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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 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 pairs, one pair per dimension, containing the minimum and maximum values for that dimension, respectively. A 1-dimensional R*Tree thus has 3 columns. A 2-dimensional R*Tree has 5 columns. A 3-dimensional R*Tree has 7 columns. A 4-dimensional R*Tree has 9 columns. And 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 stored as 32-bit floating point values for "rtree" virtual tables or as 32-bit signed integers in "rtree_i32" virtual tables. Unlike regular SQLite tables which can store data in a variety of datatypes and formats, the R*Tree indices rigidly enforce these storage types. Attempts to insert something other than an integer into the first column, or something other than a numeric 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 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 also do a query against its integer primary key efficiently, so the previous is no big deal. The real reason for using an R*Tree 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 it 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 programmer can query for all R*Tree entries that have a spatial relationship (they intersect with or are contained within) a particular bounding-box. Custom R*Tree queries, using the MATCH operator in the WHERE clause of a SELECT, allow the programmer to query for the set of R*Tree entries that intersect any arbitrary region or shape, not just a box. This capability is useful, for example, in computing the subset of objects in the R*Tree that are visible from a camera positioned in 3-D space.

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 one of the following two APIs:

>     int sqlite3_rtree_query_callback(
>       sqlite3 *db,
>       const char *zQueryFunc,
>       int (*xQueryFunc)(sqlite3_rtree_query_info*),
>       void *pContext,
>       void (*xDestructor)(void*)
>     );
>     int sqlite3_rtree_geometry_callback(
>       sqlite3 *db,
>       const char *zGeom,
>       int (*xGeom)(sqlite3_rtree_geometry *, int nCoord, double *aCoord, int *pRes),
>       void *pContext
>     );

The sqlite3_rtree_query_callback() became available with SQLite version 3.8.5 and is the preferred interface. The sqlite3_rtree_geometry_callback() is an older and less flexible interface that is supported for backwards compatibility.

A call to one of the above APIs creates a new SQL function named by the second parameter (zQueryFunc or zGeom). When that SQL function appears on the right-hand side of the MATCH operator and the left-hand side of the MATCH operator is any column in the R*Tree virtual table, then the callback defined by the third argument (xQueryFunc or xGeom) is invoked to determine if a particular object or subtree overlaps the desired region.

For example, a query like the following might be used to find all R*Tree entries that overlap with a circle centered a 45.3,22.9 with a radius of 5.0:

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

The SQL syntax for custom queries is the same regardless of which interface, sqlite3_rtree_geometry_callback() or sqlite3_rtree_query_callback(), is used to register the SQL function. However, the newer query-style callbacks give the application greater control over how the query proceeds.

### 6.1 The Legacy xGeom Callback

The legacy xGeom callback is invoked with four arguments. The first argument is a pointer to an sqlite3_rtree_geometry structure which provides information about how the SQL function was invoked. The second argument is the number of coordinates in each r-tree entry, and is always the same for any given R*Tree. The number of coordinates is 2 for a 1-dimensional R*Tree, 4 for a 2-dimensional R*Tree, 6 for a 3-dimensional R*Tree, and so forth. The third argument, aCoord[], is an array of nCoord coordinates that defines a bounding box to be tested. The last argument is a pointer into which the callback result should be written. The result is zero if the bounding-box defined by aCoord[] is completely outside the region defined by the xGeom callback and the result is non-zero if the bounding-box is inside or overlaps with the xGeom region. The xGeom callback should normally return SQLITE_OK. If xGeom returns anything other than SQLITE_OK, then the r-tree query will abort with an error.

The sqlite3_rtree_geometry structure that the first argument to the xGeom callback points to has a structure shown below. The exact same sqlite3_rtree_geometry structure is used for every callback for same MATCH operator in the same query. The contents of the sqlite3_rtree_geometry structure are initialized by SQLite but are not subsequently modified. The callback is free to make changes to the pUser and xDelUser elements of the structure if desired.

>     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 sqlite3_rtree_geometry structure is always set to a copy of the pContext argument passed to sqlite3_rtree_geometry_callback() when the callback is registered. The aParam[] array (size nParam) contains the parameter values passed to the SQL function on the right-hand side of the MATCH operator. 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 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 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.

The xGeom callback always does a depth-first search of the r-tree.

