency>

###Instantiate an R-Tree Use the static builder methods on the RTree class:

// create an R-tree using Quadratic split with max
// children per node 4, min children 2 (the threshold
// at which members are redistributed)
RTree<String, Geometry> tree = RTree.create();

You can specify a few parameters to the builder, including minChildren, maxChildren, splitter, selector:

RTree<String, Geometry> tree = RTree.�minChildren(3).maxChildren(6).create();

###Generic typing

If for instance you know that the entry geometry is always Point then create an RTree specifying that generic type to gain more type safety:

RTree<String, Point> tree = RTree.create();

###R*-tree If you'd like an R*-tree (which uses a topological splitter on minimal margin, overlap area and area and a selector combination of minimal area increase, minimal overlap, and area):

RTree<String, Geometry> tree = RTree.star().maxChildren(6).create();

See benchmarks below for some of the performance differences.

###Add items to the R-tree When you add an item to the R-tree you need to provide a geometry that represents the 2D physical location or extension of the item. The Geometries builder provides these factory methods:

• Geometries.rectangle
• Geometries.circle
• Geometries.point

To add an item to an R-tree:

RTree<T,Geometry> tree = RTree.create();
tree = tree.add(item, Geometries.point(10,20));

or

tree = tree.add(Entry.entry(item, Geometries.point(10,20));

Important note: being an immutable data structure, calling tree.add(item, geometry) does nothing to tree, it returns a new RTree containing the addition. Make sure you use the result of the add!

###Remove an item in the R-tree To remove an item from an R-tree, you need to match the item and its geometry:

tree = tree.delete(item, Geometries.point(10,20));

or

tree = tree.delete(entry);

Important note: being an immutable data structure, calling tree.delete(item, geometry) does nothing to tree, it returns a new RTree without the deleted item. Make sure you use the result of the delete!

###Custom geometries You can also write your own implementation of Geometry. An implementation of Geometry needs to specify methods to:

• check intersection with a rectangle (you can reuse the distance method here if you want but it might affect performance)
• provide a minimum bounding rectangle
• implement equals and hashCode for consistent equality checking
• measure distance to a rectangle (0 means they intersect). Note that this method is only used for search within a distance so implementing this method is optional. If you don't want to implement this method just throw a RuntimeException.

For the R-tree to be well-behaved, the distance function if implemented needs to satisfy these properties:

• distance(r) >= 0 for all rectangles r
• if rectangle r1 contains r2 then distance(r1)<=distance(r2)
• distance(r) = 0 if and only if the geometry intersects the rectangle r

###Searching The advantage of an R-tree is the ability to search for items in a region reasonably quickly. On average search is O(log(n)) but worst case is O(n).

Search methods return Observable sequences:

Observable<Entry<T, Geometry>> results =
    tree.search(Geometries.rectangle(0,0,2,2));

or search for items within a distance from the given geometry:

Observable<Entry<T, Geometry>> results =
    tree.search(Geometries.rectangle(0,0,2,2),5.0);

To return all entries from an R-tree:

Observable<Entry<T, Geometry>> results = tree.entries();

Suppose you make a custom geometry like Polygon and you want to search an RTree<String,Point> for points inside the polygon. This is how you do it:

RTree<String, Point> tree = RTree.create();
Func2<Point, Polygon, Boolean> pointInPolygon = ...
Polygon polygon = ...
...
entries = tree.search(polygon, pointInPolygon);

The key is that you need to supply the intersects function (pointInPolygon) to the search. It is on you to implement that for all types of geometry present in the RTree. This is one reason that the generic Geometry type was added in rtree 0.5 (so the type system could tell you what geometry types you needed to calculate intersection for) .

As per the example above to do a proximity search you need to specify how to calculate distance between the geometry you are searching and the entry geometries:

RTree<String, Point> tree = RTree.create();
Func2<Point, Polygon, Boolean> distancePointToPolygon = ...
Polygon polygon = ...
...
entries = tree.search(polygon, 10, distancePointToPolygon);

import com.github.davidmoten.rtree.RTree;
import static com.github.davidmoten.rtree.geometry.Geometries.*;

RTree<String, Point> tree = RTree.maxChildren(5).create();
tree = tree.add("DAVE", point(10, 20))
           .add("FRED", point(12, 25))
           .add("MARY", point(97, 125));
 
Observable<Entry<String, Point>> entries =
    tree.search(Rectangle.create(8, 15, 30, 35));

See LatLongExampleTest.java for an example. The example depends on grumpy-core artifact which is also on Maven Central.

See LatLongExampleTest.testSearchLatLongCircles() for an example of searching circles around geographic points (using great circle distance).

Very useful, see RxJava.

As an example, suppose you want to filter the search results then apply a function on each and reduce to some best answer:

import rx.Observable;
import rx.functions.*;
import rx.schedulers.Schedulers;

Func1<Entry<String, Geometry>, Character> firstCharacter =
    entry -> entry.value().charAt(0);
Func2<Character,Character,Character> firstAlphabetically 
    = (x,y) -> x <=y ? x : y;

Character result = 
    tree.search(Geometries.rectangle(8, 15, 30, 35))
        // filter for names alphabetically less than M
        .filter(entry -> entry.value() < "M")
        // get the first character of the name
        .map(entry -> firstCharacter(entry.value()))
        // reduce to the first character alphabetically 
        .reduce((x,y) -> firstAlphabetically(x,y))
        // subscribe to the stream and block for the result
        .toBlocking().single();
System.out.println(list);

output:

D

Check out the benchmarks below, but I recommend you do your own benchmarks because every data set will behave differently. If you don't want to benchmark then use the defaults. General rules based on the benchmarks:

• for data sets of <10,000 entries use the default R-tree (quadratic splitter with maxChildren=4)
• for data sets of >=10,000 entries use the star R-tree (R*-tree heuristics with maxChildren=4 by default)

Watch out though, the benchmark data sets had quite specific characteristics. The 1000 entry dataset was randomly generated (so is more or less uniformly distributed) and the Greek dataset was earthquake data with its own clustering characteristics.

