Data Structures
Sep 22, 2014
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Version (Dec 14, 2017)
Aug 26, 2014
Feb 16, 2018
Dave Moten (davidmoten)
Dave Moten (davidmoten)
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In-memory immutable 2D R-tree implementation in java using RxJava Observables for reactive processing of search results.

Status: released to Maven Central

An R-tree is a commonly used spatial index.

This was fun to make, has an elegant concise algorithm, is thread-safe, fast, and reasonably memory efficient (uses structural sharing).

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
  • serialization using FlatBuffers
  • high unit test code coverage
  • R*-tree performs 900,000 searches/second returning 22 entries from a tree of 38,377 Greek earthquake locations on i7-920@2.67Ghz (maxChildren=4, minChildren=1). Insert at 240,000 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).

Getting started

Add this maven dependency to your pom.xml:


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();


The following geometries are supported for insertion in an RTree:

  • Rectangle
  • Point
  • Circle
  • Line (requires JTS dependency, look at pom.xml)

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();


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
  • Geometries.line (requires jts-core dependency)

To add an item to an R-tree:

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


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));


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!

Geospatial geometries (lats and longs)

To handle wraparounds of longitude values on the earth (180/-180 boundary trickiness) there are special factory methods in the Geometries class. If you want to do geospatial searches then you should use these methods to build Points and Rectangles:

Point point = Geometries.pointGeographic(lon, lat);
Rectangle rectangle = Geometries.rectangleGeographic(lon1, lat1, lon2, lat2);

Under the covers these methods normalize the longitude value to be in the interval [-180, 180) and for rectangles the rightmost longitude has 360 added to it if it is less than the leftmost longitude.

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


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 =

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

Observable<Entry<T, Geometry>> results =

To return all entries from an R-tree:

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

Search with a custom geometry

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) .

Search with a custom geometry and maxDistance

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(Geometries.rectangle(8, 15, 30, 35));

Searching by distance on lat longs

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

Another lat long example searching geo circles

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

What do I do with the Observable thing?

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;

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 -> entry.value().charAt(0))
        // reduce to the first character alphabetically 
        .reduce((x,y) -> x <= y ? x : y)
        // subscribe to the stream and block for the result



How to configure the R-tree for best performance

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.

What about memory use?

To minimize memory use you can use geometries that store single precision decimal values ( float) instead of double precision ( double). Here are examples:

// create geometry using double precision 
Rectangle r = Geometries.rectangle(1.0, 2.0, 3.0, 4.0);

// create geometry using single precision
Rectangel r = Geometries.rectangle(1.0f, 2.0f, 3.0f, 4.0f);

The same creation methods exist for Circle and Line.

How do I just get an Iterable back from a search?

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))


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:


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

Visualize as text

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]]


Release 0.8 includes flatbuffers support as a serialization format and as a lower performance but lower memory consumption (approximately one third) option for an RTree.

The greek earthquake data (38,377 entries) when placed in a default RTree with maxChildren=10 takes up 4,548,133 bytes in memory. If that data is serialized then reloaded into memory using the InternalStructure.FLATBUFFERS_SINGLE_ARRAY option then the RTree takes up 1,431,772 bytes in memory (approximately one third the memory usage). Bear in mind though that searches are much more expensive (at the moment) with this data structure because of object creation and gc pressures (see benchmarks). Further work would be to enable direct searching of the underlying array without object creation expenses required to match the current search routines.

As of 5 March 2016, indicative RTree metrics using flatbuffers data structure are:

  • one third the memory use with log(N) object creations per search
  • one third the speed with backpressure (e.g. if flatMap or observeOn is downstream)
  • one tenth the speed without backpressure

Note that serialization uses an optional dependency on flatbuffers. Add the following to your pom dependencies:


Serialization example

Write an RTree to an OutputStream:

RTree<String, Point> tree = ...;
OutputStream os = ...;
Serializer<String, Point> serializer = 
serializer.write(tree, os); 

Read an RTree from an InputStream into a low-memory flatbuffers based structure:

RTree<String, Point> tree = 
  serializer.read(is, lengthBytes, InnerStructure.SINGLE_ARRAY);

Read an RTree from an InputStream into a default structure:

RTree<String, Point> tree = 
  serializer.read(is, lengthBytes, InnerStructure.DEFAULT);


As of 0.7.5 this library does not depend on guava (>2M) but rather depends on guava-mini (11K). The nearest search used to depend on MinMaxPriorityQueue from guava but now uses a backport of Java 8 PriorityQueue inside a custom BoundedPriorityQueue class that gives about 1.7x the throughput as the guava class.

