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Java library implementing immutable R-tree and R*-tree for n dimensions

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rtree-multi
=========
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In-memory immutable [R-tree](http://en.wikipedia.org/wiki/R-tree) implementation for n dimensions.

Status: *in development*

An [R-tree](http://en.wikipedia.org/wiki/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](src/main/java/com/github/davidmoten/rtree-multi/internal/LeafDefault.java) and [NonLeaf.java](src/main/java/com/github/davidmoten/rtree-multi/internal/NonLeafDefault.java).

Iterator support requires a bookmark 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](https://davidmoten.github.io/rtree-multi/index.html) including [javadoc](https://davidmoten.github.io/rtree-multi/apidocs/index.html).

Features
------------
* supports n dimensions
* immutable R-tree suitable for concurrency
* Guttman's heuristics (Quadratic splitter) ([paper](https://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CB8QFjAA&url=http%3A%2F%2Fpostgis.org%2Fsupport%2Frtree.pdf&ei=ieEQVJuKGdK8uATpgoKQCg&usg=AFQjCNED9w2KjgiAa9UI-UO_0eWjcADTng&sig2=rZ_dzKHBHY62BlkBuw3oCw&bvm=bv.74894050,d.c2E))
* R*-tree heuristics ([paper](http://dbs.mathematik.uni-marburg.de/publications/myPapers/1990/BKSS90.pdf))
* Customizable [splitter](src/main/java/com/github/davidmoten/rtree-multi/Splitter.java) and [selector](src/main/java/com/github/davidmoten/rtree-multi/Selector.java)
* search is ```O(log(n))``` on average
* insert, delete are ```O(n)``` worst case
* balanced delete
* uses structural sharing
* JMH benchmarks
* visualizer included
* decent unit test [code coverage](http://davidmoten.github.io/rtree-multi/cobertura/index.html) 
* R*-tree performs ? 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 ? entries per second.
* requires java 1.8 or later

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

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).

# 3 dimensions
Given the 38,377 data points of greek earthquakes (lat, long, time) from 1964 to 2000, the data is scanned to establish the ranges for each coordinate then normalized to a [0,1] range. The points are shuffled then added to an R-tree with `minChildren=2` and `maxChildren=4` using either the R* heuristics or standard R-tree heuristics. Visualization of the bounding boxes at nodes by method and depth is below.

The plots below are generated from the same shuffle of data points.

| Quadratic split | R*-tree split |
| :-------------: | :-----------: |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot0-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot0.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot1-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot1.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot2-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot2.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot3-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot3.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot4-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot4.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot5-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot5.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot6-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot6.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot7-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot7.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot8-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot8.png" /> |
| <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot9-q.png" /> | <img src="https://raw.githubusercontent.com/davidmoten/davidmoten.github.io/master/resources/rtree-3d/plot9.png" /> |

Getting started
----------------
Add this maven dependency to your pom.xml:

```xml
<dependency>
  <groupId>com.github.davidmoten</groupId>
  <artifactId>rtree-multi</artifactId>
  <version>VERSION_HERE</version>
</dependency>
```
### Instantiate an R-Tree
Use the static builder methods on the ```RTree``` class:

```java
// 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*:

```java
RTree<String, Geometry> tree = RTree.minChildren(3).maxChildren(6).create();
```
### Geometries
The following geometries are supported for insertion in an RTree:

* `Rectangle`
* `Point`

### 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:

```java
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. You can use these factory methods to create your geometries:

* ```Rectangle.create```
* ```Point.create```

To add an item to an R-tree:

```java
RTree<T,Geometry> tree = RTree.dimensions(3).create();
tree = tree.add(item, Point.create(10, 20, 30));
```
or 
```java
tree = tree.add(Entry.entry(item, Point.create(10, 20, 30));
```

