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third_party/sqlite/src/ext/rtree/rtree.c

+/*
+** 2001 September 15
+**
+** The author disclaims copyright to this source code.  In place of
+** a legal notice, here is a blessing:
+**
+**    May you do good and not evil.
+**    May you find forgiveness for yourself and forgive others.
+**    May you share freely, never taking more than you give.
+**
+*************************************************************************
+** This file contains code for implementations of the r-tree and r*-tree
+** algorithms packaged as an SQLite virtual table module.
+*/
+
+/*
+** Database Format of R-Tree Tables
+** --------------------------------
+**
+** The data structure for a single virtual r-tree table is stored in three 
+** native SQLite tables declared as follows. In each case, the '%' character
+** in the table name is replaced with the user-supplied name of the r-tree
+** table.
+**
+**   CREATE TABLE %_node(nodeno INTEGER PRIMARY KEY, data BLOB)
+**   CREATE TABLE %_parent(nodeno INTEGER PRIMARY KEY, parentnode INTEGER)
+**   CREATE TABLE %_rowid(rowid INTEGER PRIMARY KEY, nodeno INTEGER)
+**
+** The data for each node of the r-tree structure is stored in the %_node
+** table. For each node that is not the root node of the r-tree, there is
+** an entry in the %_parent table associating the node with its parent.
+** And for each row of data in the table, there is an entry in the %_rowid
+** table that maps from the entries rowid to the id of the node that it
+** is stored on.
+**
+** The root node of an r-tree always exists, even if the r-tree table is
+** empty. The nodeno of the root node is always 1. All other nodes in the
+** table must be the same size as the root node. The content of each node
+** is formatted as follows:
+**
+**   1. If the node is the root node (node 1), then the first 2 bytes
+**      of the node contain the tree depth as a big-endian integer.
+**      For non-root nodes, the first 2 bytes are left unused.
+**
+**   2. The next 2 bytes contain the number of entries currently 
+**      stored in the node.
+**
+**   3. The remainder of the node contains the node entries. Each entry
+**      consists of a single 8-byte integer followed by an even number
+**      of 4-byte coordinates. For leaf nodes the integer is the rowid
+**      of a record. For internal nodes it is the node number of a
+**      child page.
+*/
+
+#if !defined(SQLITE_CORE) || defined(SQLITE_ENABLE_RTREE)
+
+/*
+** This file contains an implementation of a couple of different variants
+** of the r-tree algorithm. See the README file for further details. The 
+** same data-structure is used for all, but the algorithms for insert and
+** delete operations vary. The variants used are selected at compile time 
+** by defining the following symbols:
+*/
+
+/* Either, both or none of the following may be set to activate 
+** r*tree variant algorithms.
+*/
+#define VARIANT_RSTARTREE_CHOOSESUBTREE 0
+#define VARIANT_RSTARTREE_REINSERT      1
+
+/* 
+** Exactly one of the following must be set to 1.
+*/
+#define VARIANT_GUTTMAN_QUADRATIC_SPLIT 0
+#define VARIANT_GUTTMAN_LINEAR_SPLIT    0
+#define VARIANT_RSTARTREE_SPLIT         1
+
+#define VARIANT_GUTTMAN_SPLIT \
+        (VARIANT_GUTTMAN_LINEAR_SPLIT||VARIANT_GUTTMAN_QUADRATIC_SPLIT)
+
+#if VARIANT_GUTTMAN_QUADRATIC_SPLIT
+  #define PickNext QuadraticPickNext
+  #define PickSeeds QuadraticPickSeeds
+  #define AssignCells splitNodeGuttman
+#endif
+#if VARIANT_GUTTMAN_LINEAR_SPLIT
+  #define PickNext LinearPickNext
+  #define PickSeeds LinearPickSeeds
+  #define AssignCells splitNodeGuttman
+#endif
+#if VARIANT_RSTARTREE_SPLIT
+  #define AssignCells splitNodeStartree
+#endif
+
+#if !defined(NDEBUG) && !defined(SQLITE_DEBUG) 
+# define NDEBUG 1
+#endif
+
+#ifndef SQLITE_CORE
+  #include "sqlite3ext.h"
+  SQLITE_EXTENSION_INIT1
+#else
+  #include "sqlite3.h"
+#endif
+
+#include <string.h>
+#include <assert.h>
+
+#ifndef SQLITE_AMALGAMATION
+#include "sqlite3rtree.h"
+typedef sqlite3_int64 i64;
+typedef unsigned char u8;
+typedef unsigned int u32;
+#endif
+
+/*  The following macro is used to suppress compiler warnings.
+*/
+#ifndef UNUSED_PARAMETER
+# define UNUSED_PARAMETER(x) (void)(x)
+#endif
+
+typedef struct Rtree Rtree;
+typedef struct RtreeCursor RtreeCursor;
+typedef struct RtreeNode RtreeNode;
+typedef struct RtreeCell RtreeCell;
+typedef struct RtreeConstraint RtreeConstraint;
+typedef struct RtreeMatchArg RtreeMatchArg;
+typedef struct RtreeGeomCallback RtreeGeomCallback;
+typedef union RtreeCoord RtreeCoord;
+
+/* The rtree may have between 1 and RTREE_MAX_DIMENSIONS dimensions. */
+#define RTREE_MAX_DIMENSIONS 5
+
+/* Size of hash table Rtree.aHash. This hash table is not expected to
+** ever contain very many entries, so a fixed number of buckets is 
+** used.
+*/
+#define HASHSIZE 128
+
+/* 
+** An rtree virtual-table object.
+*/
+struct Rtree {
+  sqlite3_vtab base;
+  sqlite3 *db;                /* Host database connection */
+  int iNodeSize;              /* Size in bytes of each node in the node table */
+  int nDim;                   /* Number of dimensions */
+  int nBytesPerCell;          /* Bytes consumed per cell */
+  int iDepth;                 /* Current depth of the r-tree structure */
+  char *zDb;                  /* Name of database containing r-tree table */
+  char *zName;                /* Name of r-tree table */ 
+  RtreeNode *aHash[HASHSIZE]; /* Hash table of in-memory nodes. */ 
+  int nBusy;                  /* Current number of users of this structure */
+
+  /* List of nodes removed during a CondenseTree operation. List is
+  ** linked together via the pointer normally used for hash chains -
+  ** RtreeNode.pNext. RtreeNode.iNode stores the depth of the sub-tree 
+  ** headed by the node (leaf nodes have RtreeNode.iNode==0).
+  */
+  RtreeNode *pDeleted;
+  int iReinsertHeight;        /* Height of sub-trees Reinsert() has run on */
+
+  /* Statements to read/write/delete a record from xxx_node */
+  sqlite3_stmt *pReadNode;
+  sqlite3_stmt *pWriteNode;
+  sqlite3_stmt *pDeleteNode;
+
+  /* Statements to read/write/delete a record from xxx_rowid */
+  sqlite3_stmt *pReadRowid;
+  sqlite3_stmt *pWriteRowid;
+  sqlite3_stmt *pDeleteRowid;
+
+  /* Statements to read/write/delete a record from xxx_parent */
+  sqlite3_stmt *pReadParent;
+  sqlite3_stmt *pWriteParent;
+  sqlite3_stmt *pDeleteParent;
+
+  int eCoordType;
+};
+
+/* Possible values for eCoordType: */
+#define RTREE_COORD_REAL32 0
+#define RTREE_COORD_INT32  1
+
+/*
+** The minimum number of cells allowed for a node is a third of the 
+** maximum. In Gutman's notation:
+**
+**     m = M/3
+**
+** If an R*-tree "Reinsert" operation is required, the same number of
+** cells are removed from the overfull node and reinserted into the tree.
+*/
+#define RTREE_MINCELLS(p) ((((p)->iNodeSize-4)/(p)->nBytesPerCell)/3)
+#define RTREE_REINSERT(p) RTREE_MINCELLS(p)
+#define RTREE_MAXCELLS 51
+
+/*
+** The smallest possible node-size is (512-64)==448 bytes. And the largest
+** supported cell size is 48 bytes (8 byte rowid + ten 4 byte coordinates).
+** Therefore all non-root nodes must contain at least 3 entries. Since 
+** 2^40 is greater than 2^64, an r-tree structure always has a depth of
+** 40 or less.
+*/
+#define RTREE_MAX_DEPTH 40
+
+/* 
+** An rtree cursor object.
+*/
+struct RtreeCursor {
+  sqlite3_vtab_cursor base;
+  RtreeNode *pNode;                 /* Node cursor is currently pointing at */
+  int iCell;                        /* Index of current cell in pNode */
+  int iStrategy;                    /* Copy of idxNum search parameter */
+  int nConstraint;                  /* Number of entries in aConstraint */
+  RtreeConstraint *aConstraint;     /* Search constraints. */
+};
+
+union RtreeCoord {
+  float f;
+  int i;
+};
+
+/*
+** The argument is an RtreeCoord. Return the value stored within the RtreeCoord
+** formatted as a double. This macro assumes that local variable pRtree points
+** to the Rtree structure associated with the RtreeCoord.
+*/
+#define DCOORD(coord) (                           \
+  (pRtree->eCoordType==RTREE_COORD_REAL32) ?      \
+    ((double)coord.f) :                           \
+    ((double)coord.i)                             \
+)
+
+/*
+** A search constraint.
+*/
+struct RtreeConstraint {
+  int iCoord;                     /* Index of constrained coordinate */
+  int op;                         /* Constraining operation */
+  double rValue;                  /* Constraint value. */
+  int (*xGeom)(sqlite3_rtree_geometry *, int, double *, int *);
+  sqlite3_rtree_geometry *pGeom;  /* Constraint callback argument for a MATCH */
+};
+
+/* Possible values for RtreeConstraint.op */
+#define RTREE_EQ    0x41
+#define RTREE_LE    0x42
+#define RTREE_LT    0x43
+#define RTREE_GE    0x44
+#define RTREE_GT    0x45
+#define RTREE_MATCH 0x46
+
+/* 
+** An rtree structure node.
+*/
+struct RtreeNode {
+  RtreeNode *pParent;               /* Parent node */
+  i64 iNode;
+  int nRef;
+  int isDirty;
+  u8 *zData;
+  RtreeNode *pNext;                 /* Next node in this hash chain */
+};
+#define NCELL(pNode) readInt16(&(pNode)->zData[2])
+
+/* 
+** Structure to store a deserialized rtree record.
+*/
+struct RtreeCell {
+  i64 iRowid;
+  RtreeCoord aCoord[RTREE_MAX_DIMENSIONS*2];
+};
+
+
+/*
+** Value for the first field of every RtreeMatchArg object. The MATCH
+** operator tests that the first field of a blob operand matches this
+** value to avoid operating on invalid blobs (which could cause a segfault).
+*/
+#define RTREE_GEOMETRY_MAGIC 0x891245AB
+
+/*
+** An instance of this structure must be supplied as a blob argument to
+** the right-hand-side of an SQL MATCH operator used to constrain an
+** r-tree query.
+*/
+struct RtreeMatchArg {
+  u32 magic;                      /* Always RTREE_GEOMETRY_MAGIC */
+  int (*xGeom)(sqlite3_rtree_geometry *, int, double *, int *);
+  void *pContext;
+  int nParam;
+  double aParam[1];
+};
+
+/*
+** When a geometry callback is created (see sqlite3_rtree_geometry_callback),
+** a single instance of the following structure is allocated. It is used
+** as the context for the user-function created by by s_r_g_c(). The object
+** is eventually deleted by the destructor mechanism provided by
+** sqlite3_create_function_v2() (which is called by s_r_g_c() to create
+** the geometry callback function).
+*/
+struct RtreeGeomCallback {
+  int (*xGeom)(sqlite3_rtree_geometry *, int, double *, int *);
+  void *pContext;
+};
+
+#ifndef MAX
+# define MAX(x,y) ((x) < (y) ? (y) : (x))
+#endif
+#ifndef MIN
+# define MIN(x,y) ((x) > (y) ? (y) : (x))
+#endif
+
+/*
+** Functions to deserialize a 16 bit integer, 32 bit real number and
+** 64 bit integer. The deserialized value is returned.
+*/
+static int readInt16(u8 *p){
+  return (p[0]<<8) + p[1];
+}
+static void readCoord(u8 *p, RtreeCoord *pCoord){
+  u32 i = (
+    (((u32)p[0]) << 24) + 
+    (((u32)p[1]) << 16) + 
+    (((u32)p[2]) <<  8) + 
+    (((u32)p[3]) <<  0)
+  );
+  *(u32 *)pCoord = i;
+}
+static i64 readInt64(u8 *p){
+  return (
+    (((i64)p[0]) << 56) + 
+    (((i64)p[1]) << 48) + 
+    (((i64)p[2]) << 40) + 
+    (((i64)p[3]) << 32) + 
+    (((i64)p[4]) << 24) + 
+    (((i64)p[5]) << 16) + 
+    (((i64)p[6]) <<  8) + 
+    (((i64)p[7]) <<  0)
+  );
+}
+
+/*
+** Functions to serialize a 16 bit integer, 32 bit real number and
+** 64 bit integer. The value returned is the number of bytes written
+** to the argument buffer (always 2, 4 and 8 respectively).
+*/
+static int writeInt16(u8 *p, int i){
+  p[0] = (i>> 8)&0xFF;
+  p[1] = (i>> 0)&0xFF;
+  return 2;
+}
+static int writeCoord(u8 *p, RtreeCoord *pCoord){
+  u32 i;
+  assert( sizeof(RtreeCoord)==4 );
+  assert( sizeof(u32)==4 );
+  i = *(u32 *)pCoord;
+  p[0] = (i>>24)&0xFF;
+  p[1] = (i>>16)&0xFF;
+  p[2] = (i>> 8)&0xFF;
+  p[3] = (i>> 0)&0xFF;
+  return 4;
+}
+static int writeInt64(u8 *p, i64 i){
+  p[0] = (i>>56)&0xFF;
+  p[1] = (i>>48)&0xFF;
+  p[2] = (i>>40)&0xFF;
+  p[3] = (i>>32)&0xFF;
+  p[4] = (i>>24)&0xFF;
+  p[5] = (i>>16)&0xFF;
+  p[6] = (i>> 8)&0xFF;
+  p[7] = (i>> 0)&0xFF;
+  return 8;
+}
+
+/*
+** Increment the reference count of node p.
+*/
+static void nodeReference(RtreeNode *p){
+  if( p ){
+    p->nRef++;
+  }
+}
+
+/*
+** Clear the content of node p (set all bytes to 0x00).
+*/
+static void nodeZero(Rtree *pRtree, RtreeNode *p){
+  memset(&p->zData[2], 0, pRtree->iNodeSize-2);
+  p->isDirty = 1;
+}
+
+/*
+** Given a node number iNode, return the corresponding key to use
+** in the Rtree.aHash table.
+*/
+static int nodeHash(i64 iNode){
+  return (
+    (iNode>>56) ^ (iNode>>48) ^ (iNode>>40) ^ (iNode>>32) ^ 
+    (iNode>>24) ^ (iNode>>16) ^ (iNode>> 8) ^ (iNode>> 0)
+  ) % HASHSIZE;
+}
+
+/*
+** Search the node hash table for node iNode. If found, return a pointer
+** to it. Otherwise, return 0.
+*/
+static RtreeNode *nodeHashLookup(Rtree *pRtree, i64 iNode){
+  RtreeNode *p;
+  for(p=pRtree->aHash[nodeHash(iNode)]; p && p->iNode!=iNode; p=p->pNext);
+  return p;
+}
+
+/*
+** Add node pNode to the node hash table.
