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# The SQLite R*Tree Module

## 1.0 Overview

An R-Tree is a special index that is designed for doing range queries. R-Trees are most commonly used in geospatial systems where each entry is a rectangle with minimum and maximum X and Y coordinates. Given a query rectangle, an R-Tree is able to quickly find all entries that are contained within the query rectangle or which overlap the query rectangle. This idea is easily extended to three dimensions for use in CAD systems. R-Trees also find use in time-domain range look-ups. For example, suppose a database records the starting and ending times for a large number of events. A R-Tree is able to quickly find all events, for example, that were active at any time during a given time interval, or all events that started during a particular time interval, or all events that both started and ended within a given time interval. And so forth.

The R-Tree concept originated with Toni Guttman: R-Trees: A Dynamic Index Structure for Spatial Searching, Proc. 1984 ACM SIGMOD International Conference on Management of Data, pp. 47-57. The implementation found in SQLite is a refinement of Guttman's original idea, commonly called "R*Trees", that was described by Norbert Beckmann, Hans-Peter Kriegel, Ralf Schneider, Bernhard Seeger: The R*-Tree: An Efficient and Robust Access Method for Points and Rectangles. SIGMOD Conference 1990: 322-331.

## 2.0 Compiling The R*Tree Module

The source code to the SQLite R*Tree module is included as part of the amalgamation but is disabled by default. To enable the R*Tree module, simply compile with the SQLITE_ENABLE_RTREE C-preprocessor macro defined. With many compilers, this is accomplished by adding the option "-DSQLITE_ENABLE_RTREE=1" to the compiler command-line.

## 3.0 Using the R*Tree Module

The SQLite R*Tree module is implemented as a virtual table. Each R*Tree index is a virtual table with an odd number of columns between 3 and 11. The first column is always a 64-bit signed integer primary key. The other columns are minimum- and maximum-value pairs (stored as 32-bit floating point numbers) for each dimension. A 1-dimensional R*Tree thus has 3 columns. A 2-dimensional R*Tree (the most common case) has 5 columns. A 5-dimensional R*Tree has 11 columns. The SQLite R*Tree implementation does not support R*Trees wider than 5 dimensions.

The first column of an SQLite R*Tree must always be an integer primary key. The min/max-value pair columns are always stored as 32-bit floating point values. Unlike regular SQLite tables which can store data in a variety of datatypes and formats, the R*Tree indices rigidly enforce these two storage types. Attempts to insert something other than an integer into the first column, or something other than a floating point value into the other columns, will result in an error.

### 3.1 Creating An R*Tree Index

A new R*Tree index is created as follows:

>     CREATE VIRTUAL TABLE <name> USING rtree(<column-names>);

The <name> is the name your application chooses for the R*Tree index and <column-names> is a comma separated list of between 3 and 11 columns. The virtual <name> table creates three "shadow" tables to actually store its content. The names of these shadow tables are:

> <name>_node
> <name>_rowid
> <name>_parent

The shadow tables are ordinary SQLite data tables. You can query them directly if you like, though this unlikely to reveal anything particularly useful. And you can UPDATE, DELETE, INSERT or even DROP the shadow tables, though doing so will corrupt your R*Tree index. So it is best to simply ignore the shadow tables. Recognize that they are there to hold your R*Tree index information and let it go as that.

As an example, consider creating a two-dimensional R*Tree index for use in spatial queries:

>     CREATE VIRTUAL TABLE demo_index USING rtree(
>        id,              -- Integer primary key
>        minX, maxX,      -- Minimum and maximum X coordinate
>        minY, maxY       -- Minimum and maximum Y coordinate
>     );

### 3.2 Populating An R*Tree Index

The usual INSERT, UPDATE, and DELETE commands work on an R*Tree index just like on regular tables. So to insert some data into our sample R*Tree index, we can do something like this:

>     INSERT INTO demo_index VALUES(
>         1,                   -- Primary key -- SQLite.org headquarters
>         -80.7749, -80.7747,  -- Longitude range
>         35.3776, 35.3778     -- Latitude range
>     );
>     INSERT INTO demo_index VALUES(
>         2,                   -- NC 12th Congressional District in 2010
>         -81.0, -79.6,
>         35.0, 36.2
>     );

The entries above might represent (for example) a bounding box around the main office for SQLite.org and bounding box around the 12th Congressional District of North Carolina (prior to the 2011 redistricting) in which SQLite.org was located.

