Pluggable R-tree implementation in pure Python.
Pluggable R-tree implementation in pure Python.
Since the original R-tree data structure has been initially proposed in 1984, there have been many variations introduced over the years optimized for various use cases . However, when working in Python (one of the most popular languages for spatial data processing), there is no easy way to quickly compare how these various implementations behave on real data.
The aim of this library is to provide a "pluggable" R-tree implementation that allows swapping out the various strategies for insertion, node deletion, and other behaviors so that their impact can be easily compared (without having to install separate libraries and having to make code changes to accommodate for API differences). Several of the more common R-tree variations will soon be provided as ready-built implementations (see the Status section below).
In addition, this library also provides utilities for inspecting the R-tree structure. It allows creating diagrams (using matplotlib and graphviz) that show the R-tree nodes and entries (including all the intermediate, non-leaf nodes), along with plots of their corresponding bounding boxes. It also allows exporting the R-tree to PostGIS so it could be examined using a GIS viewer like QGIS.
This library is currently in early development. The table below shows which R-tree variants have been implemented, along with which operations they currently support:
The library has a framework in place for swapping out the various strategies, making it possible to add a new R-tree variant. However, given that this library is still early in development, it is anticipated that this framework may need to be extended, resulting in breaking changes.
Contributions for implementing additional strategies and operations are welcome. See the section on Extending below.
This package is available on PyPI and can be installed using pip:
pip install rtreelib
This package requires Python 3.6+.
There are additional optional dependencies you can install if you want to be able to create diagrams or export the R-tree data to PostGIS. See the corresponding sections below for additional setup information.
To instantiate the default implementation and insert some entries:
from rtreelib import RTree, Rect t = RTree() t.insert('a', Rect(0, 0, 3, 3)) t.insert('b', Rect(2, 2, 4, 4)) t.insert('c', Rect(1, 1, 2, 4)) t.insert('d', Rect(8, 8, 10, 10)) t.insert('e', Rect(7, 7, 9, 9))
The first parameter to the
insert method represents the data, and can be of any data type
(though you will want to stick to strings, numbers, and other basic data types that can be
easily and succintly represented as a string if you want to create diagrams). The second
parameter represents the minimum bounding rectangle (MBR) of the associated data element.
The default implementation uses Guttman's original strategies for insertion, node splitting, and deletion, as outlined in his paper from 1984 .
To use the R* implementation instead:
from rtreelib import RStarTree, Rect t = RStarTree() t.insert('a', Rect(0, 0, 3, 3)) t.insert('b', Rect(2, 2, 4, 4)) t.insert('c', Rect(1, 1, 2, 4)) t.insert('d', Rect(8, 8, 10, 10)) t.insert('e', Rect(7, 7, 9, 9))
You can also create a custom implementation by inheriting from
RTreeBase and providing
your own implementations for the various behaviors (insert, overflow, etc.). See the
following section for more information.
query method to find entries at a given location. The library supports querying
by either a point or a rectangle, and returns an iterable of matching entries that
intersect the given location.
To query using
entries = t.query(Point(2, 4))
Alternatively, you can also pass a tuple or list of 2 coordinates (
entries = t.query((2, 4))
When querying by point, note that points that lie on the border (rather than the interior) of a bounding rectangle are considered to intersect the rectangle.
To query using
entries = t.query(Rect(2, 1, 4, 5))
Alternatively, you can also pass a tuple or list of 4 coordinates (the order is the
same as when using
entries = t.query((2, 1, 4, 5))
When querying by rectangle, note that the rectangles must have a non-zero intersection area. Rectangles that intersect at the border but whose interiors do not overlap will not match the query.
Note the above methods return entries rather than nodes. To get an iterable of leaf
nodes instead, use
nodes = t.query_nodes(Rect(2, 1, 4, 5))
By default, this method will only return leaf-level nodes. To include all
intermediate-level nodes (including the root), set the optional
False (it defaults to
True if not passed in):
all_nodes = t.query_nodes(Rect(2, 1, 4, 5), leaves=False)
As noted above, the purpose of this library is to provide a pluggable R-tree implementation where the various behaviors can be swapped out and customized to allow comparison. To that end, this library provides a framework for achieving this.