*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:

```java
tree = tree.delete(item, Point.create(10, 20, 30));
```
or 
```java
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```](src/main/java/com/github/davidmoten/rtree/geometry/Geometry.java). 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 ```Iterable``` sequences:
```java
Iterable<Entry<T, Geometry>> results =
    tree.search(Rectangle.create(0, 0, 2, 2));
```
or search for items within a distance from the given geometry:
```java
Iterable<Entry<T, Geometry>> results =
    tree.search(Rectangle.create(0, 0, 2, 2), 5.0);
```
To return all entries from an R-tree:
```java
Iterable<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:

```java
RTree<String, Point> tree = RTree.dimensions(2).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:

```java
RTree<String, Point> tree = RTree.create();
Func2<Point, Polygon, Boolean> distancePointToPolygon = ...
Polygon polygon = ...
...
entries = tree.search(polygon, 10, distancePointToPolygon);
```
Example
--------------
```java
import com.github.davidmoten.rtree-multi.RTree;

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

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:

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

// create geometry using single precision
Rectangle r = Rectangle.create(1.0f, 2.0f, 3.0f, 4.0f);
```

Visualizer
--------------
To visualize a 2D R-tree (only) in a PNG file of size 600 by 600 pixels just call:
```java
tree.visualize(600,600)
    .save("target/mytree.png");
```
The result is like the images in the Features section above.

Visualize as text
--------------------
The ```RTree.asString()``` method returns output like this:

```
mbr=Rectangle [mins=[10.0, 4.0], maxes=[62.0, 85.0]]
  mbr=Rectangle [mins=[28.0, 4.0], maxes=[34.0, 85.0]]
    entry=Entry [value=2, geometry=Point [29.0, 4.0]]
    entry=Entry [value=1, geometry=Point [28.0, 19.0]]
    entry=Entry [value=4, geometry=Point [34.0, 85.0]]
  mbr=Rectangle [mins=[10.0, 45.0], maxes=[62.0, 63.0]]
    entry=Entry [value=5, geometry=Point [62.0, 45.0]]
    entry=Entry [value=3, geometry=Point [10.0, 63.0]]
```

Serialization
------------------
Serialization is not supported directly in *rtree-multi*. Your best bet is to serialize entries from and `RTree` as you like (just the entries) and then use bulk loading when deserializing. Bulk loading is performed like this:

```java
// deserialize the entries from disk (for example)
List<Entry<Thing, Point> entries = ...
// bulk load
RTree<Thing, Point> tree = RTree.maxChildren(28).star().create(entries); 
```

How to build
----------------
```
git clone https://github.com/davidmoten/rtree-multi.git
cd rtree-multi
mvn clean install
```

How to run benchmarks
--------------------------
Benchmarks are provided by 
```
mvn clean install -P benchmark
```

### Notes
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.

### Results

These were run on i7-920 @2.67GHz with *rtree* version 0.9-RC1:

```
# JMH version: 1.21
# VM version: JDK 1.8.0_201, Java HotSpot(TM) 64-Bit Server VM, 25.201-b09
# VM invoker: /usr/lib/jvm/java-8-oracle/jre/bin/java
# VM options: -Xmx512m
# Warmup: 3 iterations, 10 s each
# Measurement: 10 iterations, 10 s each
# Timeout: 10 min per iteration
# Threads: 1 thread, will synchronize iterations
# Benchmark mode: Throughput, ops/time

Benchmark                                                        Mode  Cnt        Score       Error  Units
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren004       thrpt   10  1021844.129 ± 16977.542  ops/s
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren010       thrpt   10   931789.301 ± 33477.764  ops/s
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren032       thrpt   10   304816.886 ±  5674.364  ops/s
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren128       thrpt   10   606188.199 ±  7553.533  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren004  thrpt   10   527595.512 ± 34400.389  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren010  thrpt   10   194793.663 ±  5385.855  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren032  thrpt   10   297597.166 ± 40175.027  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren128  thrpt   10   221605.832 ±  9332.555  ops/s
BenchmarksRTree.searchNearestGreek                              thrpt   10     3534.733 ±   177.370  ops/s
```

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