How to build

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

How to run benchmarks

Benchmarks are provided by

mvn clean install -Pbenchmark

Coverity scan

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 i7-920 @2.67GHz with rtree version 0.8-RC7:

Benchmark                                                               Mode  Cnt        Score       Error  Units

defaultRTreeInsertOneEntryInto1000EntriesMaxChildren004                thrpt   10   262260.993 ±  2767.035  ops/s
defaultRTreeInsertOneEntryInto1000EntriesMaxChildren010                thrpt   10   296264.913 ±  2836.358  ops/s
defaultRTreeInsertOneEntryInto1000EntriesMaxChildren032                thrpt   10   135118.271 ±  1722.039  ops/s
defaultRTreeInsertOneEntryInto1000EntriesMaxChildren128                thrpt   10   315851.452 ±  3097.496  ops/s
defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren004           thrpt   10   278761.674 ±  4182.761  ops/s
defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren010           thrpt   10   315254.478 ±  4104.206  ops/s
defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren032           thrpt   10   214509.476 ±  1555.816  ops/s
defaultRTreeInsertOneEntryIntoGreekDataEntriesMaxChildren128           thrpt   10   118094.486 ±  1118.983  ops/s
defaultRTreeSearchOf1000PointsMaxChildren004                           thrpt   10  1122140.598 ±  8509.106  ops/s
defaultRTreeSearchOf1000PointsMaxChildren010                           thrpt   10   569779.807 ±  4206.544  ops/s
defaultRTreeSearchOf1000PointsMaxChildren032                           thrpt   10   238251.898 ±  3916.281  ops/s
defaultRTreeSearchOf1000PointsMaxChildren128                           thrpt   10   702437.901 ±  5108.786  ops/s
defaultRTreeSearchOfGreekDataPointsMaxChildren004                      thrpt   10   462243.509 ±  7076.045  ops/s
defaultRTreeSearchOfGreekDataPointsMaxChildren010                      thrpt   10   326395.724 ±  1699.043  ops/s
defaultRTreeSearchOfGreekDataPointsMaxChildren032                      thrpt   10   156978.822 ±  1993.372  ops/s
defaultRTreeSearchOfGreekDataPointsMaxChildren128                      thrpt   10    68267.160 ±   929.236  ops/s
rStarTreeDeleteOneEveryOccurrenceFromGreekDataChildren010              thrpt   10   211881.061 ±  3246.693  ops/s
rStarTreeInsertOneEntryInto1000EntriesMaxChildren004                   thrpt   10   187062.089 ±  3005.413  ops/s
rStarTreeInsertOneEntryInto1000EntriesMaxChildren010                   thrpt   10   186767.045 ±  2291.196  ops/s
rStarTreeInsertOneEntryInto1000EntriesMaxChildren032                   thrpt   10    37940.625 ±   743.789  ops/s
rStarTreeInsertOneEntryInto1000EntriesMaxChildren128                   thrpt   10   151897.089 ±   674.941  ops/s
rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren004              thrpt   10   237708.825 ±  1644.611  ops/s
rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren010              thrpt   10   229577.905 ±  4234.760  ops/s
rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren032              thrpt   10    78290.971 ±   393.030  ops/s
rStarTreeInsertOneEntryIntoGreekDataEntriesMaxChildren128              thrpt   10     6521.010 ±    50.798  ops/s
rStarTreeSearchOf1000PointsMaxChildren004                              thrpt   10  1330510.951 ± 18289.410  ops/s
rStarTreeSearchOf1000PointsMaxChildren010                              thrpt   10  1204347.202 ± 17403.105  ops/s
rStarTreeSearchOf1000PointsMaxChildren032                              thrpt   10   576765.468 ±  8909.880  ops/s
rStarTreeSearchOf1000PointsMaxChildren128                              thrpt   10  1028316.856 ± 13747.282  ops/s
rStarTreeSearchOfGreekDataPointsMaxChildren004                         thrpt   10   904494.751 ± 15640.005  ops/s
rStarTreeSearchOfGreekDataPointsMaxChildren010                         thrpt   10   649636.969 ± 16383.786  ops/s
rStarTreeSearchOfGreekDataPointsMaxChildren010FlatBuffers              thrpt   10    84230.053 ±  1869.345  ops/s
rStarTreeSearchOfGreekDataPointsMaxChildren010FlatBuffersBackpressure  thrpt   10    36420.500 ±  1572.298  ops/s
rStarTreeSearchOfGreekDataPointsMaxChildren010WithBackpressure         thrpt   10   116970.445 ±  1955.659  ops/s
rStarTreeSearchOfGreekDataPointsMaxChildren032                         thrpt   10   224874.016 ± 14462.325  ops/s
rStarTreeSearchOfGreekDataPointsMaxChildren128                         thrpt   10   358636.637 ±  4886.459  ops/s
searchNearestGreek                                                     thrpt   10     3715.020 ±    46.570  ops/s