As an example, the
class (aliased as
RTree) simply inherits from
RTreeBase, providing an implementation
overflow_strategy behaviors as follows:
class RTreeGuttman(RTreeBase[T]): """R-Tree implementation that uses Guttman's strategies for insertion, splitting, and deletion.""" def __init__(self, max_entries: int = DEFAULT_MAX_ENTRIES, min_entries: int = None): """ Initializes the R-Tree using Guttman's strategies for insertion, splitting, and deletion. :param max_entries: Maximum number of entries per node. :param min_entries: Minimum number of entries per node. Defaults to ceil(max_entries/2). """ super().__init__( max_entries=max_entries, min_entries=min_entries, insert=insert, choose_leaf=guttman_choose_leaf, adjust_tree=adjust_tree_strategy, overflow_strategy=quadratic_split )
Each behavior should be a function that implements a specific signature and performs a given task. Here are the behaviors that are currently required to be specified:
insert: Strategy used for inserting a single new entry into the tree.
(tree: RTreeBase[T], data: T, rect: Rect) → RTreeEntry[T]
tree: RTreeBase[T]: R-tree instance.
data: T: Data stored in this entry.
rect: Rect: Bounding rectangle.
- This function should return the newly inserted entry.
choose_leaf: Strategy used for choosing a leaf node when inserting a new entry.
(tree: RTreeBase[T], entry: RTreeEntry[T]) → RTreeNode[T]
tree: RTreeBase[T]: R-tree instance.
entry: RTreeEntry[T]: Entry being inserted.
- This function should return the leaf node where the new entry should be inserted. This
node may or may not have the capacity for the new entry. If the insertion of the new node
results in the node overflowing, then
overflow_strategywill be invoked on the node.
- This function should return the leaf node where the new entry should be inserted. This node may or may not have the capacity for the new entry. If the insertion of the new node results in the node overflowing, then
adjust_tree: Strategy used for balancing the tree, including propagating node splits, updating bounding boxes on all nodes and entries as necessary, and growing the tree by creating a new root if necessary. This strategy is executed after inserting or deleting an entry.
(tree: RTreeBase[T], node: RTreeNode[T], split_node: RTreeNode[T]) → None
tree: RTreeBase[T]: R-tree instance.
node: RTreeNode[T]: Node where a newly-inserted entry has just been added.
split_node: RTreeNode[T]: If the insertion of a new entry has caused the node to split, this is the newly-created split node. Otherwise, this will be
overflow_strategy: Strategy used for handling an overflowing node (a node that contains more than
max_entries). Depending on the implementation, this may involve splitting the node and potentially growing the tree (Guttman), performing a forced reinsert of entries (R*), or some other strategy.
(tree: RTreeBase[T], node: RTreeNode[T]) → RTreeNode[T]
tree: RTreeBase[T]: R-tree instance.
node: RTreeNode[T]: Overflowing node.
- Depending on the implementation, this function may return a newly-created split
node whose entries are a subset of the original node's entries (Guttman), or simply
- Depending on the implementation, this function may return a newly-created split node whose entries are a subset of the original node's entries (Guttman), or simply return
Creating R-tree Diagrams
This library provides a set of utility functions that can be used to create diagrams of the entire R-tree structure, including the root and all intermediate and leaf level nodes and entries.