+*/
+static void nodeHashInsert(Rtree *pRtree, RtreeNode *pNode){
+  int iHash;
+  assert( pNode->pNext==0 );
+  iHash = nodeHash(pNode->iNode);
+  pNode->pNext = pRtree->aHash[iHash];
+  pRtree->aHash[iHash] = pNode;
+}
+
+/*
+** Remove node pNode from the node hash table.
+*/
+static void nodeHashDelete(Rtree *pRtree, RtreeNode *pNode){
+  RtreeNode **pp;
+  if( pNode->iNode!=0 ){
+    pp = &pRtree->aHash[nodeHash(pNode->iNode)];
+    for( ; (*pp)!=pNode; pp = &(*pp)->pNext){ assert(*pp); }
+    *pp = pNode->pNext;
+    pNode->pNext = 0;
+  }
+}
+
+/*
+** Allocate and return new r-tree node. Initially, (RtreeNode.iNode==0),
+** indicating that node has not yet been assigned a node number. It is
+** assigned a node number when nodeWrite() is called to write the
+** node contents out to the database.
+*/
+static RtreeNode *nodeNew(Rtree *pRtree, RtreeNode *pParent){
+  RtreeNode *pNode;
+  pNode = (RtreeNode *)sqlite3_malloc(sizeof(RtreeNode) + pRtree->iNodeSize);
+  if( pNode ){
+    memset(pNode, 0, sizeof(RtreeNode) + pRtree->iNodeSize);
+    pNode->zData = (u8 *)&pNode[1];
+    pNode->nRef = 1;
+    pNode->pParent = pParent;
+    pNode->isDirty = 1;
+    nodeReference(pParent);
+  }
+  return pNode;
+}
+
+/*
+** Obtain a reference to an r-tree node.
+*/
+static int
+nodeAcquire(
+  Rtree *pRtree,             /* R-tree structure */
+  i64 iNode,                 /* Node number to load */
+  RtreeNode *pParent,        /* Either the parent node or NULL */
+  RtreeNode **ppNode         /* OUT: Acquired node */
+){
+  int rc;
+  int rc2 = SQLITE_OK;
+  RtreeNode *pNode;
+
+  /* Check if the requested node is already in the hash table. If so,
+  ** increase its reference count and return it.
+  */
+  if( (pNode = nodeHashLookup(pRtree, iNode)) ){
+    assert( !pParent || !pNode->pParent || pNode->pParent==pParent );
+    if( pParent && !pNode->pParent ){
+      nodeReference(pParent);
+      pNode->pParent = pParent;
+    }
+    pNode->nRef++;
+    *ppNode = pNode;
+    return SQLITE_OK;
+  }
+
+  sqlite3_bind_int64(pRtree->pReadNode, 1, iNode);
+  rc = sqlite3_step(pRtree->pReadNode);
+  if( rc==SQLITE_ROW ){
+    const u8 *zBlob = sqlite3_column_blob(pRtree->pReadNode, 0);
+    if( pRtree->iNodeSize==sqlite3_column_bytes(pRtree->pReadNode, 0) ){
+      pNode = (RtreeNode *)sqlite3_malloc(sizeof(RtreeNode)+pRtree->iNodeSize);
+      if( !pNode ){
+        rc2 = SQLITE_NOMEM;
+      }else{
+        pNode->pParent = pParent;
+        pNode->zData = (u8 *)&pNode[1];
+        pNode->nRef = 1;
+        pNode->iNode = iNode;
+        pNode->isDirty = 0;
+        pNode->pNext = 0;
+        memcpy(pNode->zData, zBlob, pRtree->iNodeSize);
+        nodeReference(pParent);
+      }
+    }
+  }
+  rc = sqlite3_reset(pRtree->pReadNode);
+  if( rc==SQLITE_OK ) rc = rc2;
+
+  /* If the root node was just loaded, set pRtree->iDepth to the height
+  ** of the r-tree structure. A height of zero means all data is stored on
+  ** the root node. A height of one means the children of the root node
+  ** are the leaves, and so on. If the depth as specified on the root node
+  ** is greater than RTREE_MAX_DEPTH, the r-tree structure must be corrupt.
+  */
+  if( pNode && iNode==1 ){
+    pRtree->iDepth = readInt16(pNode->zData);
+    if( pRtree->iDepth>RTREE_MAX_DEPTH ){
+      rc = SQLITE_CORRUPT;
+    }
+  }
+
+  /* If no error has occurred so far, check if the "number of entries"
+  ** field on the node is too large. If so, set the return code to 
+  ** SQLITE_CORRUPT.
+  */
+  if( pNode && rc==SQLITE_OK ){
+    if( NCELL(pNode)>((pRtree->iNodeSize-4)/pRtree->nBytesPerCell) ){
+      rc = SQLITE_CORRUPT;
+    }
+  }
+
+  if( rc==SQLITE_OK ){
+    if( pNode!=0 ){
+      nodeHashInsert(pRtree, pNode);
+    }else{
+      rc = SQLITE_CORRUPT;
+    }
+    *ppNode = pNode;
+  }else{
+    sqlite3_free(pNode);
+    *ppNode = 0;
+  }
+
+  return rc;
+}
+
+/*
+** Overwrite cell iCell of node pNode with the contents of pCell.
+*/
+static void nodeOverwriteCell(
+  Rtree *pRtree, 
+  RtreeNode *pNode,  
+  RtreeCell *pCell, 
+  int iCell
+){
+  int ii;
+  u8 *p = &pNode->zData[4 + pRtree->nBytesPerCell*iCell];
+  p += writeInt64(p, pCell->iRowid);
+  for(ii=0; ii<(pRtree->nDim*2); ii++){
+    p += writeCoord(p, &pCell->aCoord[ii]);
+  }
+  pNode->isDirty = 1;
+}
+
+/*
+** Remove cell the cell with index iCell from node pNode.
+*/
+static void nodeDeleteCell(Rtree *pRtree, RtreeNode *pNode, int iCell){
+  u8 *pDst = &pNode->zData[4 + pRtree->nBytesPerCell*iCell];
+  u8 *pSrc = &pDst[pRtree->nBytesPerCell];
+  int nByte = (NCELL(pNode) - iCell - 1) * pRtree->nBytesPerCell;
+  memmove(pDst, pSrc, nByte);
+  writeInt16(&pNode->zData[2], NCELL(pNode)-1);
+  pNode->isDirty = 1;
+}
+
+/*
+** Insert the contents of cell pCell into node pNode. If the insert
+** is successful, return SQLITE_OK.
+**
+** If there is not enough free space in pNode, return SQLITE_FULL.
+*/
+static int
+nodeInsertCell(
+  Rtree *pRtree, 
+  RtreeNode *pNode, 
+  RtreeCell *pCell 
+){
+  int nCell;                    /* Current number of cells in pNode */
+  int nMaxCell;                 /* Maximum number of cells for pNode */
+
+  nMaxCell = (pRtree->iNodeSize-4)/pRtree->nBytesPerCell;
+  nCell = NCELL(pNode);
+
+  assert( nCell<=nMaxCell );
+  if( nCell<nMaxCell ){
+    nodeOverwriteCell(pRtree, pNode, pCell, nCell);
+    writeInt16(&pNode->zData[2], nCell+1);
+    pNode->isDirty = 1;
+  }
+
+  return (nCell==nMaxCell);
+}
+
+/*
+** If the node is dirty, write it out to the database.
+*/
+static int
+nodeWrite(Rtree *pRtree, RtreeNode *pNode){
+  int rc = SQLITE_OK;
+  if( pNode->isDirty ){
+    sqlite3_stmt *p = pRtree->pWriteNode;
+    if( pNode->iNode ){
+      sqlite3_bind_int64(p, 1, pNode->iNode);
+    }else{
+      sqlite3_bind_null(p, 1);
+    }
+    sqlite3_bind_blob(p, 2, pNode->zData, pRtree->iNodeSize, SQLITE_STATIC);
+    sqlite3_step(p);
+    pNode->isDirty = 0;
+    rc = sqlite3_reset(p);
+    if( pNode->iNode==0 && rc==SQLITE_OK ){
+      pNode->iNode = sqlite3_last_insert_rowid(pRtree->db);
+      nodeHashInsert(pRtree, pNode);
+    }
+  }
+  return rc;
+}
+
+/*
+** Release a reference to a node. If the node is dirty and the reference
+** count drops to zero, the node data is written to the database.
+*/
+static int
+nodeRelease(Rtree *pRtree, RtreeNode *pNode){
+  int rc = SQLITE_OK;
+  if( pNode ){
+    assert( pNode->nRef>0 );
+    pNode->nRef--;
+    if( pNode->nRef==0 ){
+      if( pNode->iNode==1 ){
+        pRtree->iDepth = -1;
+      }
+      if( pNode->pParent ){
+        rc = nodeRelease(pRtree, pNode->pParent);
+      }
+      if( rc==SQLITE_OK ){
+        rc = nodeWrite(pRtree, pNode);
+      }
+      nodeHashDelete(pRtree, pNode);
+      sqlite3_free(pNode);
+    }
+  }
+  return rc;
+}
+
+/*
+** Return the 64-bit integer value associated with cell iCell of
+** node pNode. If pNode is a leaf node, this is a rowid. If it is
+** an internal node, then the 64-bit integer is a child page number.
+*/
+static i64 nodeGetRowid(
+  Rtree *pRtree, 
+  RtreeNode *pNode, 
+  int iCell
+){
+  assert( iCell<NCELL(pNode) );
+  return readInt64(&pNode->zData[4 + pRtree->nBytesPerCell*iCell]);
+}
+
+/*
+** Return coordinate iCoord from cell iCell in node pNode.
+*/
+static void nodeGetCoord(
+  Rtree *pRtree, 
+  RtreeNode *pNode, 
+  int iCell,
+  int iCoord,
+  RtreeCoord *pCoord           /* Space to write result to */
+){
+  readCoord(&pNode->zData[12 + pRtree->nBytesPerCell*iCell + 4*iCoord], pCoord);
+}
+
+/*
+** Deserialize cell iCell of node pNode. Populate the structure pointed
+** to by pCell with the results.
+*/
+static void nodeGetCell(
+  Rtree *pRtree, 
+  RtreeNode *pNode, 
+  int iCell,
+  RtreeCell *pCell
+){
+  int ii;
+  pCell->iRowid = nodeGetRowid(pRtree, pNode, iCell);
+  for(ii=0; ii<pRtree->nDim*2; ii++){
+    nodeGetCoord(pRtree, pNode, iCell, ii, &pCell->aCoord[ii]);
+  }
+}
+
+
+/* Forward declaration for the function that does the work of
+** the virtual table module xCreate() and xConnect() methods.
+*/
+static int rtreeInit(
+  sqlite3 *, void *, int, const char *const*, sqlite3_vtab **, char **, int
+);
+
+/* 
+** Rtree virtual table module xCreate method.
+*/
+static int rtreeCreate(
+  sqlite3 *db,
+  void *pAux,
+  int argc, const char *const*argv,
+  sqlite3_vtab **ppVtab,
+  char **pzErr
+){
+  return rtreeInit(db, pAux, argc, argv, ppVtab, pzErr, 1);
+}
+
+/* 
+** Rtree virtual table module xConnect method.
+*/
+static int rtreeConnect(
+  sqlite3 *db,
+  void *pAux,
+  int argc, const char *const*argv,
+  sqlite3_vtab **ppVtab,
+  char **pzErr
+){
+  return rtreeInit(db, pAux, argc, argv, ppVtab, pzErr, 0);
+}
+
+/*
+** Increment the r-tree reference count.
+*/
+static void rtreeReference(Rtree *pRtree){
+  pRtree->nBusy++;
+}
+
+/*
+** Decrement the r-tree reference count. When the reference count reaches
+** zero the structure is deleted.
+*/
+static void rtreeRelease(Rtree *pRtree){
+  pRtree->nBusy--;
+  if( pRtree->nBusy==0 ){
+    sqlite3_finalize(pRtree->pReadNode);
+    sqlite3_finalize(pRtree->pWriteNode);
+    sqlite3_finalize(pRtree->pDeleteNode);
+    sqlite3_finalize(pRtree->pReadRowid);
+    sqlite3_finalize(pRtree->pWriteRowid);
+    sqlite3_finalize(pRtree->pDeleteRowid);
+    sqlite3_finalize(pRtree->pReadParent);
+    sqlite3_finalize(pRtree->pWriteParent);
+    sqlite3_finalize(pRtree->pDeleteParent);
+    sqlite3_free(pRtree);
+  }
+}
+
+/* 
+** Rtree virtual table module xDisconnect method.
+*/
+static int rtreeDisconnect(sqlite3_vtab *pVtab){
+  rtreeRelease((Rtree *)pVtab);
+  return SQLITE_OK;
+}
+
+/* 
+** Rtree virtual table module xDestroy method.
+*/
+static int rtreeDestroy(sqlite3_vtab *pVtab){
+  Rtree *pRtree = (Rtree *)pVtab;
+  int rc;
+  char *zCreate = sqlite3_mprintf(
+    "DROP TABLE '%q'.'%q_node';"
+    "DROP TABLE '%q'.'%q_rowid';"
+    "DROP TABLE '%q'.'%q_parent';",
+    pRtree->zDb, pRtree->zName, 
+    pRtree->zDb, pRtree->zName,
+    pRtree->zDb, pRtree->zName
+  );
+  if( !zCreate ){
+    rc = SQLITE_NOMEM;
+  }else{
+    rc = sqlite3_exec(pRtree->db, zCreate, 0, 0, 0);
+    sqlite3_free(zCreate);
+  }
+  if( rc==SQLITE_OK ){
+    rtreeRelease(pRtree);
+  }
+
+  return rc;
+}
+
+/* 
+** Rtree virtual table module xOpen method.
+*/
+static int rtreeOpen(sqlite3_vtab *pVTab, sqlite3_vtab_cursor **ppCursor){
+  int rc = SQLITE_NOMEM;
+  RtreeCursor *pCsr;
+
+  pCsr = (RtreeCursor *)sqlite3_malloc(sizeof(RtreeCursor));
+  if( pCsr ){
+    memset(pCsr, 0, sizeof(RtreeCursor));
+    pCsr->base.pVtab = pVTab;
+    rc = SQLITE_OK;
+  }
+  *ppCursor = (sqlite3_vtab_cursor *)pCsr;
+
+  return rc;
+}
+
+
+/*
+** Free the RtreeCursor.aConstraint[] array and its contents.
+*/
+static void freeCursorConstraints(RtreeCursor *pCsr){
+  if( pCsr->aConstraint ){
+    int i;                        /* Used to iterate through constraint array */
+    for(i=0; i<pCsr->nConstraint; i++){
+      sqlite3_rtree_geometry *pGeom = pCsr->aConstraint[i].pGeom;
+      if( pGeom ){
+        if( pGeom->xDelUser ) pGeom->xDelUser(pGeom->pUser);
+        sqlite3_free(pGeom);
+      }
+    }
+    sqlite3_free(pCsr->aConstraint);
+    pCsr->aConstraint = 0;
+  }
+}
+
+/* 
+** Rtree virtual table module xClose method.
+*/
+static int rtreeClose(sqlite3_vtab_cursor *cur){
+  Rtree *pRtree = (Rtree *)(cur->pVtab);
+  int rc;
+  RtreeCursor *pCsr = (RtreeCursor *)cur;
+  freeCursorConstraints(pCsr);
+  rc = nodeRelease(pRtree, pCsr->pNode);
+  sqlite3_free(pCsr);
+  return rc;
+}
+
+/*
+** Rtree virtual table module xEof method.
+**
+** Return non-zero if the cursor does not currently point to a valid 
+** record (i.e if the scan has finished), or zero otherwise.
