rtreelib

Pluggable R-tree implementation in pure Python.

Overview

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 [1]. 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.

Status

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:

R-Tree Variant	Insert	Update	Delete
Guttman [2]	✔️	🔲	🔲
R*-Tree [3]	✔️	🔲	🔲

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.

Setup

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.

Usage

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 [2].

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.

Querying

Use the 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 Point:

entries = t.query(Point(2, 4))

Alternatively, you can also pass a tuple or list of 2 coordinates (x and y):

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 Rect:

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 Rect, namely min_x, min_y, max_x, and max_y):

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 query_nodes:

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 leaves parameter to False (it defaults to True if not passed in):

all_nodes = t.query_nodes(Rect(2, 1, 4, 5), leaves=False)

Extending

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 RTreeGuttman class (aliased as RTree) simply inherits from RTreeBase, providing an implementation for the insert, choose_leaf, adjust_tree, and 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.
  • Signature: (tree: RTreeBase[T], data: T, rect: Rect) → RTreeEntry[T]
  • Arguments:
    • tree: RTreeBase[T]: R-tree instance.
    • data: T: Data stored in this entry.
    • rect: Rect: Bounding rectangle.
  • Returns: RTreeEntry[T]
    • This function should return the newly inserted entry.
• choose_leaf: Strategy used for choosing a leaf node when inserting a new entry.
  • Signature: (tree: RTreeBase[T], entry: RTreeEntry[T]) → RTreeNode[T]
  • Arguments:
    • tree: RTreeBase[T]: R-tree instance.
    • entry: RTreeEntry[T]: Entry being inserted.
  • Returns: RTreeNode[T]
    • 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_strategy will be invoked on the node.
• 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.
  • Signature: (tree: RTreeBase[T], node: RTreeNode[T], split_node: RTreeNode[T]) → None
  • Arguments:
    • 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 None.
  • Returns: None
• 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.
  • Signature: (tree: RTreeBase[T], node: RTreeNode[T]) → RTreeNode[T]
  • Arguments:
    • tree: RTreeBase[T]: R-tree instance.
    • node: RTreeNode[T]: Overflowing node.
  • Returns: RTreeNode[T]
    • 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 None.

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:

• TkInter
• Graphviz

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 the --no-binary flag to ensure that it is built from source, and also to avoid a console warning when using psycopg2):

pip install psycopg2 --no-binary psycopg2

Once psycopg2 is installed, you should be able to import the functions you need from the rtreelib.pg module:

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 connection information:

1. 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.
2. Manually open the connection yourself, and pass in the connection object to the function.
3. 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 psycopg2.connect function. For example, you can pass in a connection string:

init_db_pool("dbname=mydb user=postgres password=temp123!")

Alternatively, using the URL syntax:

init_db_pool("postgresql://localhost/mydb?user=postgres&password=temp123!")

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 clear_rtree_tables).
• 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 rtree that 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 (srid). 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:

create_rtree_tables(srid=4326)

You can also choose to create the tables in a different schema (other than public):

create_rtree_tables(srid=4326, schema="temp")

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 RTree). This can either be a string containing a PostgreSQL column type:

create_rtree_tables(srid=4326, datatype='VARCHAR(255)')

Or a Python type, in which case an appropriate PostgreSQL data type will be inferred:

create_rtree_tables(srid=4326, datatype=int)

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, rtree_node, and rtree_entry tables with the data from the R-tree, and returns the ID of the newly-inserted R-tree in the rtree table.

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 export_to_postgis:

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_node or rtree_entry table, depending on which part of the structure you wish to visualize. For this example, we will be looking at the nodes, so select rtree_node.
• 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 rtree_id=1).
• 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 R*.

Guttman:

R*-Tree:

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:

tree	level	OverlapArea
Guttman	1	7.89e+11
Guttman	2	9.12e+11
Guttman	3	4.75e+11
R*	1	3.97e+11
R*	2	4.35e+11
R*	3	1.80e+11

In the above example, the R*-Tree (id=2) achieved a smaller overlap area at every level of the tree compared to Guttman (id=1).

Cleaning Up

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 clear_rtree_tables:

clear_rtree_tables()

This will perform a SQL TRUNCATE on all R-tree tables.

Note that if you created the tables in a different schema (other than public), you will need to pass in that same schema to this function:

clear_rtree_tables(schema='temp')

You may also wish to completely drop all the tables that were created by create_rtree_tables. To do so, call drop_rtree_tables:

drop_rtree_tables()

Again, you may need to pass in a schema if it is something other than public:

drop_rtree_tables(schema='temp')

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")

References

[1]: Nanopoulos, Alexandros & Papadopoulos, Apostolos (2003): "R-Trees Have Grown Everywhere"

[2]: 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.

[3]: 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.