These features are optional, and the required dependencies are not automatically installed when installing this library. Therefore, you must install them manually. This includes the following Python dependencies which can be installed using pip:
pip install matplotlib pydot tqdm
This also includes the following system-level dependencies:
On Ubuntu, these can be installed using:
sudo apt install python3-tk graphviz
Once the above dependencies are installed, you can create an R-tree diagram as follows:
from rtreelib import RTree, Rect from rtreelib.diagram import create_rtree_diagram # Create an RTree instance with some sample data t = RTree(max_entries=4) t.insert('a', Rect(0, 0, 3, 3)) t.insert('b', Rect(2, 2, 4, 4)) t.insert('c', Rect(1, 1, 2, 4)) t.insert('d', Rect(8, 8, 10, 10)) t.insert('e', Rect(7, 7, 9, 9)) # Create a diagram of the R-tree structure create_rtree_diagram(t)
This creates a diagram like the following:
The diagram is created in a temp directory as a PNG file, and the default viewer is automatically launched for convenience. Each box in the main diagram represents a node (except at the leaf level, where it represents the leaf entry), and contains a plot that depicts all of the data spatially. The bounding boxes of each node are represented using tan rectangles with a dashed outline. The bounding box corresponding to the current node is highlighted in pink.
The bounding boxes for the original data entries themselves are depicted in blue, and are
labeled using the value that was passed in to
insert. At the leaf level, the corresponding
data element is highlighted in pink.
The entries contained in each node are depicted along the bottom of the node's box, and point to either a child node (for non-leaf nodes), or to the data entries (for leaf nodes).
As can be seen in the above screenshot, the diagram depicts the entire tree structure, which can be quite large depending on the number of nodes and entries. It may also take a while to generate, since it launches matplotlib to plot the data spatially for each node and entry, and then graphviz to generate the overall diagram. Given the size and execution time required to generate these diagrams, it's only practical for R-trees containing a relatively small amount of data (e.g., no more than about a dozen total entries). To analyze the resulting R-tree structure when working with a large amount of data, it is recommended to export the data to PostGIS and use a viewer like QGIS (as explained in the following section).
Exporting to PostGIS
In addition to creating diagrams, this library also allows exporting R-trees to a PostGIS database.
To do so, you will first need to install the psycopg2 driver.
This is an optional dependency, so it is not automatically installed when you install
this package. Refer to the
installation instructions for psycopg2 to
ensure that you have all the necessary system-wide prerequisites installed (C compiler,
Python header files, etc.). Then, install
psycopg2 using the following command (passing
--no-binary flag to ensure that it is built from source, and also to avoid a console
warning when using
pip install psycopg2 --no-binary psycopg2
psycopg2 is installed, you should be able to import the functions you need from the
from rtreelib.pg import init_db_pool, create_rtree_tables, export_to_postgis
The subsections below guide you throw how to use this library to export R-trees to the database. You will first need to decide on your preferred method for connecting to the database, as well as create the necessary tables to store the R-tree data. Once these prerequisites are met, exporting the R-tree can be done using a simple function call. Finally, this guide shows how you can visualize the exported data using QGIS, a popular and freely-available GIS viewer.
Initializing a Connection Pool
When working with the
rtreelib.pg module, there are three ways of passing database
- Initialize a connection pool by calling
init_db_pool. This allows using the other functions in this module without having to pass around connection info.
- Manually open the connection yourself, and pass in the connection object to the function.
- Pass in keyword arguments that can be used to establish the database connection.
The first method is generally the easiest - you just have to call it once, and not have to worry about passing in connection information to the other functions. This section explains this method, and the following sections assume that you are using it. However, the other methods are also explained later on in this guide.
init_db_pool accepts the same parameters as the
For example, you can pass in a connection string:
init_db_pool("dbname=mydb user=postgres password=temp123!")
Alternatively, using the URL syntax:
Or keyword arguments:
init_db_pool(user="postgres", password="temp123!", host="localhost", database="mydb")
Next, before you can export an R-tree, you first need to create a few database tables to store the data. The following section explains how to achieve this.
Creating Tables to Store R-tree Data
When exporting an R-tree using this library, the data is populated inside three tables:
rtree: This tables simply contains the ID of each R-tree that was exported. This library allows you to export multiple R-trees at once, and they are differentiated by ID (you can also clear the contents of all tables using
rtree_node: Contains information about each node in the R-tree, including its bounding box (as a PostGIS geometry column), a pointer to the parent entry containing this node, and the level of this node (starting at 0 for the root). The node also contains a reference to the
rtreethat it is a part of.
rtree_entry: Contains information about each entry in the R-tree, including its bounding box (as a PostGIS geometry column) and a pointer to the node containing this entry. For leaf entries, this also contains the value of the data element.