+*/
+static int rtreeEof(sqlite3_vtab_cursor *cur){
+  RtreeCursor *pCsr = (RtreeCursor *)cur;
+  return (pCsr->pNode==0);
+}
+
+/*
+** The r-tree constraint passed as the second argument to this function is
+** guaranteed to be a MATCH constraint.
+*/
+static int testRtreeGeom(
+  Rtree *pRtree,                  /* R-Tree object */
+  RtreeConstraint *pConstraint,   /* MATCH constraint to test */
+  RtreeCell *pCell,               /* Cell to test */
+  int *pbRes                      /* OUT: Test result */
+){
+  int i;
+  double aCoord[RTREE_MAX_DIMENSIONS*2];
+  int nCoord = pRtree->nDim*2;
+
+  assert( pConstraint->op==RTREE_MATCH );
+  assert( pConstraint->pGeom );
+
+  for(i=0; i<nCoord; i++){
+    aCoord[i] = DCOORD(pCell->aCoord[i]);
+  }
+  return pConstraint->xGeom(pConstraint->pGeom, nCoord, aCoord, pbRes);
+}
+
+/* 
+** Cursor pCursor currently points to a cell in a non-leaf page.
+** Set *pbEof to true if the sub-tree headed by the cell is filtered
+** (excluded) by the constraints in the pCursor->aConstraint[] 
+** array, or false otherwise.
+**
+** Return SQLITE_OK if successful or an SQLite error code if an error
+** occurs within a geometry callback.
+*/
+static int testRtreeCell(Rtree *pRtree, RtreeCursor *pCursor, int *pbEof){
+  RtreeCell cell;
+  int ii;
+  int bRes = 0;
+  int rc = SQLITE_OK;
+
+  nodeGetCell(pRtree, pCursor->pNode, pCursor->iCell, &cell);
+  for(ii=0; bRes==0 && ii<pCursor->nConstraint; ii++){
+    RtreeConstraint *p = &pCursor->aConstraint[ii];
+    double cell_min = DCOORD(cell.aCoord[(p->iCoord>>1)*2]);
+    double cell_max = DCOORD(cell.aCoord[(p->iCoord>>1)*2+1]);
+
+    assert(p->op==RTREE_LE || p->op==RTREE_LT || p->op==RTREE_GE 
+        || p->op==RTREE_GT || p->op==RTREE_EQ || p->op==RTREE_MATCH
+    );
+
+    switch( p->op ){
+      case RTREE_LE: case RTREE_LT: 
+        bRes = p->rValue<cell_min; 
+        break;
+
+      case RTREE_GE: case RTREE_GT: 
+        bRes = p->rValue>cell_max; 
+        break;
+
+      case RTREE_EQ:
+        bRes = (p->rValue>cell_max || p->rValue<cell_min);
+        break;
+
+      default: {
+        assert( p->op==RTREE_MATCH );
+        rc = testRtreeGeom(pRtree, p, &cell, &bRes);
+        bRes = !bRes;
+        break;
+      }
+    }
+  }
+
+  *pbEof = bRes;
+  return rc;
+}
+
+/* 
+** Test if the cell that cursor pCursor currently points to
+** would be filtered (excluded) by the constraints in the 
+** pCursor->aConstraint[] array. If so, set *pbEof to true before
+** returning. If the cell is not filtered (excluded) by the constraints,
+** set pbEof to zero.
+**
+** Return SQLITE_OK if successful or an SQLite error code if an error
+** occurs within a geometry callback.
+**
+** This function assumes that the cell is part of a leaf node.
+*/
+static int testRtreeEntry(Rtree *pRtree, RtreeCursor *pCursor, int *pbEof){
+  RtreeCell cell;
+  int ii;
+  *pbEof = 0;
+
+  nodeGetCell(pRtree, pCursor->pNode, pCursor->iCell, &cell);
+  for(ii=0; ii<pCursor->nConstraint; ii++){
+    RtreeConstraint *p = &pCursor->aConstraint[ii];
+    double coord = DCOORD(cell.aCoord[p->iCoord]);
+    int res;
+    assert(p->op==RTREE_LE || p->op==RTREE_LT || p->op==RTREE_GE 
+        || p->op==RTREE_GT || p->op==RTREE_EQ || p->op==RTREE_MATCH
+    );
+    switch( p->op ){
+      case RTREE_LE: res = (coord<=p->rValue); break;
+      case RTREE_LT: res = (coord<p->rValue);  break;
+      case RTREE_GE: res = (coord>=p->rValue); break;
+      case RTREE_GT: res = (coord>p->rValue);  break;
+      case RTREE_EQ: res = (coord==p->rValue); break;
+      default: {
+        int rc;
+        assert( p->op==RTREE_MATCH );
+        rc = testRtreeGeom(pRtree, p, &cell, &res);
+        if( rc!=SQLITE_OK ){
+          return rc;
+        }
+        break;
+      }
+    }
+
+    if( !res ){
+      *pbEof = 1;
+      return SQLITE_OK;
+    }
+  }
+
+  return SQLITE_OK;
+}
+
+/*
+** Cursor pCursor currently points at a node that heads a sub-tree of
+** height iHeight (if iHeight==0, then the node is a leaf). Descend
+** to point to the left-most cell of the sub-tree that matches the 
+** configured constraints.
+*/
+static int descendToCell(
+  Rtree *pRtree, 
+  RtreeCursor *pCursor, 
+  int iHeight,
+  int *pEof                 /* OUT: Set to true if cannot descend */
+){
+  int isEof;
+  int rc;
+  int ii;
+  RtreeNode *pChild;
+  sqlite3_int64 iRowid;
+
+  RtreeNode *pSavedNode = pCursor->pNode;
+  int iSavedCell = pCursor->iCell;
+
+  assert( iHeight>=0 );
+
+  if( iHeight==0 ){
+    rc = testRtreeEntry(pRtree, pCursor, &isEof);
+  }else{
+    rc = testRtreeCell(pRtree, pCursor, &isEof);
+  }
+  if( rc!=SQLITE_OK || isEof || iHeight==0 ){
+    goto descend_to_cell_out;
+  }
+
+  iRowid = nodeGetRowid(pRtree, pCursor->pNode, pCursor->iCell);
+  rc = nodeAcquire(pRtree, iRowid, pCursor->pNode, &pChild);
+  if( rc!=SQLITE_OK ){
+    goto descend_to_cell_out;
+  }
+
+  nodeRelease(pRtree, pCursor->pNode);
+  pCursor->pNode = pChild;
+  isEof = 1;
+  for(ii=0; isEof && ii<NCELL(pChild); ii++){
+    pCursor->iCell = ii;
+    rc = descendToCell(pRtree, pCursor, iHeight-1, &isEof);
+    if( rc!=SQLITE_OK ){
+      goto descend_to_cell_out;
+    }
+  }
+
+  if( isEof ){
+    assert( pCursor->pNode==pChild );
+    nodeReference(pSavedNode);
+    nodeRelease(pRtree, pChild);
+    pCursor->pNode = pSavedNode;
+    pCursor->iCell = iSavedCell;
+  }
+
+descend_to_cell_out:
+  *pEof = isEof;
+  return rc;
+}
+
+/*
+** One of the cells in node pNode is guaranteed to have a 64-bit 
+** integer value equal to iRowid. Return the index of this cell.
+*/
+static int nodeRowidIndex(
+  Rtree *pRtree, 
+  RtreeNode *pNode, 
+  i64 iRowid,
+  int *piIndex
+){
+  int ii;
+  int nCell = NCELL(pNode);
+  for(ii=0; ii<nCell; ii++){
+    if( nodeGetRowid(pRtree, pNode, ii)==iRowid ){
+      *piIndex = ii;
+      return SQLITE_OK;
+    }
+  }
+  return SQLITE_CORRUPT;
+}
+
+/*
+** Return the index of the cell containing a pointer to node pNode
+** in its parent. If pNode is the root node, return -1.
+*/
+static int nodeParentIndex(Rtree *pRtree, RtreeNode *pNode, int *piIndex){
+  RtreeNode *pParent = pNode->pParent;
+  if( pParent ){
+    return nodeRowidIndex(pRtree, pParent, pNode->iNode, piIndex);
+  }
+  *piIndex = -1;
+  return SQLITE_OK;
+}
+
+/* 
+** Rtree virtual table module xNext method.
+*/
+static int rtreeNext(sqlite3_vtab_cursor *pVtabCursor){
+  Rtree *pRtree = (Rtree *)(pVtabCursor->pVtab);
+  RtreeCursor *pCsr = (RtreeCursor *)pVtabCursor;
+  int rc = SQLITE_OK;
+
+  /* RtreeCursor.pNode must not be NULL. If is is NULL, then this cursor is
+  ** already at EOF. It is against the rules to call the xNext() method of
+  ** a cursor that has already reached EOF.
+  */
+  assert( pCsr->pNode );
+
+  if( pCsr->iStrategy==1 ){
+    /* This "scan" is a direct lookup by rowid. There is no next entry. */
+    nodeRelease(pRtree, pCsr->pNode);
+    pCsr->pNode = 0;
+  }else{
+    /* Move to the next entry that matches the configured constraints. */
+    int iHeight = 0;
+    while( pCsr->pNode ){
+      RtreeNode *pNode = pCsr->pNode;
+      int nCell = NCELL(pNode);
+      for(pCsr->iCell++; pCsr->iCell<nCell; pCsr->iCell++){
+        int isEof;
+        rc = descendToCell(pRtree, pCsr, iHeight, &isEof);
+        if( rc!=SQLITE_OK || !isEof ){
+          return rc;
+        }
+      }
+      pCsr->pNode = pNode->pParent;
+      rc = nodeParentIndex(pRtree, pNode, &pCsr->iCell);
+      if( rc!=SQLITE_OK ){
+        return rc;
+      }
+      nodeReference(pCsr->pNode);
+      nodeRelease(pRtree, pNode);
+      iHeight++;
+    }
+  }
+
+  return rc;
+}
+
+/* 
+** Rtree virtual table module xRowid method.
+*/
+static int rtreeRowid(sqlite3_vtab_cursor *pVtabCursor, sqlite_int64 *pRowid){
+  Rtree *pRtree = (Rtree *)pVtabCursor->pVtab;
+  RtreeCursor *pCsr = (RtreeCursor *)pVtabCursor;
+
+  assert(pCsr->pNode);
+  *pRowid = nodeGetRowid(pRtree, pCsr->pNode, pCsr->iCell);
+
+  return SQLITE_OK;
+}
+
+/* 
+** Rtree virtual table module xColumn method.
+*/
+static int rtreeColumn(sqlite3_vtab_cursor *cur, sqlite3_context *ctx, int i){
+  Rtree *pRtree = (Rtree *)cur->pVtab;
+  RtreeCursor *pCsr = (RtreeCursor *)cur;
+
+  if( i==0 ){
+    i64 iRowid = nodeGetRowid(pRtree, pCsr->pNode, pCsr->iCell);
+    sqlite3_result_int64(ctx, iRowid);
+  }else{
+    RtreeCoord c;
+    nodeGetCoord(pRtree, pCsr->pNode, pCsr->iCell, i-1, &c);
+    if( pRtree->eCoordType==RTREE_COORD_REAL32 ){
+      sqlite3_result_double(ctx, c.f);
+    }else{
+      assert( pRtree->eCoordType==RTREE_COORD_INT32 );
+      sqlite3_result_int(ctx, c.i);
+    }
+  }
+
+  return SQLITE_OK;
+}
+
+/* 
+** Use nodeAcquire() to obtain the leaf node containing the record with 
+** rowid iRowid. If successful, set *ppLeaf to point to the node and
+** return SQLITE_OK. If there is no such record in the table, set
+** *ppLeaf to 0 and return SQLITE_OK. If an error occurs, set *ppLeaf
+** to zero and return an SQLite error code.
+*/
+static int findLeafNode(Rtree *pRtree, i64 iRowid, RtreeNode **ppLeaf){
+  int rc;
+  *ppLeaf = 0;
+  sqlite3_bind_int64(pRtree->pReadRowid, 1, iRowid);
+  if( sqlite3_step(pRtree->pReadRowid)==SQLITE_ROW ){
+    i64 iNode = sqlite3_column_int64(pRtree->pReadRowid, 0);
+    rc = nodeAcquire(pRtree, iNode, 0, ppLeaf);
+    sqlite3_reset(pRtree->pReadRowid);
+  }else{
+    rc = sqlite3_reset(pRtree->pReadRowid);
+  }
+  return rc;
+}
+
+/*
+** This function is called to configure the RtreeConstraint object passed
+** as the second argument for a MATCH constraint. The value passed as the
+** first argument to this function is the right-hand operand to the MATCH
+** operator.
+*/
+static int deserializeGeometry(sqlite3_value *pValue, RtreeConstraint *pCons){
+  RtreeMatchArg *p;
+  sqlite3_rtree_geometry *pGeom;
+  int nBlob;
+
+  /* Check that value is actually a blob. */
+  if( !sqlite3_value_type(pValue)==SQLITE_BLOB ) return SQLITE_ERROR;
+
+  /* Check that the blob is roughly the right size. */
+  nBlob = sqlite3_value_bytes(pValue);
+  if( nBlob<(int)sizeof(RtreeMatchArg) 
+   || ((nBlob-sizeof(RtreeMatchArg))%sizeof(double))!=0
+  ){
+    return SQLITE_ERROR;
+  }
+
+  pGeom = (sqlite3_rtree_geometry *)sqlite3_malloc(
+      sizeof(sqlite3_rtree_geometry) + nBlob
+  );
+  if( !pGeom ) return SQLITE_NOMEM;
+  memset(pGeom, 0, sizeof(sqlite3_rtree_geometry));
+  p = (RtreeMatchArg *)&pGeom[1];
+
+  memcpy(p, sqlite3_value_blob(pValue), nBlob);
+  if( p->magic!=RTREE_GEOMETRY_MAGIC 
+   || nBlob!=(int)(sizeof(RtreeMatchArg) + (p->nParam-1)*sizeof(double))
+  ){
+    sqlite3_free(pGeom);
+    return SQLITE_ERROR;
+  }
+
+  pGeom->pContext = p->pContext;
+  pGeom->nParam = p->nParam;
+  pGeom->aParam = p->aParam;
+
+  pCons->xGeom = p->xGeom;
+  pCons->pGeom = pGeom;
+  return SQLITE_OK;
+}
+
+/* 
+** Rtree virtual table module xFilter method.
+*/
+static int rtreeFilter(
+  sqlite3_vtab_cursor *pVtabCursor, 
+  int idxNum, const char *idxStr,
+  int argc, sqlite3_value **argv
+){
+  Rtree *pRtree = (Rtree *)pVtabCursor->pVtab;
+  RtreeCursor *pCsr = (RtreeCursor *)pVtabCursor;
+
+  RtreeNode *pRoot = 0;
+  int ii;
+  int rc = SQLITE_OK;
+
+  rtreeReference(pRtree);
+
+  freeCursorConstraints(pCsr);
+  pCsr->iStrategy = idxNum;
+
+  if( idxNum==1 ){
+    /* Special case - lookup by rowid. */
+    RtreeNode *pLeaf;        /* Leaf on which the required cell resides */
+    i64 iRowid = sqlite3_value_int64(argv[0]);
+    rc = findLeafNode(pRtree, iRowid, &pLeaf);
+    pCsr->pNode = pLeaf; 
+    if( pLeaf ){
+      assert( rc==SQLITE_OK );
+      rc = nodeRowidIndex(pRtree, pLeaf, iRowid, &pCsr->iCell);
+    }
+  }else{
+    /* Normal case - r-tree scan. Set up the RtreeCursor.aConstraint array 
+    ** with the configured constraints. 
