//----------------------------------------------------------------------
// File: kd_split.cpp
// Programmer: Sunil Arya and David Mount
// Description: Methods for splitting kd-trees
// Last modified: 01/04/05 (Version 1.0)
//----------------------------------------------------------------------
// Copyright (c) 1997-2005 University of Maryland and Sunil Arya and
// David Mount. All Rights Reserved.
//
// This software and related documentation is part of the Approximate
// Nearest Neighbor Library (ANN). This software is provided under
// the provisions of the Lesser GNU Public License (LGPL). See the
// file ../ReadMe.txt for further information.
//
// The University of Maryland (U.M.) and the authors make no
// representations about the suitability or fitness of this software for
// any purpose. It is provided "as is" without express or implied
// warranty.
//----------------------------------------------------------------------
// History:
// Revision 0.1 03/04/98
// Initial release
// Revision 1.0 04/01/05
//----------------------------------------------------------------------
#include "kd_tree.h" // kd-tree definitions
#include "kd_util.h" // kd-tree utilities
#include "kd_split.h" // splitting functions
//----------------------------------------------------------------------
// Constants
//----------------------------------------------------------------------
const double ERR = 0.001; // a small value
const double FS_ASPECT_RATIO = 3.0; // maximum allowed aspect ratio
// in fair split. Must be >= 2.
//----------------------------------------------------------------------
// kd_split - Bentley's standard splitting routine for kd-trees
// Find the dimension of the greatest spread, and split
// just before the median point along this dimension.
//----------------------------------------------------------------------
void kd_split(
ANNpointArray pa, // point array (permuted on return)
ANNidxArray pidx, // point indices
const ANNorthRect &bnds, // bounding rectangle for cell
int n, // number of points
int dim, // dimension of space
int &cut_dim, // cutting dimension (returned)
ANNcoord &cut_val, // cutting value (returned)
int &n_lo) // num of points on low side (returned)
{
// find dimension of maximum spread
cut_dim = annMaxSpread(pa, pidx, n, dim);
n_lo = n/2; // median rank
// split about median
annMedianSplit(pa, pidx, n, cut_dim, cut_val, n_lo);
}
//----------------------------------------------------------------------
// midpt_split - midpoint splitting rule for box-decomposition trees
//
// This is the simplest splitting rule that guarantees boxes
// of bounded aspect ratio. It simply cuts the box with the
// longest side through its midpoint. If there are ties, it
// selects the dimension with the maximum point spread.
//
// WARNING: This routine (while simple) doesn't seem to work
// well in practice in high dimensions, because it tends to
// generate a large number of trivial and/or unbalanced splits.
// Either kd_split(), sl_midpt_split(), or fair_split() are
// recommended, instead.
//----------------------------------------------------------------------
void midpt_split(
ANNpointArray pa, // point array
ANNidxArray pidx, // point indices (permuted on return)
const ANNorthRect &bnds, // bounding rectangle for cell
int n, // number of points
int dim, // dimension of space
int &cut_dim, // cutting dimension (returned)
ANNcoord &cut_val, // cutting value (returned)
int &n_lo) // num of points on low side (returned)
{
int d;
ANNcoord max_length = bnds.hi[0] - bnds.lo[0];
for (d = 1; d < dim; d++) { // find length of longest box side
ANNcoord length = bnds.hi[d] - bnds.lo[d];
if (length > max_length) {
max_length = length;
}
}
ANNcoord max_spread = -1; // find long side with most spread
for (d = 0; d < dim; d++) {
// is it among longest?
if (double(bnds.hi[d] - bnds.lo[d]) >= (1-ERR)*max_length) {
// compute its spread
ANNcoord spr = annSpread(pa, pidx, n, d);
if (spr > max_spread) { // is it max so far?
max_spread = spr;
cut_dim = d;
}
}
}
// split along cut_dim at midpoint
cut_val = (bnds.lo[cut_dim] + bnds.hi[cut_dim]) / 2;
// permute points accordingly
int br1, br2;
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
//------------------------------------------------------------------
// On return: pa[0..br1-1] < cut_val
// pa[br1..br2-1] == cut_val
// pa[br2..n-1] > cut_val
//
// We can set n_lo to any value in the range [br1..br2].