These tables can be created using the
create_rtree_tables function. This is
something you only need to do once.
This function can be called without any arguments if you have established the
connection pool, and your data does not use a spatial reference system (
However, generally when working with spatial data, you will have a particular
SRID that your data is in, in which case you should pass it in to ensure that
all geometry columns use the correct SRID:
You can also choose to create the tables in a different schema (other than
However, in this case, be sure to pass in the same schema to the other functions in this module.
You can also pass in a
datatype, which indicates the type of data stored in the leaf
entries (i.e., the type of the data you pass in to the
insert method of
This can either be a string containing a PostgreSQL column type:
Or a Python type, in which case an appropriate PostgreSQL data type will be inferred:
If you don't pass anything in, or an appropriate PostgreSQL data type cannot be
determined from the Python type, the column type will default to
TEXT, which allows
storing arbitrary-length strings.
When passing a string containing a PostgreSQL column type, you also have the option
of adding a modifier such as
NOT NULL, or even a foreign key constraint:
create_rtree_tables(srid=4326, datatype='INT REFERENCES my_other_table (my_id_column)')
Exporting the R-tree
To export the R-tree once the tables have been created, simply call the
export_to_postgis function, passing in the R-tree instance (and optionally an SRID):
rtree_id = export_to_postgis(tree, srid=4326)
This function populates the
rtree_entry tables with
the data from the R-tree, and returns the ID of the newly-inserted R-tree in the
Note that if you used a schema other than
public when calling
create_rtree_tables, you will need to pass in the same schema when calling
rtree_id = export_to_postgis(tree, srid=4326, schema='temp')
Viewing the Data Using QGIS
QGIS is a popular and freely-available GIS viewer which can be used to visualize the exported R-tree data. To do so, launch QGIS and create a new project. Then, follow these steps to add the exported R-tree data as a layer:
- Go to Layer → Add Layer → Add PostGIS Layers
- Connect to the database where you exported the data
- Select either the
rtree_entrytable, depending on which part of the structure you wish to visualize. For this example, we will be looking at the nodes, so select
- Optionally, you can set a layer filter to only include the nodes belonging to a
particular tree (if you exported multiple R-trees). To do so, click the
Set Filter button, and enter a filter expression (such as
- Click Add
At this point, the layer will be displaying all nodes at every level of the tree, which may be a bit hard to decipher if you have a lot of data. After adjusting the layer style to make it partially transparent, here is an example of what an R-tree with a couple hundred leaf entries might look like (41 nodes across 3 levels):
To make it easier to understand the structure, it might help to be able to view each level of the tree independently. To do this, double click the layer in the Layers panel, switch to the Style tab, and change the style type at the top from "Single symbol" (the default) to "Categorized". Then in the Column dropdown, select the "level" column. You can optionally assign a color ramp or use random colors so that each level gets a different color. Then click Classify to automatically create a separate style for each layer:
Now in the layers panel, each level will be shown as a separate entry and can be toggled on and off, making it possible to explore the R-tree structure one level at a time:
The advantage with exporting the data to QGIS is you can also bring in your original dataset as a layer to see how it was partitioned spatially. Further, you can import multiple R-trees as separate layers and be able to compare them side by side.
Below, I am using a subset of the FAA airspace data for a portion of the
Northeastern US, and then toggling each level of the
rtree_node layer individually
so we can examine the resulting R-tree structure one level at a time. After
compositing these together, you can see how the Guttman R-Tree performs against
It is evident that R* has resulted in more square-like bounding rectangles with less overlap at the intermediate levels, compared to Guttman. The areas of overlap are made especially evident when using a partially transparent fill. Ideally, the spatial partitioning scheme should aim to minimize this overlap, since a query to find the leaf entry for a given point would require visiting multiple subtrees if that point happens to land in one of these darker shaded areas of overlap.