+    */
+    if( argc>0 ){
+      pCsr->aConstraint = sqlite3_malloc(sizeof(RtreeConstraint)*argc);
+      pCsr->nConstraint = argc;
+      if( !pCsr->aConstraint ){
+        rc = SQLITE_NOMEM;
+      }else{
+        memset(pCsr->aConstraint, 0, sizeof(RtreeConstraint)*argc);
+        assert( (idxStr==0 && argc==0) || (int)strlen(idxStr)==argc*2 );
+        for(ii=0; ii<argc; ii++){
+          RtreeConstraint *p = &pCsr->aConstraint[ii];
+          p->op = idxStr[ii*2];
+          p->iCoord = idxStr[ii*2+1]-'a';
+          if( p->op==RTREE_MATCH ){
+            /* A MATCH operator. The right-hand-side must be a blob that
+            ** can be cast into an RtreeMatchArg object. One created using
+            ** an sqlite3_rtree_geometry_callback() SQL user function.
+            */
+            rc = deserializeGeometry(argv[ii], p);
+            if( rc!=SQLITE_OK ){
+              break;
+            }
+          }else{
+            p->rValue = sqlite3_value_double(argv[ii]);
+          }
+        }
+      }
+    }
+  
+    if( rc==SQLITE_OK ){
+      pCsr->pNode = 0;
+      rc = nodeAcquire(pRtree, 1, 0, &pRoot);
+    }
+    if( rc==SQLITE_OK ){
+      int isEof = 1;
+      int nCell = NCELL(pRoot);
+      pCsr->pNode = pRoot;
+      for(pCsr->iCell=0; rc==SQLITE_OK && pCsr->iCell<nCell; pCsr->iCell++){
+        assert( pCsr->pNode==pRoot );
+        rc = descendToCell(pRtree, pCsr, pRtree->iDepth, &isEof);
+        if( !isEof ){
+          break;
+        }
+      }
+      if( rc==SQLITE_OK && isEof ){
+        assert( pCsr->pNode==pRoot );
+        nodeRelease(pRtree, pRoot);
+        pCsr->pNode = 0;
+      }
+      assert( rc!=SQLITE_OK || !pCsr->pNode || pCsr->iCell<NCELL(pCsr->pNode) );
+    }
+  }
+
+  rtreeRelease(pRtree);
+  return rc;
+}
+
+/*
+** Rtree virtual table module xBestIndex method. There are three
+** table scan strategies to choose from (in order from most to 
+** least desirable):
+**
+**   idxNum     idxStr        Strategy
+**   ------------------------------------------------
+**     1        Unused        Direct lookup by rowid.
+**     2        See below     R-tree query or full-table scan.
+**   ------------------------------------------------
+**
+** If strategy 1 is used, then idxStr is not meaningful. If strategy
+** 2 is used, idxStr is formatted to contain 2 bytes for each 
+** constraint used. The first two bytes of idxStr correspond to 
+** the constraint in sqlite3_index_info.aConstraintUsage[] with
+** (argvIndex==1) etc.
+**
+** The first of each pair of bytes in idxStr identifies the constraint
+** operator as follows:
+**
+**   Operator    Byte Value
+**   ----------------------
+**      =        0x41 ('A')
+**     <=        0x42 ('B')
+**      <        0x43 ('C')
+**     >=        0x44 ('D')
+**      >        0x45 ('E')
+**   MATCH       0x46 ('F')
+**   ----------------------
+**
+** The second of each pair of bytes identifies the coordinate column
+** to which the constraint applies. The leftmost coordinate column
+** is 'a', the second from the left 'b' etc.
+*/
+static int rtreeBestIndex(sqlite3_vtab *tab, sqlite3_index_info *pIdxInfo){
+  int rc = SQLITE_OK;
+  int ii;
+
+  int iIdx = 0;
+  char zIdxStr[RTREE_MAX_DIMENSIONS*8+1];
+  memset(zIdxStr, 0, sizeof(zIdxStr));
+  UNUSED_PARAMETER(tab);
+
+  assert( pIdxInfo->idxStr==0 );
+  for(ii=0; ii<pIdxInfo->nConstraint && iIdx<(int)(sizeof(zIdxStr)-1); ii++){
+    struct sqlite3_index_constraint *p = &pIdxInfo->aConstraint[ii];
+
+    if( p->usable && p->iColumn==0 && p->op==SQLITE_INDEX_CONSTRAINT_EQ ){
+      /* We have an equality constraint on the rowid. Use strategy 1. */
+      int jj;
+      for(jj=0; jj<ii; jj++){
+        pIdxInfo->aConstraintUsage[jj].argvIndex = 0;
+        pIdxInfo->aConstraintUsage[jj].omit = 0;
+      }
+      pIdxInfo->idxNum = 1;
+      pIdxInfo->aConstraintUsage[ii].argvIndex = 1;
+      pIdxInfo->aConstraintUsage[jj].omit = 1;
+
+      /* This strategy involves a two rowid lookups on an B-Tree structures
+      ** and then a linear search of an R-Tree node. This should be 
+      ** considered almost as quick as a direct rowid lookup (for which 
+      ** sqlite uses an internal cost of 0.0).
+      */ 
+      pIdxInfo->estimatedCost = 10.0;
+      return SQLITE_OK;
+    }
+
+    if( p->usable && (p->iColumn>0 || p->op==SQLITE_INDEX_CONSTRAINT_MATCH) ){
+      u8 op;
+      switch( p->op ){
+        case SQLITE_INDEX_CONSTRAINT_EQ: op = RTREE_EQ; break;
+        case SQLITE_INDEX_CONSTRAINT_GT: op = RTREE_GT; break;
+        case SQLITE_INDEX_CONSTRAINT_LE: op = RTREE_LE; break;
+        case SQLITE_INDEX_CONSTRAINT_LT: op = RTREE_LT; break;
+        case SQLITE_INDEX_CONSTRAINT_GE: op = RTREE_GE; break;
+        default:
+          assert( p->op==SQLITE_INDEX_CONSTRAINT_MATCH );
+          op = RTREE_MATCH; 
+          break;
+      }
+      zIdxStr[iIdx++] = op;
+      zIdxStr[iIdx++] = p->iColumn - 1 + 'a';
+      pIdxInfo->aConstraintUsage[ii].argvIndex = (iIdx/2);
+      pIdxInfo->aConstraintUsage[ii].omit = 1;
+    }
+  }
+
+  pIdxInfo->idxNum = 2;
+  pIdxInfo->needToFreeIdxStr = 1;
+  if( iIdx>0 && 0==(pIdxInfo->idxStr = sqlite3_mprintf("%s", zIdxStr)) ){
+    return SQLITE_NOMEM;
+  }
+  assert( iIdx>=0 );
+  pIdxInfo->estimatedCost = (2000000.0 / (double)(iIdx + 1));
+  return rc;
+}
+
+/*
+** Return the N-dimensional volumn of the cell stored in *p.
+*/
+static float cellArea(Rtree *pRtree, RtreeCell *p){
+  float area = 1.0;
+  int ii;
+  for(ii=0; ii<(pRtree->nDim*2); ii+=2){
+    area = area * (DCOORD(p->aCoord[ii+1]) - DCOORD(p->aCoord[ii]));
+  }
+  return area;
+}
+
+/*
+** Return the margin length of cell p. The margin length is the sum
+** of the objects size in each dimension.
+*/
+static float cellMargin(Rtree *pRtree, RtreeCell *p){
+  float margin = 0.0;
+  int ii;
+  for(ii=0; ii<(pRtree->nDim*2); ii+=2){
+    margin += (DCOORD(p->aCoord[ii+1]) - DCOORD(p->aCoord[ii]));
+  }
+  return margin;
+}
+
+/*
+** Store the union of cells p1 and p2 in p1.
+*/
+static void cellUnion(Rtree *pRtree, RtreeCell *p1, RtreeCell *p2){
+  int ii;
+  if( pRtree->eCoordType==RTREE_COORD_REAL32 ){
+    for(ii=0; ii<(pRtree->nDim*2); ii+=2){
+      p1->aCoord[ii].f = MIN(p1->aCoord[ii].f, p2->aCoord[ii].f);
+      p1->aCoord[ii+1].f = MAX(p1->aCoord[ii+1].f, p2->aCoord[ii+1].f);
+    }
+  }else{
+    for(ii=0; ii<(pRtree->nDim*2); ii+=2){
+      p1->aCoord[ii].i = MIN(p1->aCoord[ii].i, p2->aCoord[ii].i);
+      p1->aCoord[ii+1].i = MAX(p1->aCoord[ii+1].i, p2->aCoord[ii+1].i);
+    }
+  }
+}
+
+/*
+** Return true if the area covered by p2 is a subset of the area covered
+** by p1. False otherwise.
+*/
+static int cellContains(Rtree *pRtree, RtreeCell *p1, RtreeCell *p2){
+  int ii;
+  int isInt = (pRtree->eCoordType==RTREE_COORD_INT32);
+  for(ii=0; ii<(pRtree->nDim*2); ii+=2){
+    RtreeCoord *a1 = &p1->aCoord[ii];
+    RtreeCoord *a2 = &p2->aCoord[ii];
+    if( (!isInt && (a2[0].f<a1[0].f || a2[1].f>a1[1].f)) 
+     || ( isInt && (a2[0].i<a1[0].i || a2[1].i>a1[1].i)) 
+    ){
+      return 0;
+    }
+  }
+  return 1;
+}
+
+/*
+** Return the amount cell p would grow by if it were unioned with pCell.
+*/
+static float cellGrowth(Rtree *pRtree, RtreeCell *p, RtreeCell *pCell){
+  float area;
+  RtreeCell cell;
+  memcpy(&cell, p, sizeof(RtreeCell));
+  area = cellArea(pRtree, &cell);
+  cellUnion(pRtree, &cell, pCell);
+  return (cellArea(pRtree, &cell)-area);
+}
+
+#if VARIANT_RSTARTREE_CHOOSESUBTREE || VARIANT_RSTARTREE_SPLIT
+static float cellOverlap(
+  Rtree *pRtree, 
+  RtreeCell *p, 
+  RtreeCell *aCell, 
+  int nCell, 
+  int iExclude
+){
+  int ii;
+  float overlap = 0.0;
+  for(ii=0; ii<nCell; ii++){
+#if VARIANT_RSTARTREE_CHOOSESUBTREE
+    if( ii!=iExclude )
+#else
+    assert( iExclude==-1 );
+    UNUSED_PARAMETER(iExclude);
+#endif
+    {
+      int jj;
+      float o = 1.0;
+      for(jj=0; jj<(pRtree->nDim*2); jj+=2){
+        double x1;
+        double x2;
+
+        x1 = MAX(DCOORD(p->aCoord[jj]), DCOORD(aCell[ii].aCoord[jj]));
+        x2 = MIN(DCOORD(p->aCoord[jj+1]), DCOORD(aCell[ii].aCoord[jj+1]));
+
+        if( x2<x1 ){
+          o = 0.0;
+          break;
+        }else{
+          o = o * (x2-x1);
+        }
+      }
+      overlap += o;
+    }
+  }
+  return overlap;
+}
+#endif
+
+#if VARIANT_RSTARTREE_CHOOSESUBTREE
+static float cellOverlapEnlargement(
+  Rtree *pRtree, 
+  RtreeCell *p, 
+  RtreeCell *pInsert, 
+  RtreeCell *aCell, 
+  int nCell, 
+  int iExclude
+){
+  float before;
+  float after;
+  before = cellOverlap(pRtree, p, aCell, nCell, iExclude);
+  cellUnion(pRtree, p, pInsert);
+  after = cellOverlap(pRtree, p, aCell, nCell, iExclude);
+  return after-before;
+}
+#endif
+
+
+/*
+** This function implements the ChooseLeaf algorithm from Gutman[84].
+** ChooseSubTree in r*tree terminology.
+*/
+static int ChooseLeaf(
+  Rtree *pRtree,               /* Rtree table */
+  RtreeCell *pCell,            /* Cell to insert into rtree */
+  int iHeight,                 /* Height of sub-tree rooted at pCell */
+  RtreeNode **ppLeaf           /* OUT: Selected leaf page */
+){
+  int rc;
+  int ii;
+  RtreeNode *pNode;
+  rc = nodeAcquire(pRtree, 1, 0, &pNode);
+
+  for(ii=0; rc==SQLITE_OK && ii<(pRtree->iDepth-iHeight); ii++){
+    int iCell;
+    sqlite3_int64 iBest;
+
+    float fMinGrowth;
+    float fMinArea;
+    float fMinOverlap;
+
+    int nCell = NCELL(pNode);
+    RtreeCell cell;
+    RtreeNode *pChild;
+
+    RtreeCell *aCell = 0;
+
+#if VARIANT_RSTARTREE_CHOOSESUBTREE
+    if( ii==(pRtree->iDepth-1) ){
+      int jj;
+      aCell = sqlite3_malloc(sizeof(RtreeCell)*nCell);
+      if( !aCell ){
+        rc = SQLITE_NOMEM;
+        nodeRelease(pRtree, pNode);
+        pNode = 0;
+        continue;
+      }
+      for(jj=0; jj<nCell; jj++){
+        nodeGetCell(pRtree, pNode, jj, &aCell[jj]);
+      }
+    }
+#endif
+
+    /* Select the child node which will be enlarged the least if pCell
+    ** is inserted into it. Resolve ties by choosing the entry with
+    ** the smallest area.
+    */
+    for(iCell=0; iCell<nCell; iCell++){
+      int bBest = 0;
+      float growth;
+      float area;
+      float overlap = 0.0;
+      nodeGetCell(pRtree, pNode, iCell, &cell);
+      growth = cellGrowth(pRtree, &cell, pCell);
+      area = cellArea(pRtree, &cell);
+
+#if VARIANT_RSTARTREE_CHOOSESUBTREE
+      if( ii==(pRtree->iDepth-1) ){
+        overlap = cellOverlapEnlargement(pRtree,&cell,pCell,aCell,nCell,iCell);
+      }
+      if( (iCell==0) 
+       || (overlap<fMinOverlap) 
+       || (overlap==fMinOverlap && growth<fMinGrowth)
+       || (overlap==fMinOverlap && growth==fMinGrowth && area<fMinArea)
+      ){
+        bBest = 1;
+      }
+#else
+      if( iCell==0||growth<fMinGrowth||(growth==fMinGrowth && area<fMinArea) ){
+        bBest = 1;
+      }
+#endif
+      if( bBest ){
+        fMinOverlap = overlap;
+        fMinGrowth = growth;
+        fMinArea = area;
+        iBest = cell.iRowid;
+      }
+    }
+
+    sqlite3_free(aCell);
+    rc = nodeAcquire(pRtree, iBest, pNode, &pChild);
+    nodeRelease(pRtree, pNode);
+    pNode = pChild;
+  }
+
+  *ppLeaf = pNode;
+  return rc;
+}
+
+/*
+** A cell with the same content as pCell has just been inserted into
+** the node pNode. This function updates the bounding box cells in
+** all ancestor elements.
+*/
+static int AdjustTree(
+  Rtree *pRtree,                    /* Rtree table */
+  RtreeNode *pNode,                 /* Adjust ancestry of this node. */
+  RtreeCell *pCell                  /* This cell was just inserted */
+){
+  RtreeNode *p = pNode;
+  while( p->pParent ){
+    RtreeNode *pParent = p->pParent;
+    RtreeCell cell;
+    int iCell;
+
+    if( nodeParentIndex(pRtree, p, &iCell) ){
+      return SQLITE_CORRUPT;
+    }
+
+    nodeGetCell(pRtree, pParent, iCell, &cell);
+    if( !cellContains(pRtree, &cell, pCell) ){
+      cellUnion(pRtree, &cell, pCell);
+      nodeOverwriteCell(pRtree, pParent, &cell, iCell);
+    }
+ 
+    p = pParent;
+  }
+  return SQLITE_OK;
+}
+
+/*
+** Write mapping (iRowid->iNode) to the <rtree>_rowid table.