### 3.3 Querying An R*Tree Index

Any valid query will work against an R*Tree index. But the R*Tree implementation is designed to make two kinds of queries especially efficient. First, queries against the primary key are efficient:

>     SELECT * FROM demo_index WHERE id=1;

Of course, an ordinary SQLite table will do a query against its integer primary key efficiently, so the previous is no big deal. The real reason for using an R*Tree in the first place is so that you can efficiently do inequality queries against the coordinate ranges. To find all elements of the index that are contained within the vicinity of Charlotte, North Carolina, one might do:

>     SELECT id FROM demo_index
>      WHERE minX>=-81.08 AND maxX<=-80.58
>        AND minY>=35.00  AND maxY<=35.44;

The query above would very quickly locate the id of 1 even if the R*Tree contained millions of entries. The previous is an example of a "contained-within" query. The R*Tree also supports "overlapping" queries. For example, to find all bounding boxes that overlap the Charlotte area:

>     SELECT id FROM demo_index
>      WHERE maxX>=-81.08 AND minX<=-80.58
>        AND maxY>=35.00  AND minY<=35.44;

This second query would find both entry 1 (the SQLite.org office) which is entirely contained within the query box and also the 12th Congressional District which extends well outside the query box but still overlaps the query box.

Note that is not necessary for all coordinates in an R*Tree index to be constrained in order for the index search to be efficient. One might, for example, want to query all objects that overlap with the 35th parallel:

>     SELECT id FROM demo_index
>      WHERE maxY>=35.0  AND minY<=35.0;

But, generally speaking, the more constraints that the R*Tree module has to work with, and the smaller the bounding box, the faster the results will come back.

### 3.3 Roundoff Error

By default, coordinates are stored in an R*Tree using 32-bit floating point values. When a coordinate cannot be exactly represented by a 32-bit floating point number, the lower-bound coordinates are rounded down and the upper-bound coordinates are rounded up. Thus, bounding boxes might be slightly larger than specified, but will never be any smaller. This is exactly what is desired for doing the more common "overlapping" queries where the application wants to find every entry in the R*Tree that overlaps a query bounding box. Rounding the entry bounding boxes outward might cause a few extra entries to appears in an overlapping query if the edge of the entry bounding box corresponds to an edge of the query bounding box. But the overlapping query will never miss a valid table entry.

However, for a "contained-within" style query, rounding the bounding boxes outward might cause some entries to be excluded from the result set if the edge of the entry bounding box corresponds to the edge of the query bounding box. To guard against this, applications should expand their contained-within query boxes slightly (by 0.000012%) by rounding down the lower coordinates and rounding up the top coordinates, in each dimension.

## 4.0 Using R*Trees Effectively

The only information that an R*Tree index stores about an object is its integer ID and its bounding box. Additional information needs to be stored in separate tables and related to the R*Tree index using the primary key. For the example above, one might create an auxiliary table as follows:

>     CREATE TABLE demo_data(
>       id INTEGER PRIMARY KEY,  -- primary key
>       objname TEXT,            -- name of the object
>       objtype TEXT,            -- object type
>       boundary BLOB            -- detailed boundary of object
>     );

In this example, the demo_data.boundary field is intended to hold some kind of binary representation of the precise boundaries of the object. The R*Tree index only holds an axis-aligned rectangular boundary for the object. The R*Tree boundary is just an approximation of the true object boundary. So what typically happens is that the R*Tree index is used to narrow a search down to a list of candidate objects and then more detailed and expensive computations are done on each candidate to find if the candidate truly meets the search criteria.