If you are not familiar with the Observable API and want to skip the reactive stuff then here's how to get an Iterable from a search:

Iterable<T> it = tree.search(Geometries.point(4,5))
                     .toBlocking().toIterable();

The backpressure slow path may be enabled by some RxJava operators. This may slow search performance by a factor of 3 but avoids possible out of memory errors and thread starvation due to asynchronous buffering. Backpressure is benchmarked below.

To visualize the R-tree in a PNG file of size 600 by 600 pixels just call:

tree.visualize(600,600)
    .save("target/mytree.png");

The result is like the images in the Features section above.

The RTree.asString() method returns output like this:

mbr=Rectangle [x1=10.0, y1=4.0, x2=62.0, y2=85.0]
  mbr=Rectangle [x1=28.0, y1=4.0, x2=34.0, y2=85.0]
    entry=Entry [value=2, geometry=Point [x=29.0, y=4.0]]
    entry=Entry [value=1, geometry=Point [x=28.0, y=19.0]]
    entry=Entry [value=4, geometry=Point [x=34.0, y=85.0]]
  mbr=Rectangle [x1=10.0, y1=45.0, x2=62.0, y2=63.0]
    entry=Entry [value=5, geometry=Point [x=62.0, y=45.0]]
    entry=Entry [value=3, geometry=Point [x=10.0, y=63.0]]

This library has a dependency on guava 18.0 which is about 2.2M. If you are coding for Android you may want to use ProGuard to trim the final application size. The dependency is driven by extensive use of Optional,Preconditions,Objects, and the use of MinMaxPriorityQueue for the nearest-k search. I'm open to the possibility of internalizing these dependencies if people care about the dependency size a lot. Let me know.

git clone https://github.com/davidmoten/rtree.git
cd rtree
mvn clean install

Benchmarks are provided by

mvn clean install -Pbenchmark

This codebase is scanned by Coverity scan whenever the branch coverity_scan is updated.

For the project committers if a coverity scan is desired just do this:

git checkout coverity_scan
git pull origin master
git push origin coverity_scan

The Greek data referred to in the benchmarks is a collection of some 38,377 entries corresponding to the epicentres of earthquakes in Greece between 1964 and 2000. This data set is used by multiple studies on R-trees as a test case.

These were run on [email protected] with rtree version 0.6.1:

Benchmark                                                                                  Mode  Samples       Score       Error  Units
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryInto1000EntriesMaxChildren004           thrpt       10  170629.988 ±  1992.468  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryInto1000EntriesMaxChildren010           thrpt       10  199225.527 ±  3687.661  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryInto1000EntriesMaxChildren032           thrpt       10   92269.915 ±  1777.963  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryInto1000EntriesMaxChildren128           thrpt       10  282006.364 ±  4718.936  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren004      thrpt       10  178533.217 ±  1397.901  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren010      thrpt       10  211693.060 ±  2160.555  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren032      thrpt       10  147575.589 ±  1309.416  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren128      thrpt       10   81962.980 ±  1114.323  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOf1000PointsMaxChildren004                      thrpt       10  841284.193 ± 12172.013  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOf1000PointsMaxChildren010                      thrpt       10  331521.896 ± 15179.039  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOf1000PointsMaxChildren032                      thrpt       10  339449.893 ±  1768.640  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOf1000PointsMaxChildren128                      thrpt       10  430664.624 ±  7939.214  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOfGreekDataPointsMaxChildren004                 thrpt       10  332686.163 ±  7647.466  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOfGreekDataPointsMaxChildren010                 thrpt       10  213191.627 ±  2998.905  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOfGreekDataPointsMaxChildren032                 thrpt       10  122136.799 ±  3024.498  ops/s
c.g.d.r.BenchmarksRTree.defaultRTreeSearchOfGreekDataPointsMaxChildren128                 thrpt       10   49765.765 ±   391.503  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeDeleteOneEveryOccurrenceFromGreekDataChildren010         thrpt       10  195698.148 ±  1650.701  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryInto1000EntriesMaxChildren004              thrpt       10  155788.206 ±  1976.180  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryInto1000EntriesMaxChildren010              thrpt       10  143670.205 ±  1525.071  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryInto1000EntriesMaxChildren032              thrpt       10   37517.491 ±   634.409  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryInto1000EntriesMaxChildren128              thrpt       10  116450.918 ±  1461.198  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren004         thrpt       10  135777.256 ±  1003.535  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren010         thrpt       10  108184.558 ±  1210.154  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren032         thrpt       10   31833.529 ±   349.885  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren128         thrpt       10    3884.568 ±    34.335  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren004                         thrpt       10  438527.764 ±  5882.232  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren010                         thrpt       10  707710.764 ± 13940.099  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren032                         thrpt       10  426923.809 ±  6349.023  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren128                         thrpt       10  698529.029 ±  9430.641  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren004                    thrpt       10  531461.541 ±  8004.566  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren010                    thrpt       10  407222.646 ±  4319.595  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren010WithBackpressure    thrpt       10  109341.392 ±  1624.047  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren032                    thrpt       10  359887.451 ±  2771.129  ops/s
c.g.d.r.BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren128                    thrpt       10  247366.614 ±  3056.753  ops/s