+*/
+static int rowidWrite(Rtree *pRtree, sqlite3_int64 iRowid, sqlite3_int64 iNode){
+  sqlite3_bind_int64(pRtree->pWriteRowid, 1, iRowid);
+  sqlite3_bind_int64(pRtree->pWriteRowid, 2, iNode);
+  sqlite3_step(pRtree->pWriteRowid);
+  return sqlite3_reset(pRtree->pWriteRowid);
+}
+
+/*
+** Write mapping (iNode->iPar) to the <rtree>_parent table.
+*/
+static int parentWrite(Rtree *pRtree, sqlite3_int64 iNode, sqlite3_int64 iPar){
+  sqlite3_bind_int64(pRtree->pWriteParent, 1, iNode);
+  sqlite3_bind_int64(pRtree->pWriteParent, 2, iPar);
+  sqlite3_step(pRtree->pWriteParent);
+  return sqlite3_reset(pRtree->pWriteParent);
+}
+
+static int rtreeInsertCell(Rtree *, RtreeNode *, RtreeCell *, int);
+
+#if VARIANT_GUTTMAN_LINEAR_SPLIT
+/*
+** Implementation of the linear variant of the PickNext() function from
+** Guttman[84].
+*/
+static RtreeCell *LinearPickNext(
+  Rtree *pRtree,
+  RtreeCell *aCell, 
+  int nCell, 
+  RtreeCell *pLeftBox, 
+  RtreeCell *pRightBox,
+  int *aiUsed
+){
+  int ii;
+  for(ii=0; aiUsed[ii]; ii++);
+  aiUsed[ii] = 1;
+  return &aCell[ii];
+}
+
+/*
+** Implementation of the linear variant of the PickSeeds() function from
+** Guttman[84].
+*/
+static void LinearPickSeeds(
+  Rtree *pRtree,
+  RtreeCell *aCell, 
+  int nCell, 
+  int *piLeftSeed, 
+  int *piRightSeed
+){
+  int i;
+  int iLeftSeed = 0;
+  int iRightSeed = 1;
+  float maxNormalInnerWidth = 0.0;
+
+  /* Pick two "seed" cells from the array of cells. The algorithm used
+  ** here is the LinearPickSeeds algorithm from Gutman[1984]. The 
+  ** indices of the two seed cells in the array are stored in local
+  ** variables iLeftSeek and iRightSeed.
+  */
+  for(i=0; i<pRtree->nDim; i++){
+    float x1 = DCOORD(aCell[0].aCoord[i*2]);
+    float x2 = DCOORD(aCell[0].aCoord[i*2+1]);
+    float x3 = x1;
+    float x4 = x2;
+    int jj;
+
+    int iCellLeft = 0;
+    int iCellRight = 0;
+
+    for(jj=1; jj<nCell; jj++){
+      float left = DCOORD(aCell[jj].aCoord[i*2]);
+      float right = DCOORD(aCell[jj].aCoord[i*2+1]);
+
+      if( left<x1 ) x1 = left;
+      if( right>x4 ) x4 = right;
+      if( left>x3 ){
+        x3 = left;
+        iCellRight = jj;
+      }
+      if( right<x2 ){
+        x2 = right;
+        iCellLeft = jj;
+      }
+    }
+
+    if( x4!=x1 ){
+      float normalwidth = (x3 - x2) / (x4 - x1);
+      if( normalwidth>maxNormalInnerWidth ){
+        iLeftSeed = iCellLeft;
+        iRightSeed = iCellRight;
+      }
+    }
+  }
+
+  *piLeftSeed = iLeftSeed;
+  *piRightSeed = iRightSeed;
+}
+#endif /* VARIANT_GUTTMAN_LINEAR_SPLIT */
+
+#if VARIANT_GUTTMAN_QUADRATIC_SPLIT
+/*
+** Implementation of the quadratic variant of the PickNext() function from
+** Guttman[84].
+*/
+static RtreeCell *QuadraticPickNext(
+  Rtree *pRtree,
+  RtreeCell *aCell, 
+  int nCell, 
+  RtreeCell *pLeftBox, 
+  RtreeCell *pRightBox,
+  int *aiUsed
+){
+  #define FABS(a) ((a)<0.0?-1.0*(a):(a))
+
+  int iSelect = -1;
+  float fDiff;
+  int ii;
+  for(ii=0; ii<nCell; ii++){
+    if( aiUsed[ii]==0 ){
+      float left = cellGrowth(pRtree, pLeftBox, &aCell[ii]);
+      float right = cellGrowth(pRtree, pLeftBox, &aCell[ii]);
+      float diff = FABS(right-left);
+      if( iSelect<0 || diff>fDiff ){
+        fDiff = diff;
+        iSelect = ii;
+      }
+    }
+  }
+  aiUsed[iSelect] = 1;
+  return &aCell[iSelect];
+}
+
+/*
+** Implementation of the quadratic variant of the PickSeeds() function from
+** Guttman[84].
+*/
+static void QuadraticPickSeeds(
+  Rtree *pRtree,
+  RtreeCell *aCell, 
+  int nCell, 
+  int *piLeftSeed, 
+  int *piRightSeed
+){
+  int ii;
+  int jj;
+
+  int iLeftSeed = 0;
+  int iRightSeed = 1;
+  float fWaste = 0.0;
+
+  for(ii=0; ii<nCell; ii++){
+    for(jj=ii+1; jj<nCell; jj++){
+      float right = cellArea(pRtree, &aCell[jj]);
+      float growth = cellGrowth(pRtree, &aCell[ii], &aCell[jj]);
+      float waste = growth - right;
+
+      if( waste>fWaste ){
+        iLeftSeed = ii;
+        iRightSeed = jj;
+        fWaste = waste;
+      }
+    }
+  }
+
+  *piLeftSeed = iLeftSeed;
+  *piRightSeed = iRightSeed;
+}
+#endif /* VARIANT_GUTTMAN_QUADRATIC_SPLIT */
+
+/*
+** Arguments aIdx, aDistance and aSpare all point to arrays of size
+** nIdx. The aIdx array contains the set of integers from 0 to 
+** (nIdx-1) in no particular order. This function sorts the values
+** in aIdx according to the indexed values in aDistance. For
+** example, assuming the inputs:
+**
+**   aIdx      = { 0,   1,   2,   3 }
+**   aDistance = { 5.0, 2.0, 7.0, 6.0 }
+**
+** this function sets the aIdx array to contain:
+**
+**   aIdx      = { 0,   1,   2,   3 }
+**
+** The aSpare array is used as temporary working space by the
+** sorting algorithm.
+*/
+static void SortByDistance(
+  int *aIdx, 
+  int nIdx, 
+  float *aDistance, 
+  int *aSpare
+){
+  if( nIdx>1 ){
+    int iLeft = 0;
+    int iRight = 0;
+
+    int nLeft = nIdx/2;
+    int nRight = nIdx-nLeft;
+    int *aLeft = aIdx;
+    int *aRight = &aIdx[nLeft];
+
+    SortByDistance(aLeft, nLeft, aDistance, aSpare);
+    SortByDistance(aRight, nRight, aDistance, aSpare);
+
+    memcpy(aSpare, aLeft, sizeof(int)*nLeft);
+    aLeft = aSpare;
+
+    while( iLeft<nLeft || iRight<nRight ){
+      if( iLeft==nLeft ){
+        aIdx[iLeft+iRight] = aRight[iRight];
+        iRight++;
+      }else if( iRight==nRight ){
+        aIdx[iLeft+iRight] = aLeft[iLeft];
+        iLeft++;
+      }else{
+        float fLeft = aDistance[aLeft[iLeft]];
+        float fRight = aDistance[aRight[iRight]];
+        if( fLeft<fRight ){
+          aIdx[iLeft+iRight] = aLeft[iLeft];
+          iLeft++;
+        }else{
+          aIdx[iLeft+iRight] = aRight[iRight];
+          iRight++;
+        }
+      }
+    }
+
+#if 0
+    /* Check that the sort worked */
+    {
+      int jj;
+      for(jj=1; jj<nIdx; jj++){
+        float left = aDistance[aIdx[jj-1]];
+        float right = aDistance[aIdx[jj]];
+        assert( left<=right );
+      }
+    }
+#endif
+  }
+}
+
+/*
+** Arguments aIdx, aCell and aSpare all point to arrays of size
+** nIdx. The aIdx array contains the set of integers from 0 to 
+** (nIdx-1) in no particular order. This function sorts the values
+** in aIdx according to dimension iDim of the cells in aCell. The
+** minimum value of dimension iDim is considered first, the
+** maximum used to break ties.
+**
+** The aSpare array is used as temporary working space by the
+** sorting algorithm.
+*/
+static void SortByDimension(
+  Rtree *pRtree,
+  int *aIdx, 
+  int nIdx, 
+  int iDim, 
+  RtreeCell *aCell, 
+  int *aSpare
+){
+  if( nIdx>1 ){
+
+    int iLeft = 0;
+    int iRight = 0;
+
+    int nLeft = nIdx/2;
+    int nRight = nIdx-nLeft;
+    int *aLeft = aIdx;
+    int *aRight = &aIdx[nLeft];
+
+    SortByDimension(pRtree, aLeft, nLeft, iDim, aCell, aSpare);
+    SortByDimension(pRtree, aRight, nRight, iDim, aCell, aSpare);
+
+    memcpy(aSpare, aLeft, sizeof(int)*nLeft);
+    aLeft = aSpare;
+    while( iLeft<nLeft || iRight<nRight ){
+      double xleft1 = DCOORD(aCell[aLeft[iLeft]].aCoord[iDim*2]);
+      double xleft2 = DCOORD(aCell[aLeft[iLeft]].aCoord[iDim*2+1]);
+      double xright1 = DCOORD(aCell[aRight[iRight]].aCoord[iDim*2]);
+      double xright2 = DCOORD(aCell[aRight[iRight]].aCoord[iDim*2+1]);
+      if( (iLeft!=nLeft) && ((iRight==nRight)
+       || (xleft1<xright1)
+       || (xleft1==xright1 && xleft2<xright2)
+      )){
+        aIdx[iLeft+iRight] = aLeft[iLeft];
+        iLeft++;
+      }else{
+        aIdx[iLeft+iRight] = aRight[iRight];
+        iRight++;
+      }
+    }
+
+#if 0
+    /* Check that the sort worked */
+    {
+      int jj;
+      for(jj=1; jj<nIdx; jj++){
+        float xleft1 = aCell[aIdx[jj-1]].aCoord[iDim*2];
+        float xleft2 = aCell[aIdx[jj-1]].aCoord[iDim*2+1];
+        float xright1 = aCell[aIdx[jj]].aCoord[iDim*2];
+        float xright2 = aCell[aIdx[jj]].aCoord[iDim*2+1];
+        assert( xleft1<=xright1 && (xleft1<xright1 || xleft2<=xright2) );
+      }
+    }
+#endif
+  }
+}
+
+#if VARIANT_RSTARTREE_SPLIT
+/*
+** Implementation of the R*-tree variant of SplitNode from Beckman[1990].
+*/
+static int splitNodeStartree(
+  Rtree *pRtree,
+  RtreeCell *aCell,
+  int nCell,
+  RtreeNode *pLeft,
+  RtreeNode *pRight,
+  RtreeCell *pBboxLeft,
+  RtreeCell *pBboxRight
+){
+  int **aaSorted;
+  int *aSpare;
+  int ii;
+
+  int iBestDim;
+  int iBestSplit;
+  float fBestMargin;
+
+  int nByte = (pRtree->nDim+1)*(sizeof(int*)+nCell*sizeof(int));
+
+  aaSorted = (int **)sqlite3_malloc(nByte);
+  if( !aaSorted ){
+    return SQLITE_NOMEM;
+  }
+
+  aSpare = &((int *)&aaSorted[pRtree->nDim])[pRtree->nDim*nCell];
+  memset(aaSorted, 0, nByte);
+  for(ii=0; ii<pRtree->nDim; ii++){
+    int jj;
+    aaSorted[ii] = &((int *)&aaSorted[pRtree->nDim])[ii*nCell];
+    for(jj=0; jj<nCell; jj++){
+      aaSorted[ii][jj] = jj;
+    }
+    SortByDimension(pRtree, aaSorted[ii], nCell, ii, aCell, aSpare);
+  }
+
+  for(ii=0; ii<pRtree->nDim; ii++){
+    float margin = 0.0;
+    float fBestOverlap;
+    float fBestArea;
+    int iBestLeft;
+    int nLeft;
+
+    for(
+      nLeft=RTREE_MINCELLS(pRtree); 
+      nLeft<=(nCell-RTREE_MINCELLS(pRtree)); 
+      nLeft++
+    ){
+      RtreeCell left;
+      RtreeCell right;
+      int kk;
+      float overlap;
+      float area;
+
+      memcpy(&left, &aCell[aaSorted[ii][0]], sizeof(RtreeCell));
+      memcpy(&right, &aCell[aaSorted[ii][nCell-1]], sizeof(RtreeCell));
+      for(kk=1; kk<(nCell-1); kk++){
+        if( kk<nLeft ){
+          cellUnion(pRtree, &left, &aCell[aaSorted[ii][kk]]);
+        }else{
+          cellUnion(pRtree, &right, &aCell[aaSorted[ii][kk]]);
+        }
+      }
+      margin += cellMargin(pRtree, &left);
+      margin += cellMargin(pRtree, &right);
+      overlap = cellOverlap(pRtree, &left, &right, 1, -1);
+      area = cellArea(pRtree, &left) + cellArea(pRtree, &right);
+      if( (nLeft==RTREE_MINCELLS(pRtree))
+       || (overlap<fBestOverlap)
+       || (overlap==fBestOverlap && area<fBestArea)
+      ){
+        iBestLeft = nLeft;
+        fBestOverlap = overlap;
+        fBestArea = area;
+      }
+    }
+
+    if( ii==0 || margin<fBestMargin ){
+      iBestDim = ii;
+      fBestMargin = margin;
+      iBestSplit = iBestLeft;
+    }
+  }
+
+  memcpy(pBboxLeft, &aCell[aaSorted[iBestDim][0]], sizeof(RtreeCell));
+  memcpy(pBboxRight, &aCell[aaSorted[iBestDim][iBestSplit]], sizeof(RtreeCell));
+  for(ii=0; ii<nCell; ii++){
+    RtreeNode *pTarget = (ii<iBestSplit)?pLeft:pRight;
+    RtreeCell *pBbox = (ii<iBestSplit)?pBboxLeft:pBboxRight;
+    RtreeCell *pCell = &aCell[aaSorted[iBestDim][ii]];
+    nodeInsertCell(pRtree, pTarget, pCell);
+    cellUnion(pRtree, pBbox, pCell);
+  }
+
+  sqlite3_free(aaSorted);
+  return SQLITE_OK;
+}
+#endif
+
+#if VARIANT_GUTTMAN_SPLIT
+/*
+** Implementation of the regular R-tree SplitNode from Guttman[1984].