### 6.2 The New xQueryFunc Callback

The newer xQueryFunc callback receives more information from the r-tree query engine on each call, and it sends more information back to the query engine before it returns. To help keep the interface manageable, the xQueryFunc callback sends and receives information from the query engine as fields in the sqlite3_rtree_query_info structure:

>     struct sqlite3_rtree_query_info {
>       void *pContext;                   /* pContext from when function registered */
>       int nParam;                       /* Number of function parameters */
>       sqlite3_rtree_dbl *aParam;        /* value of function parameters */
>       void *pUser;                      /* callback can use this, if desired */
>       void (*xDelUser)(void*);          /* function to free pUser */
>       sqlite3_rtree_dbl *aCoord;        /* Coordinates of node or entry to check */
>       unsigned int *anQueue;            /* Number of pending entries in the queue */
>       int nCoord;                       /* Number of coordinates */
>       int iLevel;                       /* Level of current node or entry */
>       int mxLevel;                      /* The largest iLevel value in the tree */
>       sqlite3_int64 iRowid;             /* Rowid for current entry */
>       sqlite3_rtree_dbl rParentScore;   /* Score of parent node */
>       int eParentWithin;                /* Visibility of parent node */
>       int eWithin;                      /* OUT: Visiblity */
>       sqlite3_rtree_dbl rScore;         /* OUT: Write the score here */
>     };

The first five fields of the sqlite3_rtree_query_info structure are identical to the sqlite3_rtree_geometry structure, and have exactly the same meaning. The sqlite3_rtree_query_info structure also contains nCoord and aCoord fields which have the same meaning as the parameter of the same name in the xGeom callback.

The xQueryFunc must set the eWithin field of sqlite3_rtree_query_info to on of the values NOT_WITHIN, PARTLY_WITHIN, or FULLY_WITHIN depending on whether or not the bounding box defined by aCoord[] is completely outside the region, overlaps the region, or is completely inside the region, respectively. In addition, the xQueryFunc must set the rScore field to a non-negative value that indicates the order in which subtrees and entries of the query should be analyzed and returned. Smaller scores are processed first.

As its name implies, an R*Tree is organized as a tree. Each node of the tree is a bounding box. The root of the tree is a bounding box that encapsulates all elements of the tree. Beneath the root are a number of subtrees (typically 20 or more) each with their own smaller bounding boxes and each containing some subset of the R*Tree entries. The subtrees may have sub-subtrees, and so forth until finally one reaches the leaves of the tree which are the actual R*Tree entries.

An R*Tree query is initialized by making the root node the only entry in a priority queue sorted by rScore. The query proceeds by extracting the entry from the priority queue that has the lowest score. If that entry is a leaf (meaning that it is an actual R*Tree entry and not a subtree) then that entry is returned as one row of the query result. If the extracted priority queue entry is a node (a subtree), then sub-subtrees or leaves contained within that entry are passed to the xQueryFunc callback, one by one. Those subelements for which the xQueryFunc callback sets eWithin to PARTLY_WITHIN or FULLY_WITHIN are added to the priority queue using the score supplied by the callback. Subelements that return NOT_WITHIN are discarded. The query runs until the priority queue is empty.

Every leaf entry and node (subtree) within the R*Tree has an integer "level". The leaves have a level of 0. The first containing subtree of the leaves has a level of 1. The root of the R*Tree has the largest level value. The mxLevel entry in the sqlite3_rtree_query_info structure is the level value for the root of the R*Tree. The iLevel entry in sqlite3_rtree_query_info gives the level for the object being interrogated.

Most R*Tree queries use a depth-first search. This is accomplished by setting the rScore equal to iLevel. A depth-first search is usually preferred since it minimizes the number of elements in the priority queue, which reduces memory requirements and speeds processing. However, some application may prefer a breadth-first search, which can be accomplished by setting rScore to mxLevel-iLevel. By creating more complex formulas for rScore, applications can exercise detailed control over the order in which subtree are searched and leaf R*Tree entries are returned. For example, in an application with many millions of R*Tree entries, the rScore might be arranged so that the largest or most significant entries are returned first, allowing the application to display the most important information quickly, and filling in smaller and less important details as they become available.

Other information fields of the sqlite3_rtree_query_info structure are available for use by the xQueryFunc callback, if desired. The iRowid field is the rowid (the first of the 3 to 11 columns in the R*Tree) for the element being considered. iRowid is only valid for leaves. The eParentWithin and rParentScore values are copies of the eWithin and rScore values from the containing subtree of the current row. The anQueue field is an array of mxLevel+1 unsigned integers that tell the current number of elements in the priority queue at each level.

### 6.3 Additional Considerations for Custom Queries

The MATCH operator of a custom R*Tree query function must be a top-level AND-connected term of the WHERE clause, or else it will not be usable by the R*Tree query optimizer and the query will not be runnable. If the MATCH operator is connected to other terms of the WHERE clause via an OR operator, for example, the query will fail with an error.

Two or more MATCH operators are allowed in the same WHERE clause, as long as they are connected by AND operators. However, the R*Tree query engine only contains a single priority queue. The priority assigned to each node in the search is the lowest priority returned by any of the MATCH operators.