You can also write a query to analyze the amount of overlap that resulted in each level of the tree. For example, the query below returns the total amount of overlap area of all nodes at level 2 of an exported R-tree having ID 1:
SELECT ST_Area(ST_Union(ST_Intersection(n1.bbox, n2.bbox))) AS OverlapArea FROM temp.rtree t INNER JOIN temp.rtree_node n1 ON n1.rtree_id = t.id INNER JOIN temp.rtree_node n2 ON n2.rtree_id = t.id AND n1.level = n2.level WHERE t.id = 1 AND n1.level = 2 AND ST_Overlaps(n1.bbox, n2.bbox) AND n1.id <> n2.id;
Extending this even further, you can compare the total overlap area of multiple exported R-trees by level:
SELECT CASE t.id WHEN 1 THEN 'Guttman' WHEN 2 THEN 'R*' END AS tree, n.level, ST_Area(ST_Union(ST_Intersection(n.bbox, n2.bbox))) AS OverlapArea FROM temp.rtree t INNER JOIN temp.rtree_node n ON n.rtree_id = t.id INNER JOIN temp.rtree_node n2 ON n2.rtree_id = t.id AND n.level = n2.level WHERE ST_Overlaps(n.bbox, n2.bbox) AND n.id <> n2.id GROUP BY t.id, n.level ORDER BY t.id, n.level;
The above query may return a result like the following:
In the above example, the R*-Tree (
id=2) achieved a smaller overlap area at
every level of the tree compared to Guttman (
As mentioned above, when you call
export_to_postgis, the existing data in the
tables is not cleared. This allows you to export multiple R-trees at once and
compare them side-by-side.
However, for simplicity, you may wish to clear out the existing data prior to
exporting new data. To do so, call
This will perform a SQL
TRUNCATE on all R-tree tables.
Note that if you created the tables in a different schema (other than
you will need to pass in that same schema to this function:
You may also wish to completely drop all the tables that were created by
create_rtree_tables. To do so, call
Again, you may need to pass in a schema if it is something other than
Alternate Database Connection Handling Methods
As mentioned earlier in this guide, instead of initializing a connection pool, you have other options for how to handle establishing database connections when using this library. You can choose to handle opening and closing the connection yourself and pass in the connection object; alternatively, you can pass in the connection information as keyword arguments.
To establish the database connection yourself, the typical usage scenario might look like this:
import psycopg2 from rtreelib import RTree, Rect from rtreelib.pg import init_db_pool, create_rtree_tables, clear_rtree_tables, export_to_postgis, drop_rtree_tables # Create an RTree instance with some sample data t = RTree(max_entries=4) t.insert('a', Rect(0, 0, 3, 3)) t.insert('b', Rect(2, 2, 4, 4)) t.insert('c', Rect(1, 1, 2, 4)) t.insert('d', Rect(8, 8, 10, 10)) t.insert('e', Rect(7, 7, 9, 9)) # Export R-tree to PostGIS (using explicit connection) conn = None try: conn = psycopg2.connect(user="postgres", password="temp123!", host="localhost", database="mydb") create_rtree_tables(conn, schema='temp') rtree_id = export_to_postgis(t, conn=conn, schema='temp') print(rtree_id) finally: if conn: conn.close()
You can also pass in the database connection information separately to each method as keyword arguments. These keyword arguments should be the same ones as required by the psycopg2.connect function:
rtree_id = export_to_postgis(tree, schema='temp', user="postgres", password="temp123!", host="localhost", database="mydb")
: Nanopoulos, Alexandros & Papadopoulos, Apostolos (2003): "R-Trees Have Grown Everywhere"
: Guttman, A. (1984): "R-trees: a Dynamic Index Structure for Spatial Searching" (PDF), Proceedings of the 1984 ACM SIGMOD international conference on Management of data – SIGMOD '84. p. 47.
: Beckmann, Norbert, et al. "The R*-tree: an efficient and robust access method for points and rectangles." Proceedings of the 1990 ACM SIGMOD international conference on Management of data. 1990.
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