+*/
+static int splitNodeGuttman(
+  Rtree *pRtree,
+  RtreeCell *aCell,
+  int nCell,
+  RtreeNode *pLeft,
+  RtreeNode *pRight,
+  RtreeCell *pBboxLeft,
+  RtreeCell *pBboxRight
+){
+  int iLeftSeed = 0;
+  int iRightSeed = 1;
+  int *aiUsed;
+  int i;
+
+  aiUsed = sqlite3_malloc(sizeof(int)*nCell);
+  if( !aiUsed ){
+    return SQLITE_NOMEM;
+  }
+  memset(aiUsed, 0, sizeof(int)*nCell);
+
+  PickSeeds(pRtree, aCell, nCell, &iLeftSeed, &iRightSeed);
+
+  memcpy(pBboxLeft, &aCell[iLeftSeed], sizeof(RtreeCell));
+  memcpy(pBboxRight, &aCell[iRightSeed], sizeof(RtreeCell));
+  nodeInsertCell(pRtree, pLeft, &aCell[iLeftSeed]);
+  nodeInsertCell(pRtree, pRight, &aCell[iRightSeed]);
+  aiUsed[iLeftSeed] = 1;
+  aiUsed[iRightSeed] = 1;
+
+  for(i=nCell-2; i>0; i--){
+    RtreeCell *pNext;
+    pNext = PickNext(pRtree, aCell, nCell, pBboxLeft, pBboxRight, aiUsed);
+    float diff =  
+      cellGrowth(pRtree, pBboxLeft, pNext) - 
+      cellGrowth(pRtree, pBboxRight, pNext)
+    ;
+    if( (RTREE_MINCELLS(pRtree)-NCELL(pRight)==i)
+     || (diff>0.0 && (RTREE_MINCELLS(pRtree)-NCELL(pLeft)!=i))
+    ){
+      nodeInsertCell(pRtree, pRight, pNext);
+      cellUnion(pRtree, pBboxRight, pNext);
+    }else{
+      nodeInsertCell(pRtree, pLeft, pNext);
+      cellUnion(pRtree, pBboxLeft, pNext);
+    }
+  }
+
+  sqlite3_free(aiUsed);
+  return SQLITE_OK;
+}
+#endif
+
+static int updateMapping(
+  Rtree *pRtree, 
+  i64 iRowid, 
+  RtreeNode *pNode, 
+  int iHeight
+){
+  int (*xSetMapping)(Rtree *, sqlite3_int64, sqlite3_int64);
+  xSetMapping = ((iHeight==0)?rowidWrite:parentWrite);
+  if( iHeight>0 ){
+    RtreeNode *pChild = nodeHashLookup(pRtree, iRowid);
+    if( pChild ){
+      nodeRelease(pRtree, pChild->pParent);
+      nodeReference(pNode);
+      pChild->pParent = pNode;
+    }
+  }
+  return xSetMapping(pRtree, iRowid, pNode->iNode);
+}
+
+static int SplitNode(
+  Rtree *pRtree,
+  RtreeNode *pNode,
+  RtreeCell *pCell,
+  int iHeight
+){
+  int i;
+  int newCellIsRight = 0;
+
+  int rc = SQLITE_OK;
+  int nCell = NCELL(pNode);
+  RtreeCell *aCell;
+  int *aiUsed;
+
+  RtreeNode *pLeft = 0;
+  RtreeNode *pRight = 0;
+
+  RtreeCell leftbbox;
+  RtreeCell rightbbox;
+
+  /* Allocate an array and populate it with a copy of pCell and 
+  ** all cells from node pLeft. Then zero the original node.
+  */
+  aCell = sqlite3_malloc((sizeof(RtreeCell)+sizeof(int))*(nCell+1));
+  if( !aCell ){
+    rc = SQLITE_NOMEM;
+    goto splitnode_out;
+  }
+  aiUsed = (int *)&aCell[nCell+1];
+  memset(aiUsed, 0, sizeof(int)*(nCell+1));
+  for(i=0; i<nCell; i++){
+    nodeGetCell(pRtree, pNode, i, &aCell[i]);
+  }
+  nodeZero(pRtree, pNode);
+  memcpy(&aCell[nCell], pCell, sizeof(RtreeCell));
+  nCell++;
+
+  if( pNode->iNode==1 ){
+    pRight = nodeNew(pRtree, pNode);
+    pLeft = nodeNew(pRtree, pNode);
+    pRtree->iDepth++;
+    pNode->isDirty = 1;
+    writeInt16(pNode->zData, pRtree->iDepth);
+  }else{
+    pLeft = pNode;
+    pRight = nodeNew(pRtree, pLeft->pParent);
+    nodeReference(pLeft);
+  }
+
+  if( !pLeft || !pRight ){
+    rc = SQLITE_NOMEM;
+    goto splitnode_out;
+  }
+
+  memset(pLeft->zData, 0, pRtree->iNodeSize);
+  memset(pRight->zData, 0, pRtree->iNodeSize);
+
+  rc = AssignCells(pRtree, aCell, nCell, pLeft, pRight, &leftbbox, &rightbbox);
+  if( rc!=SQLITE_OK ){
+    goto splitnode_out;
+  }
+
+  /* Ensure both child nodes have node numbers assigned to them by calling
+  ** nodeWrite(). Node pRight always needs a node number, as it was created
+  ** by nodeNew() above. But node pLeft sometimes already has a node number.
+  ** In this case avoid the all to nodeWrite().
+  */
+  if( SQLITE_OK!=(rc = nodeWrite(pRtree, pRight))
+   || (0==pLeft->iNode && SQLITE_OK!=(rc = nodeWrite(pRtree, pLeft)))
+  ){
+    goto splitnode_out;
+  }
+
+  rightbbox.iRowid = pRight->iNode;
+  leftbbox.iRowid = pLeft->iNode;
+
+  if( pNode->iNode==1 ){
+    rc = rtreeInsertCell(pRtree, pLeft->pParent, &leftbbox, iHeight+1);
+    if( rc!=SQLITE_OK ){
+      goto splitnode_out;
+    }
+  }else{
+    RtreeNode *pParent = pLeft->pParent;
+    int iCell;
+    rc = nodeParentIndex(pRtree, pLeft, &iCell);
+    if( rc==SQLITE_OK ){
+      nodeOverwriteCell(pRtree, pParent, &leftbbox, iCell);
+      rc = AdjustTree(pRtree, pParent, &leftbbox);
+    }
+    if( rc!=SQLITE_OK ){
+      goto splitnode_out;
+    }
+  }
+  if( (rc = rtreeInsertCell(pRtree, pRight->pParent, &rightbbox, iHeight+1)) ){
+    goto splitnode_out;
+  }
+
+  for(i=0; i<NCELL(pRight); i++){
+    i64 iRowid = nodeGetRowid(pRtree, pRight, i);
+    rc = updateMapping(pRtree, iRowid, pRight, iHeight);
+    if( iRowid==pCell->iRowid ){
+      newCellIsRight = 1;
+    }
+    if( rc!=SQLITE_OK ){
+      goto splitnode_out;
+    }
+  }
+  if( pNode->iNode==1 ){
+    for(i=0; i<NCELL(pLeft); i++){
+      i64 iRowid = nodeGetRowid(pRtree, pLeft, i);
+      rc = updateMapping(pRtree, iRowid, pLeft, iHeight);
+      if( rc!=SQLITE_OK ){
+        goto splitnode_out;
+      }
+    }
+  }else if( newCellIsRight==0 ){
+    rc = updateMapping(pRtree, pCell->iRowid, pLeft, iHeight);
+  }
+
+  if( rc==SQLITE_OK ){
+    rc = nodeRelease(pRtree, pRight);
+    pRight = 0;
+  }
+  if( rc==SQLITE_OK ){
+    rc = nodeRelease(pRtree, pLeft);
+    pLeft = 0;
+  }
+
+splitnode_out:
+  nodeRelease(pRtree, pRight);
+  nodeRelease(pRtree, pLeft);
+  sqlite3_free(aCell);
+  return rc;
+}
+
+/*
+** If node pLeaf is not the root of the r-tree and its pParent pointer is 
+** still NULL, load all ancestor nodes of pLeaf into memory and populate
+** the pLeaf->pParent chain all the way up to the root node.
+**
+** This operation is required when a row is deleted (or updated - an update
+** is implemented as a delete followed by an insert). SQLite provides the
+** rowid of the row to delete, which can be used to find the leaf on which
+** the entry resides (argument pLeaf). Once the leaf is located, this 
+** function is called to determine its ancestry.
+*/
+static int fixLeafParent(Rtree *pRtree, RtreeNode *pLeaf){
+  int rc = SQLITE_OK;
+  RtreeNode *pChild = pLeaf;
+  while( rc==SQLITE_OK && pChild->iNode!=1 && pChild->pParent==0 ){
+    int rc2 = SQLITE_OK;          /* sqlite3_reset() return code */
+    sqlite3_bind_int64(pRtree->pReadParent, 1, pChild->iNode);
+    rc = sqlite3_step(pRtree->pReadParent);
+    if( rc==SQLITE_ROW ){
+      RtreeNode *pTest;           /* Used to test for reference loops */
+      i64 iNode;                  /* Node number of parent node */
+
+      /* Before setting pChild->pParent, test that we are not creating a
+      ** loop of references (as we would if, say, pChild==pParent). We don't
+      ** want to do this as it leads to a memory leak when trying to delete
+      ** the referenced counted node structures.
+      */
+      iNode = sqlite3_column_int64(pRtree->pReadParent, 0);
+      for(pTest=pLeaf; pTest && pTest->iNode!=iNode; pTest=pTest->pParent);
+      if( !pTest ){
+        rc2 = nodeAcquire(pRtree, iNode, 0, &pChild->pParent);
+      }
+    }
+    rc = sqlite3_reset(pRtree->pReadParent);
+    if( rc==SQLITE_OK ) rc = rc2;
+    if( rc==SQLITE_OK && !pChild->pParent ) rc = SQLITE_CORRUPT;
+    pChild = pChild->pParent;
+  }
+  return rc;
+}
+
+static int deleteCell(Rtree *, RtreeNode *, int, int);
+
+static int removeNode(Rtree *pRtree, RtreeNode *pNode, int iHeight){
+  int rc;
+  int rc2;
+  RtreeNode *pParent;
+  int iCell;
+
+  assert( pNode->nRef==1 );
+
+  /* Remove the entry in the parent cell. */
+  rc = nodeParentIndex(pRtree, pNode, &iCell);
+  if( rc==SQLITE_OK ){
+    pParent = pNode->pParent;
+    pNode->pParent = 0;
+    rc = deleteCell(pRtree, pParent, iCell, iHeight+1);
+  }
+  rc2 = nodeRelease(pRtree, pParent);
+  if( rc==SQLITE_OK ){
+    rc = rc2;
+  }
+  if( rc!=SQLITE_OK ){
+    return rc;
+  }
+
+  /* Remove the xxx_node entry. */
+  sqlite3_bind_int64(pRtree->pDeleteNode, 1, pNode->iNode);
+  sqlite3_step(pRtree->pDeleteNode);
+  if( SQLITE_OK!=(rc = sqlite3_reset(pRtree->pDeleteNode)) ){
+    return rc;
+  }
+
+  /* Remove the xxx_parent entry. */
+  sqlite3_bind_int64(pRtree->pDeleteParent, 1, pNode->iNode);
+  sqlite3_step(pRtree->pDeleteParent);
+  if( SQLITE_OK!=(rc = sqlite3_reset(pRtree->pDeleteParent)) ){
+    return rc;
+  }
+  
+  /* Remove the node from the in-memory hash table and link it into
+  ** the Rtree.pDeleted list. Its contents will be re-inserted later on.
+  */
+  nodeHashDelete(pRtree, pNode);
+  pNode->iNode = iHeight;
+  pNode->pNext = pRtree->pDeleted;
+  pNode->nRef++;
+  pRtree->pDeleted = pNode;
+
+  return SQLITE_OK;
+}
+
+static int fixBoundingBox(Rtree *pRtree, RtreeNode *pNode){
+  RtreeNode *pParent = pNode->pParent;
+  int rc = SQLITE_OK; 
+  if( pParent ){
+    int ii; 
+    int nCell = NCELL(pNode);
+    RtreeCell box;                            /* Bounding box for pNode */
+    nodeGetCell(pRtree, pNode, 0, &box);
+    for(ii=1; ii<nCell; ii++){
+      RtreeCell cell;
+      nodeGetCell(pRtree, pNode, ii, &cell);
+      cellUnion(pRtree, &box, &cell);
+    }
+    box.iRowid = pNode->iNode;
+    rc = nodeParentIndex(pRtree, pNode, &ii);
+    if( rc==SQLITE_OK ){
+      nodeOverwriteCell(pRtree, pParent, &box, ii);
+      rc = fixBoundingBox(pRtree, pParent);
+    }
+  }
+  return rc;
+}
+
+/*
+** Delete the cell at index iCell of node pNode. After removing the
+** cell, adjust the r-tree data structure if required.
+*/
+static int deleteCell(Rtree *pRtree, RtreeNode *pNode, int iCell, int iHeight){
+  RtreeNode *pParent;
+  int rc;
+
+  if( SQLITE_OK!=(rc = fixLeafParent(pRtree, pNode)) ){
+    return rc;
+  }
+
+  /* Remove the cell from the node. This call just moves bytes around
+  ** the in-memory node image, so it cannot fail.
+  */
+  nodeDeleteCell(pRtree, pNode, iCell);
+
+  /* If the node is not the tree root and now has less than the minimum
+  ** number of cells, remove it from the tree. Otherwise, update the
+  ** cell in the parent node so that it tightly contains the updated
+  ** node.
+  */
+  pParent = pNode->pParent;
+  assert( pParent || pNode->iNode==1 );
+  if( pParent ){
+    if( NCELL(pNode)<RTREE_MINCELLS(pRtree) ){
+      rc = removeNode(pRtree, pNode, iHeight);
+    }else{
+      rc = fixBoundingBox(pRtree, pNode);
+    }
+  }
+
+  return rc;
+}
+
+static int Reinsert(
+  Rtree *pRtree, 
+  RtreeNode *pNode, 
+  RtreeCell *pCell, 
+  int iHeight
+){
+  int *aOrder;
+  int *aSpare;
+  RtreeCell *aCell;
+  float *aDistance;
+  int nCell;
+  float aCenterCoord[RTREE_MAX_DIMENSIONS];
+  int iDim;
+  int ii;
+  int rc = SQLITE_OK;
+
+  memset(aCenterCoord, 0, sizeof(float)*RTREE_MAX_DIMENSIONS);
+
+  nCell = NCELL(pNode)+1;
+
+  /* Allocate the buffers used by this operation. The allocation is
+  ** relinquished before this function returns.