> Key Point: An R*Tree index does not normally provide the exact answer but merely reduces the set of potential answers from millions to dozens.

Suppose the demo_data.boundary field holds some proprietary data description of a complex two-dimensional boundary for an object and suppose that the application has used the sqlite3_create_function() interface to created application-defined functions "contained_in" and "overlaps" accepting two demo_data.boundary objects and return true or false. One may assume that "contained_in" and "overlaps" are relatively slow functions that we do not want to invoke too frequently. Then an efficient way to find the name of all objects located within the North Carolina 12th District, one may be to run a query like this:

>     SELECT objname FROM demo_data, demo_index
>      WHERE demo_data.id=demo_index.id
>        AND contained_in(demo_data.boundary, :boundary)
>        AND minX>=-81.0 AND maxX<=-79.6
>        AND minY>=35.0 AND maxY<=36.2;

In the query above, one would presumably bind the binary BLOB description of the precise boundary of the 12th district to the ":boundary" parameter.

Notice how the query above works: The R*Tree index runs in the outer loop to find entries that are contained within the bounding box of longitude -81..-79.6 and latitude 35.0..36.2. For each object identifier found, SQLite looks up the corresponding entry in the demo_data table. It then uses the boundary field from the demo_data table as a parameter to the contained_in() function and if that function returns true, the objname field from the demo_data table is returned as the next row of query result.

One would get the same answer without the use of the R*Tree index using the following simpler query:

>     SELECT objname FROM demo_data
>      WHERE contained_in(demo_data.boundary, :boundary);

The problem with this latter query is that it must apply the contained_in() function to millions of entries in the demo_data table. The use of the R*Tree in the penultimate query reduces the number of calls to contained_in() function to a small subset of the entire table. The R*Tree index did not find the exact answer itself, it merely limited the search space.

## 5.0 Integer-Valued R-Trees

The default virtual table ("rtree") normally stores coordinates as single-precision (4-byte) floating point numbers. If integer coordinates are desired, declare the table using "rtree_i32" instead:

>     CREATE VIRTUAL TABLE intrtree USING rtree_i32(id,x0,x1,y0,y1,z0,z1);

An rtree_i32 stores coordinates as 32-bit signed integers. But it still using floating point computations internally as part of the r-tree algorithm. For applications running on processors without hardware floating point, it might be desirable to have a pure integer implementation of r-trees. This is accomplished by compiling with the SQLITE_RTREE_INT_ONLY option. When SQLITE_RTREE_INT_ONLY is used, both the "rtree" and the "rtree_i32" virtual tables store 32-bit integers and only integer values are used for internal computations.

## 6.0 Custom R-Tree Queries

By using standard SQL expressions in the WHERE clause of a SELECT query, a user may query for all r-tree entries that intersect a specified bounding-box, or for all entries that are completely encapsulated by a specified bounding-box. Custom r-tree queries, which use the special MATCH operator in the WHERE clause of a SELECT, allow the user to query for the set of r-tree entries that intersect any arbitrarily defined region.

Regions for custom r-tree queries are defined by r-tree geometry callbacks implemented by the application and registered with SQLite via a call to the following API:

>     int sqlite3_rtree_geometry_callback(
>       sqlite3 *db,
>       const char *zGeom,
>       int (*xGeom)(sqlite3_rtree_geometry *, int nCoord, double *aCoord, int *pRes),
>       void *pContext
>     );

A call to the above API registers an r-tree geometry callback named zGeom. If an r-tree geometry callback or ordinary SQL user function named zGeom already exists when sqlite3_rtree_geometry_callback() is called, it is replaced by the new r-tree geometry callback. If the xGeom parameter is passed a NULL value, then any existing r-tree geometry callback or SQL user function is removed from the system, but no new r-tree geometry callback is registered.