+  */
+  aCell = (RtreeCell *)sqlite3_malloc(nCell * (
+    sizeof(RtreeCell) +         /* aCell array */
+    sizeof(int)       +         /* aOrder array */
+    sizeof(int)       +         /* aSpare array */
+    sizeof(float)               /* aDistance array */
+  ));
+  if( !aCell ){
+    return SQLITE_NOMEM;
+  }
+  aOrder    = (int *)&aCell[nCell];
+  aSpare    = (int *)&aOrder[nCell];
+  aDistance = (float *)&aSpare[nCell];
+
+  for(ii=0; ii<nCell; ii++){
+    if( ii==(nCell-1) ){
+      memcpy(&aCell[ii], pCell, sizeof(RtreeCell));
+    }else{
+      nodeGetCell(pRtree, pNode, ii, &aCell[ii]);
+    }
+    aOrder[ii] = ii;
+    for(iDim=0; iDim<pRtree->nDim; iDim++){
+      aCenterCoord[iDim] += DCOORD(aCell[ii].aCoord[iDim*2]);
+      aCenterCoord[iDim] += DCOORD(aCell[ii].aCoord[iDim*2+1]);
+    }
+  }
+  for(iDim=0; iDim<pRtree->nDim; iDim++){
+    aCenterCoord[iDim] = aCenterCoord[iDim]/((float)nCell*2.0);
+  }
+
+  for(ii=0; ii<nCell; ii++){
+    aDistance[ii] = 0.0;
+    for(iDim=0; iDim<pRtree->nDim; iDim++){
+      float coord = DCOORD(aCell[ii].aCoord[iDim*2+1]) - 
+          DCOORD(aCell[ii].aCoord[iDim*2]);
+      aDistance[ii] += (coord-aCenterCoord[iDim])*(coord-aCenterCoord[iDim]);
+    }
+  }
+
+  SortByDistance(aOrder, nCell, aDistance, aSpare);
+  nodeZero(pRtree, pNode);
+
+  for(ii=0; rc==SQLITE_OK && ii<(nCell-(RTREE_MINCELLS(pRtree)+1)); ii++){
+    RtreeCell *p = &aCell[aOrder[ii]];
+    nodeInsertCell(pRtree, pNode, p);
+    if( p->iRowid==pCell->iRowid ){
+      if( iHeight==0 ){
+        rc = rowidWrite(pRtree, p->iRowid, pNode->iNode);
+      }else{
+        rc = parentWrite(pRtree, p->iRowid, pNode->iNode);
+      }
+    }
+  }
+  if( rc==SQLITE_OK ){
+    rc = fixBoundingBox(pRtree, pNode);
+  }
+  for(; rc==SQLITE_OK && ii<nCell; ii++){
+    /* Find a node to store this cell in. pNode->iNode currently contains
+    ** the height of the sub-tree headed by the cell.
+    */
+    RtreeNode *pInsert;
+    RtreeCell *p = &aCell[aOrder[ii]];
+    rc = ChooseLeaf(pRtree, p, iHeight, &pInsert);
+    if( rc==SQLITE_OK ){
+      int rc2;
+      rc = rtreeInsertCell(pRtree, pInsert, p, iHeight);
+      rc2 = nodeRelease(pRtree, pInsert);
+      if( rc==SQLITE_OK ){
+        rc = rc2;
+      }
+    }
+  }
+
+  sqlite3_free(aCell);
+  return rc;
+}
+
+/*
+** Insert cell pCell into node pNode. Node pNode is the head of a 
+** subtree iHeight high (leaf nodes have iHeight==0).
+*/
+static int rtreeInsertCell(
+  Rtree *pRtree,
+  RtreeNode *pNode,
+  RtreeCell *pCell,
+  int iHeight
+){
+  int rc = SQLITE_OK;
+  if( iHeight>0 ){
+    RtreeNode *pChild = nodeHashLookup(pRtree, pCell->iRowid);
+    if( pChild ){
+      nodeRelease(pRtree, pChild->pParent);
+      nodeReference(pNode);
+      pChild->pParent = pNode;
+    }
+  }
+  if( nodeInsertCell(pRtree, pNode, pCell) ){
+#if VARIANT_RSTARTREE_REINSERT
+    if( iHeight<=pRtree->iReinsertHeight || pNode->iNode==1){
+      rc = SplitNode(pRtree, pNode, pCell, iHeight);
+    }else{
+      pRtree->iReinsertHeight = iHeight;
+      rc = Reinsert(pRtree, pNode, pCell, iHeight);
+    }
+#else
+    rc = SplitNode(pRtree, pNode, pCell, iHeight);
+#endif
+  }else{
+    rc = AdjustTree(pRtree, pNode, pCell);
+    if( rc==SQLITE_OK ){
+      if( iHeight==0 ){
+        rc = rowidWrite(pRtree, pCell->iRowid, pNode->iNode);
+      }else{
+        rc = parentWrite(pRtree, pCell->iRowid, pNode->iNode);
+      }
+    }
+  }
+  return rc;
+}
+
+static int reinsertNodeContent(Rtree *pRtree, RtreeNode *pNode){
+  int ii;
+  int rc = SQLITE_OK;
+  int nCell = NCELL(pNode);
+
+  for(ii=0; rc==SQLITE_OK && ii<nCell; ii++){
+    RtreeNode *pInsert;
+    RtreeCell cell;
+    nodeGetCell(pRtree, pNode, ii, &cell);
+
+    /* Find a node to store this cell in. pNode->iNode currently contains
+    ** the height of the sub-tree headed by the cell.
+    */
+    rc = ChooseLeaf(pRtree, &cell, pNode->iNode, &pInsert);
+    if( rc==SQLITE_OK ){
+      int rc2;
+      rc = rtreeInsertCell(pRtree, pInsert, &cell, pNode->iNode);
+      rc2 = nodeRelease(pRtree, pInsert);
+      if( rc==SQLITE_OK ){
+        rc = rc2;
+      }
+    }
+  }
+  return rc;
+}
+
+/*
+** Select a currently unused rowid for a new r-tree record.
+*/
+static int newRowid(Rtree *pRtree, i64 *piRowid){
+  int rc;
+  sqlite3_bind_null(pRtree->pWriteRowid, 1);
+  sqlite3_bind_null(pRtree->pWriteRowid, 2);
+  sqlite3_step(pRtree->pWriteRowid);
+  rc = sqlite3_reset(pRtree->pWriteRowid);
+  *piRowid = sqlite3_last_insert_rowid(pRtree->db);
+  return rc;
+}
+
+/*
+** The xUpdate method for rtree module virtual tables.
+*/
+static int rtreeUpdate(
+  sqlite3_vtab *pVtab, 
+  int nData, 
+  sqlite3_value **azData, 
+  sqlite_int64 *pRowid
+){
+  Rtree *pRtree = (Rtree *)pVtab;
+  int rc = SQLITE_OK;
+
+  rtreeReference(pRtree);
+
+  assert(nData>=1);
+
+  /* If azData[0] is not an SQL NULL value, it is the rowid of a
+  ** record to delete from the r-tree table. The following block does
+  ** just that.
+  */
+  if( sqlite3_value_type(azData[0])!=SQLITE_NULL ){
+    i64 iDelete;                /* The rowid to delete */
+    RtreeNode *pLeaf;           /* Leaf node containing record iDelete */
+    int iCell;                  /* Index of iDelete cell in pLeaf */
+    RtreeNode *pRoot;
+
+    /* Obtain a reference to the root node to initialise Rtree.iDepth */
+    rc = nodeAcquire(pRtree, 1, 0, &pRoot);
+
+    /* Obtain a reference to the leaf node that contains the entry 
+    ** about to be deleted. 
+    */
+    if( rc==SQLITE_OK ){
+      iDelete = sqlite3_value_int64(azData[0]);
+      rc = findLeafNode(pRtree, iDelete, &pLeaf);
+    }
+
+    /* Delete the cell in question from the leaf node. */
+    if( rc==SQLITE_OK ){
+      int rc2;
+      rc = nodeRowidIndex(pRtree, pLeaf, iDelete, &iCell);
+      if( rc==SQLITE_OK ){
+        rc = deleteCell(pRtree, pLeaf, iCell, 0);
+      }
+      rc2 = nodeRelease(pRtree, pLeaf);
+      if( rc==SQLITE_OK ){
+        rc = rc2;
+      }
+    }
+
+    /* Delete the corresponding entry in the <rtree>_rowid table. */
+    if( rc==SQLITE_OK ){
+      sqlite3_bind_int64(pRtree->pDeleteRowid, 1, iDelete);
+      sqlite3_step(pRtree->pDeleteRowid);
+      rc = sqlite3_reset(pRtree->pDeleteRowid);
+    }
+
+    /* Check if the root node now has exactly one child. If so, remove
+    ** it, schedule the contents of the child for reinsertion and 
+    ** reduce the tree height by one.
+    **
+    ** This is equivalent to copying the contents of the child into
+    ** the root node (the operation that Gutman's paper says to perform 
+    ** in this scenario).
+    */
+    if( rc==SQLITE_OK && pRtree->iDepth>0 && NCELL(pRoot)==1 ){
+      int rc2;
+      RtreeNode *pChild;
+      i64 iChild = nodeGetRowid(pRtree, pRoot, 0);
+      rc = nodeAcquire(pRtree, iChild, pRoot, &pChild);
+      if( rc==SQLITE_OK ){
+        rc = removeNode(pRtree, pChild, pRtree->iDepth-1);
+      }
+      rc2 = nodeRelease(pRtree, pChild);
+      if( rc==SQLITE_OK ) rc = rc2;
+      if( rc==SQLITE_OK ){
+        pRtree->iDepth--;
+        writeInt16(pRoot->zData, pRtree->iDepth);
+        pRoot->isDirty = 1;
+      }
+    }
+
+    /* Re-insert the contents of any underfull nodes removed from the tree. */
+    for(pLeaf=pRtree->pDeleted; pLeaf; pLeaf=pRtree->pDeleted){
+      if( rc==SQLITE_OK ){
+        rc = reinsertNodeContent(pRtree, pLeaf);
+      }
+      pRtree->pDeleted = pLeaf->pNext;
+      sqlite3_free(pLeaf);
+    }
+
+    /* Release the reference to the root node. */
+    if( rc==SQLITE_OK ){
+      rc = nodeRelease(pRtree, pRoot);
+    }else{
+      nodeRelease(pRtree, pRoot);
+    }
+  }
+
+  /* If the azData[] array contains more than one element, elements
+  ** (azData[2]..azData[argc-1]) contain a new record to insert into
+  ** the r-tree structure.
+  */
+  if( rc==SQLITE_OK && nData>1 ){
+    /* Insert a new record into the r-tree */
+    RtreeCell cell;
+    int ii;
+    RtreeNode *pLeaf;
+
+    /* Populate the cell.aCoord[] array. The first coordinate is azData[3]. */
+    assert( nData==(pRtree->nDim*2 + 3) );
+    if( pRtree->eCoordType==RTREE_COORD_REAL32 ){
+      for(ii=0; ii<(pRtree->nDim*2); ii+=2){
+        cell.aCoord[ii].f = (float)sqlite3_value_double(azData[ii+3]);
+        cell.aCoord[ii+1].f = (float)sqlite3_value_double(azData[ii+4]);
+        if( cell.aCoord[ii].f>cell.aCoord[ii+1].f ){
+          rc = SQLITE_CONSTRAINT;
+          goto constraint;
+        }
+      }
+    }else{
+      for(ii=0; ii<(pRtree->nDim*2); ii+=2){
+        cell.aCoord[ii].i = sqlite3_value_int(azData[ii+3]);
+        cell.aCoord[ii+1].i = sqlite3_value_int(azData[ii+4]);
+        if( cell.aCoord[ii].i>cell.aCoord[ii+1].i ){
+          rc = SQLITE_CONSTRAINT;
+          goto constraint;
+        }
+      }
+    }
+
+    /* Figure out the rowid of the new row. */
+    if( sqlite3_value_type(azData[2])==SQLITE_NULL ){
+      rc = newRowid(pRtree, &cell.iRowid);
+    }else{
+      cell.iRowid = sqlite3_value_int64(azData[2]);
+      sqlite3_bind_int64(pRtree->pReadRowid, 1, cell.iRowid);
+      if( SQLITE_ROW==sqlite3_step(pRtree->pReadRowid) ){
+        sqlite3_reset(pRtree->pReadRowid);
+        rc = SQLITE_CONSTRAINT;
+        goto constraint;
+      }
+      rc = sqlite3_reset(pRtree->pReadRowid);
+    }
+    *pRowid = cell.iRowid;
+
+    if( rc==SQLITE_OK ){
+      rc = ChooseLeaf(pRtree, &cell, 0, &pLeaf);
+    }
+    if( rc==SQLITE_OK ){
+      int rc2;
+      pRtree->iReinsertHeight = -1;
+      rc = rtreeInsertCell(pRtree, pLeaf, &cell, 0);
+      rc2 = nodeRelease(pRtree, pLeaf);
+      if( rc==SQLITE_OK ){
+        rc = rc2;
+      }
+    }
+  }
+
+constraint:
+  rtreeRelease(pRtree);
+  return rc;
+}
+
+/*
+** The xRename method for rtree module virtual tables.
+*/
+static int rtreeRename(sqlite3_vtab *pVtab, const char *zNewName){
+  Rtree *pRtree = (Rtree *)pVtab;
+  int rc = SQLITE_NOMEM;
+  char *zSql = sqlite3_mprintf(
+    "ALTER TABLE %Q.'%q_node'   RENAME TO \"%w_node\";"
+    "ALTER TABLE %Q.'%q_parent' RENAME TO \"%w_parent\";"
+    "ALTER TABLE %Q.'%q_rowid'  RENAME TO \"%w_rowid\";"
+    , pRtree->zDb, pRtree->zName, zNewName 
+    , pRtree->zDb, pRtree->zName, zNewName 
+    , pRtree->zDb, pRtree->zName, zNewName
+  );
+  if( zSql ){
+    rc = sqlite3_exec(pRtree->db, zSql, 0, 0, 0);
+    sqlite3_free(zSql);
+  }
+  return rc;
+}
+
+static sqlite3_module rtreeModule = {
+  0,                         /* iVersion */
+  rtreeCreate,                /* xCreate - create a table */
+  rtreeConnect,               /* xConnect - connect to an existing table */
+  rtreeBestIndex,             /* xBestIndex - Determine search strategy */
+  rtreeDisconnect,            /* xDisconnect - Disconnect from a table */
+  rtreeDestroy,               /* xDestroy - Drop a table */
+  rtreeOpen,                  /* xOpen - open a cursor */
+  rtreeClose,                 /* xClose - close a cursor */
+  rtreeFilter,                /* xFilter - configure scan constraints */
+  rtreeNext,                  /* xNext - advance a cursor */
+  rtreeEof,                   /* xEof */
+  rtreeColumn,                /* xColumn - read data */
+  rtreeRowid,                 /* xRowid - read data */
+  rtreeUpdate,                /* xUpdate - write data */
+  0,                          /* xBegin - begin transaction */
+  0,                          /* xSync - sync transaction */
+  0,                          /* xCommit - commit transaction */
+  0,                          /* xRollback - rollback transaction */
+  0,                          /* xFindFunction - function overloading */
+  rtreeRename                 /* xRename - rename the table */
+};
+
+static int rtreeSqlInit(
+  Rtree *pRtree, 
+  sqlite3 *db, 
+  const char *zDb, 
+  const char *zPrefix, 
+  int isCreate
+){
+  int rc = SQLITE_OK;
+
+  #define N_STATEMENT 9
+  static const char *azSql[N_STATEMENT] = {
+    /* Read and write the xxx_node table */
+    "SELECT data FROM '%q'.'%q_node' WHERE nodeno = :1",
+    "INSERT OR REPLACE INTO '%q'.'%q_node' VALUES(:1, :2)",
+    "DELETE FROM '%q'.'%q_node' WHERE nodeno = :1",
+
+    /* Read and write the xxx_rowid table */
+    "SELECT nodeno FROM '%q'.'%q_rowid' WHERE rowid = :1",
+    "INSERT OR REPLACE INTO '%q'.'%q_rowid' VALUES(:1, :2)",
+    "DELETE FROM '%q'.'%q_rowid' WHERE rowid = :1",
+
+    /* Read and write the xxx_parent table */
+    "SELECT parentnode FROM '%q'.'%q_parent' WHERE nodeno = :1",
+    "INSERT OR REPLACE INTO '%q'.'%q_parent' VALUES(:1, :2)",
+    "DELETE FROM '%q'.'%q_parent' WHERE nodeno = :1"
+  };
+  sqlite3_stmt **appStmt[N_STATEMENT];
+  int i;
+
+  pRtree->db = db;
+
+  if( isCreate ){
+    char *zCreate = sqlite3_mprintf(
+"CREATE TABLE \"%w\".\"%w_node\"(nodeno INTEGER PRIMARY KEY, data BLOB);"
+"CREATE TABLE \"%w\".\"%w_rowid\"(rowid INTEGER PRIMARY KEY, nodeno INTEGER);"
+"CREATE TABLE \"%w\".\"%w_parent\"(nodeno INTEGER PRIMARY KEY, parentnode INTEGER);"
+"INSERT INTO '%q'.'%q_node' VALUES(1, zeroblob(%d))",
+      zDb, zPrefix, zDb, zPrefix, zDb, zPrefix, zDb, zPrefix, pRtree->iNodeSize
+    );
+    if( !zCreate ){
+      return SQLITE_NOMEM;
+    }
+    rc = sqlite3_exec(db, zCreate, 0, 0, 0);
+    sqlite3_free(zCreate);
+    if( rc!=SQLITE_OK ){
+      return rc;
+    }
+  }
+
+  appStmt[0] = &pRtree->pReadNode;
+  appStmt[1] = &pRtree->pWriteNode;
+  appStmt[2] = &pRtree->pDeleteNode;
+  appStmt[3] = &pRtree->pReadRowid;
+  appStmt[4] = &pRtree->pWriteRowid;
+  appStmt[5] = &pRtree->pDeleteRowid;
+  appStmt[6] = &pRtree->pReadParent;
+  appStmt[7] = &pRtree->pWriteParent;
+  appStmt[8] = &pRtree->pDeleteParent;
+
+  for(i=0; i<N_STATEMENT && rc==SQLITE_OK; i++){
+    char *zSql = sqlite3_mprintf(azSql[i], zDb, zPrefix);
+    if( zSql ){
+      rc = sqlite3_prepare_v2(db, zSql, -1, appStmt[i], 0); 
+    }else{
+      rc = SQLITE_NOMEM;
+    }
+    sqlite3_free(zSql);
+  }
+
+  return rc;
+}
+
+/*
+** The second argument to this function contains the text of an SQL statement
+** that returns a single integer value. The statement is compiled and executed
+** using database connection db. If successful, the integer value returned
+** is written to *piVal and SQLITE_OK returned. Otherwise, an SQLite error
+** code is returned and the value of *piVal after returning is not defined.
+*/
+static int getIntFromStmt(sqlite3 *db, const char *zSql, int *piVal){
+  int rc = SQLITE_NOMEM;
+  if( zSql ){
+    sqlite3_stmt *pStmt = 0;
+    rc = sqlite3_prepare_v2(db, zSql, -1, &pStmt, 0);
+    if( rc==SQLITE_OK ){
+      if( SQLITE_ROW==sqlite3_step(pStmt) ){
+        *piVal = sqlite3_column_int(pStmt, 0);
+      }
+      rc = sqlite3_finalize(pStmt);
+    }
+  }
+  return rc;
+}
+
+/*
+** This function is called from within the xConnect() or xCreate() method to
+** determine the node-size used by the rtree table being created or connected
+** to. If successful, pRtree->iNodeSize is populated and SQLITE_OK returned.
+** Otherwise, an SQLite error code is returned.
+**
+** If this function is being called as part of an xConnect(), then the rtree
+** table already exists. In this case the node-size is determined by inspecting
+** the root node of the tree.
+**
+** Otherwise, for an xCreate(), use 64 bytes less than the database page-size. 
+** This ensures that each node is stored on a single database page. If the 
+** database page-size is so large that more than RTREE_MAXCELLS entries 
+** would fit in a single node, use a smaller node-size.
+*/
+static int getNodeSize(
+  sqlite3 *db,                    /* Database handle */
+  Rtree *pRtree,                  /* Rtree handle */
+  int isCreate                    /* True for xCreate, false for xConnect */
+){
+  int rc;
+  char *zSql;
+  if( isCreate ){
+    int iPageSize;
+    zSql = sqlite3_mprintf("PRAGMA %Q.page_size", pRtree->zDb);
+    rc = getIntFromStmt(db, zSql, &iPageSize);
+    if( rc==SQLITE_OK ){
+      pRtree->iNodeSize = iPageSize-64;
+      if( (4+pRtree->nBytesPerCell*RTREE_MAXCELLS)<pRtree->iNodeSize ){
+        pRtree->iNodeSize = 4+pRtree->nBytesPerCell*RTREE_MAXCELLS;
+      }
+    }
+  }else{
+    zSql = sqlite3_mprintf(
+        "SELECT length(data) FROM '%q'.'%q_node' WHERE nodeno = 1",
+        pRtree->zDb, pRtree->zName
+    );
+    rc = getIntFromStmt(db, zSql, &pRtree->iNodeSize);
+  }
+
+  sqlite3_free(zSql);
+  return rc;
+}
+
+/* 
+** This function is the implementation of both the xConnect and xCreate
+** methods of the r-tree virtual table.
+**
+**   argv[0]   -> module name
+**   argv[1]   -> database name
+**   argv[2]   -> table name
+**   argv[...] -> column names...
+*/
+static int rtreeInit(
+  sqlite3 *db,                        /* Database connection */
+  void *pAux,                         /* One of the RTREE_COORD_* constants */
+  int argc, const char *const*argv,   /* Parameters to CREATE TABLE statement */
+  sqlite3_vtab **ppVtab,              /* OUT: New virtual table */
+  char **pzErr,                       /* OUT: Error message, if any */
+  int isCreate                        /* True for xCreate, false for xConnect */
+){
+  int rc = SQLITE_OK;
+  Rtree *pRtree;
+  int nDb;              /* Length of string argv[1] */
+  int nName;            /* Length of string argv[2] */
+  int eCoordType = (pAux ? RTREE_COORD_INT32 : RTREE_COORD_REAL32);
+
+  const char *aErrMsg[] = {
+    0,                                                    /* 0 */
+    "Wrong number of columns for an rtree table",         /* 1 */
+    "Too few columns for an rtree table",                 /* 2 */
+    "Too many columns for an rtree table"                 /* 3 */
+  };
+
+  int iErr = (argc<6) ? 2 : argc>(RTREE_MAX_DIMENSIONS*2+4) ? 3 : argc%2;
+  if( aErrMsg[iErr] ){
+    *pzErr = sqlite3_mprintf("%s", aErrMsg[iErr]);
+    return SQLITE_ERROR;
+  }
+
+  /* Allocate the sqlite3_vtab structure */
+  nDb = strlen(argv[1]);
+  nName = strlen(argv[2]);
+  pRtree = (Rtree *)sqlite3_malloc(sizeof(Rtree)+nDb+nName+2);
+  if( !pRtree ){
+    return SQLITE_NOMEM;
+  }
+  memset(pRtree, 0, sizeof(Rtree)+nDb+nName+2);
+  pRtree->nBusy = 1;
+  pRtree->base.pModule = &rtreeModule;
+  pRtree->zDb = (char *)&pRtree[1];
+  pRtree->zName = &pRtree->zDb[nDb+1];
+  pRtree->nDim = (argc-4)/2;
+  pRtree->nBytesPerCell = 8 + pRtree->nDim*4*2;
+  pRtree->eCoordType = eCoordType;
+  memcpy(pRtree->zDb, argv[1], nDb);
+  memcpy(pRtree->zName, argv[2], nName);
+
+  /* Figure out the node size to use. */
+  rc = getNodeSize(db, pRtree, isCreate);
+
+  /* Create/Connect to the underlying relational database schema. If
+  ** that is successful, call sqlite3_declare_vtab() to configure
+  ** the r-tree table schema.
+  */
+  if( rc==SQLITE_OK ){
+    if( (rc = rtreeSqlInit(pRtree, db, argv[1], argv[2], isCreate)) ){
+      *pzErr = sqlite3_mprintf("%s", sqlite3_errmsg(db));
+    }else{
+      char *zSql = sqlite3_mprintf("CREATE TABLE x(%s", argv[3]);
+      char *zTmp;
+      int ii;
+      for(ii=4; zSql && ii<argc; ii++){
+        zTmp = zSql;
+        zSql = sqlite3_mprintf("%s, %s", zTmp, argv[ii]);
+        sqlite3_free(zTmp);
+      }
+      if( zSql ){
+        zTmp = zSql;
+        zSql = sqlite3_mprintf("%s);", zTmp);
+        sqlite3_free(zTmp);
+      }
+      if( !zSql ){
+        rc = SQLITE_NOMEM;
+      }else if( SQLITE_OK!=(rc = sqlite3_declare_vtab(db, zSql)) ){
+        *pzErr = sqlite3_mprintf("%s", sqlite3_errmsg(db));
+      }
+      sqlite3_free(zSql);
+    }
+  }
+
+  if( rc==SQLITE_OK ){
+    *ppVtab = (sqlite3_vtab *)pRtree;
+  }else{
+    rtreeRelease(pRtree);
+  }
+  return rc;
+}
+
+
+/*
+** Implementation of a scalar function that decodes r-tree nodes to
+** human readable strings. This can be used for debugging and analysis.
+**
+** The scalar function takes two arguments, a blob of data containing
+** an r-tree node, and the number of dimensions the r-tree indexes.
+** For a two-dimensional r-tree structure called "rt", to deserialize
+** all nodes, a statement like:
+**
+**   SELECT rtreenode(2, data) FROM rt_node;
+**
+** The human readable string takes the form of a Tcl list with one
+** entry for each cell in the r-tree node. Each entry is itself a
+** list, containing the 8-byte rowid/pageno followed by the 
+** <num-dimension>*2 coordinates.
+*/
+static void rtreenode(sqlite3_context *ctx, int nArg, sqlite3_value **apArg){
+  char *zText = 0;
+  RtreeNode node;
+  Rtree tree;
+  int ii;
+
+  UNUSED_PARAMETER(nArg);
+  memset(&node, 0, sizeof(RtreeNode));
+  memset(&tree, 0, sizeof(Rtree));
+  tree.nDim = sqlite3_value_int(apArg[0]);
+  tree.nBytesPerCell = 8 + 8 * tree.nDim;
+  node.zData = (u8 *)sqlite3_value_blob(apArg[1]);
+
+  for(ii=0; ii<NCELL(&node); ii++){
+    char zCell[512];
+    int nCell = 0;
+    RtreeCell cell;
+    int jj;
+
+    nodeGetCell(&tree, &node, ii, &cell);
+    sqlite3_snprintf(512-nCell,&zCell[nCell],"%lld", cell.iRowid);
+    nCell = strlen(zCell);
+    for(jj=0; jj<tree.nDim*2; jj++){
+      sqlite3_snprintf(512-nCell,&zCell[nCell]," %f",(double)cell.aCoord[jj].f);
+      nCell = strlen(zCell);
+    }
+
+    if( zText ){
+      char *zTextNew = sqlite3_mprintf("%s {%s}", zText, zCell);
+      sqlite3_free(zText);
+      zText = zTextNew;
+    }else{
+      zText = sqlite3_mprintf("{%s}", zCell);
+    }
+  }
+  
+  sqlite3_result_text(ctx, zText, -1, sqlite3_free);
+}
+
+static void rtreedepth(sqlite3_context *ctx, int nArg, sqlite3_value **apArg){
+  UNUSED_PARAMETER(nArg);
+  if( sqlite3_value_type(apArg[0])!=SQLITE_BLOB 
+   || sqlite3_value_bytes(apArg[0])<2
+  ){
+    sqlite3_result_error(ctx, "Invalid argument to rtreedepth()", -1); 
+  }else{
+    u8 *zBlob = (u8 *)sqlite3_value_blob(apArg[0]);
+    sqlite3_result_int(ctx, readInt16(zBlob));
+  }
+}
+
+/*
+** Register the r-tree module with database handle db. This creates the
+** virtual table module "rtree" and the debugging/analysis scalar 
+** function "rtreenode".
+*/
+int sqlite3RtreeInit(sqlite3 *db){
+  const int utf8 = SQLITE_UTF8;
+  int rc;
+
+  rc = sqlite3_create_function(db, "rtreenode", 2, utf8, 0, rtreenode, 0, 0);
+  if( rc==SQLITE_OK ){
+    rc = sqlite3_create_function(db, "rtreedepth", 1, utf8, 0,rtreedepth, 0, 0);
+  }
+  if( rc==SQLITE_OK ){
+    void *c = (void *)RTREE_COORD_REAL32;
+    rc = sqlite3_create_module_v2(db, "rtree", &rtreeModule, c, 0);
+  }
+  if( rc==SQLITE_OK ){
+    void *c = (void *)RTREE_COORD_INT32;
+    rc = sqlite3_create_module_v2(db, "rtree_i32", &rtreeModule, c, 0);
+  }
+
+  return rc;
+}
+
+/*
+** A version of sqlite3_free() that can be used as a callback. This is used
+** in two places - as the destructor for the blob value returned by the
+** invocation of a geometry function, and as the destructor for the geometry
+** functions themselves.
+*/
+static void doSqlite3Free(void *p){
+  sqlite3_free(p);
+}
+
+/*
+** Each call to sqlite3_rtree_geometry_callback() creates an ordinary SQLite
+** scalar user function. This C function is the callback used for all such
+** registered SQL functions.
+**
+** The scalar user functions return a blob that is interpreted by r-tree
+** table MATCH operators.
+*/
+static void geomCallback(sqlite3_context *ctx, int nArg, sqlite3_value **aArg){
+  RtreeGeomCallback *pGeomCtx = (RtreeGeomCallback *)sqlite3_user_data(ctx);
+  RtreeMatchArg *pBlob;
+  int nBlob;
+
+  nBlob = sizeof(RtreeMatchArg) + (nArg-1)*sizeof(double);
+  pBlob = (RtreeMatchArg *)sqlite3_malloc(nBlob);
+  if( !pBlob ){
+    sqlite3_result_error_nomem(ctx);
+  }else{
+    int i;
+    pBlob->magic = RTREE_GEOMETRY_MAGIC;
+    pBlob->xGeom = pGeomCtx->xGeom;
+    pBlob->pContext = pGeomCtx->pContext;
+    pBlob->nParam = nArg;
+    for(i=0; i<nArg; i++){
+      pBlob->aParam[i] = sqlite3_value_double(aArg[i]);
+    }
+    sqlite3_result_blob(ctx, pBlob, nBlob, doSqlite3Free);
+  }
+}
+
+/*
+** Register a new geometry function for use with the r-tree MATCH operator.
+*/
+int sqlite3_rtree_geometry_callback(
+  sqlite3 *db,
+  const char *zGeom,
+  int (*xGeom)(sqlite3_rtree_geometry *, int, double *, int *),
+  void *pContext
+){
+  RtreeGeomCallback *pGeomCtx;      /* Context object for new user-function */
+
+  /* Allocate and populate the context object. */
+  pGeomCtx = (RtreeGeomCallback *)sqlite3_malloc(sizeof(RtreeGeomCallback));
+  if( !pGeomCtx ) return SQLITE_NOMEM;
+  pGeomCtx->xGeom = xGeom;
+  pGeomCtx->pContext = pContext;
+
+  /* Create the new user-function. Register a destructor function to delete
+  ** the context object when it is no longer required.  */
+  return sqlite3_create_function_v2(db, zGeom, -1, SQLITE_ANY, 
+      (void *)pGeomCtx, geomCallback, 0, 0, doSqlite3Free
+  );
+}
+
+#if !SQLITE_CORE
+int sqlite3_extension_init(
+  sqlite3 *db,
+  char **pzErrMsg,
+  const sqlite3_api_routines *pApi
+){
+  SQLITE_EXTENSION_INIT2(pApi)
+  return sqlite3RtreeInit(db);
+}
+#endif
+
+#endif