When the r-tree geometry callback is used in a custom r-tree query, the registered callback is invoked one or more times by SQLite to test whether or not the user-defined region intersects with specific bounding boxes. The bounding box being tested is defined by the contents of the aCoord[] array (size nCoord) passed to the callback. The aCoord[] array always contains the same number of entries as there are coordinate columns in the r-tree table (one less than the total number of columns, since the object id column does not contain a coordinate). They define a bounding-box in the same way as each row of the r-tree table itself does - the first scalar coordinate contains the minimum value for the first dimension, followed by the maximum value for the first dimension, followed by the minimum value for the second dimension, and so on. If the specified bounding box intersects with the custom query region, then the implementation of the callback must set the output parameter *pRes to non-zero and return SQLITE_OK. If the specified bounding box does not intersect the queried region, *pRes should be set to zero before returning SQLITE_OK. If an error occurs, the callback may return an SQLite error code other than SQLITE_OK, in which case the value of *pRes is ignored by SQLite and the query abandoned.

A registered r-tree geometry callback is used in an r-tree query by adding a MATCH condition to the WHERE clause of a SELECT statement. For example, assuming a custom geometry callback named "circle" has been registered, it may be used in a query on the two-dimensional r-tree table "demo_index" defined in earlier examples as follows:

>     SELECT * FROM demo_index WHERE id MATCH circle(45.3, 22.9, 5.0)

The left-hand side of the MATCH operator may be any column from the r-tree table, including the object id column. It makes no difference which column is used. The right-hand side of the MATCH operator is passed the results of an SQL function with the same name as the r-tree geometry callback. Zero or more function parameters may be specified by the user. Parameters are always interpreted as 64-bit real values. If a text, integer or blob value is passed as a parameter to an r-tree geometry callback function, it is converted to a real value as if by a CAST expression. If an SQL NULL value is passed to an r-tree geometry callback function, it is converted to the value 0.0.

Parameters passed to r-tree geometry callback functions may be used by the implementation of the r-tree geometry callback to define the specified region in any way. For example, the three parameters passed to the "circle" geometry callback above could identify the center point and radius of the circular region of interest.

The first argument to each invocation of an r-tree geometry callback is a pointer to a structure of the following type. The contents of the structure are not modified between multiple calls to the r-tree geometry callback associated with a single query (unless the pUser or xDelUser member variables are modified by the callback implementation - see below).

>     typedef struct sqlite3_rtree_geometry sqlite3_rtree_geometry;
>     struct sqlite3_rtree_geometry {
>       void *pContext;                 /* Copy of pContext passed to s_r_g_c() */
>       int nParam;                     /* Size of array aParam */
>       double *aParam;                 /* Parameters passed to SQL geom function */
>       void *pUser;                    /* Callback implementation user data */
>       void (*xDelUser)(void *);       /* Called by SQLite to clean up pUser */
>     };

The pContext member of the structure is always set to a copy of the pContext argument passed to sqlite3_rtree_geometry_callback() when the r-tree geometry callback is registered. The aParam[] array (size nParam) contains the parameter values passed to the r-tree geometry callback as part of the SQL query. In the example "circle" query above, nParam would be set to 3 and the aParam[] array would contain the three values 45.3, 22.9 and 5.0.

The pUser and xDelUser members of the sqlite3_rtree_geometry structure are initially set to NULL. The pUser variable may be set by the callback implementation to any arbitrary value that may be useful to subsequent invocations of the callback within the same custom r-tree query (for example, a pointer to a complicated data structure used to test for region intersection). If the xDelUser variable is set to a non-NULL value, then after the custom r-tree query has finished running SQLite automatically invokes it with the value of the pUser variable as the only argument. In other words, xDelUser may be set to a destructor function for the pUser value.

Example code implementing the "circle" r-tree geometry callback may be found in the file test_rtree.c.