// We choose split so that points are most evenly divided.
//------------------------------------------------------------------
if (br1 > n/2) n_lo = br1;
else if (br2 < n/2) n_lo = br2;
else n_lo = n/2;
}
//----------------------------------------------------------------------
// sl_midpt_split - sliding midpoint splitting rule
//
// This is a modification of midpt_split, which has the nonsensical
// name "sliding midpoint". The idea is that we try to use the
// midpoint rule, by bisecting the longest side. If there are
// ties, the dimension with the maximum spread is selected. If,
// however, the midpoint split produces a trivial split (no points
// on one side of the splitting plane) then we slide the splitting
// (maintaining its orientation) until it produces a nontrivial
// split. For example, if the splitting plane is along the x-axis,
// and all the data points have x-coordinate less than the x-bisector,
// then the split is taken along the maximum x-coordinate of the
// data points.
//
// Intuitively, this rule cannot generate trivial splits, and
// hence avoids midpt_split's tendency to produce trees with
// a very large number of nodes.
//
//----------------------------------------------------------------------
void sl_midpt_split(
ANNpointArray pa, // point array
ANNidxArray pidx, // point indices (permuted on return)
const ANNorthRect &bnds, // bounding rectangle for cell
int n, // number of points
int dim, // dimension of space
int &cut_dim, // cutting dimension (returned)
ANNcoord &cut_val, // cutting value (returned)
int &n_lo) // num of points on low side (returned)
{
int d;
ANNcoord max_length = bnds.hi[0] - bnds.lo[0];
for (d = 1; d < dim; d++) { // find length of longest box side
ANNcoord length = bnds.hi[d] - bnds.lo[d];
if (length > max_length) {
max_length = length;
}
}
ANNcoord max_spread = -1; // find long side with most spread
for (d = 0; d < dim; d++) {
// is it among longest?
if ((bnds.hi[d] - bnds.lo[d]) >= (1-ERR)*max_length) {
// compute its spread
ANNcoord spr = annSpread(pa, pidx, n, d);
if (spr > max_spread) { // is it max so far?
max_spread = spr;
cut_dim = d;
}
}
}
// ideal split at midpoint
ANNcoord ideal_cut_val = (bnds.lo[cut_dim] + bnds.hi[cut_dim])/2;
ANNcoord min, max;
annMinMax(pa, pidx, n, cut_dim, min, max); // find min/max coordinates
if (ideal_cut_val < min) // slide to min or max as needed
cut_val = min;
else if (ideal_cut_val > max)
cut_val = max;
else
cut_val = ideal_cut_val;
// permute points accordingly
int br1, br2;
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
//------------------------------------------------------------------
// On return: pa[0..br1-1] < cut_val
// pa[br1..br2-1] == cut_val
// pa[br2..n-1] > cut_val
//
// We can set n_lo to any value in the range [br1..br2] to satisfy
// the exit conditions of the procedure.
//
// if ideal_cut_val < min (implying br2 >= 1),
// then we select n_lo = 1 (so there is one point on left) and
// if ideal_cut_val > max (implying br1 <= n-1),
// then we select n_lo = n-1 (so there is one point on right).
// Otherwise, we select n_lo as close to n/2 as possible within
// [br1..br2].
//------------------------------------------------------------------
if (ideal_cut_val < min) n_lo = 1;
else if (ideal_cut_val > max) n_lo = n-1;
else if (br1 > n/2) n_lo = br1;
else if (br2 < n/2) n_lo = br2;
else n_lo = n/2;
}
//----------------------------------------------------------------------
// fair_split - fair-split splitting rule
//
// This is a compromise between the kd-tree splitting rule (which
// always splits data points at their median) and the midpoint
// splitting rule (which always splits a box through its center.
// The goal of this procedure is to achieve both nicely balanced
// splits, and boxes of bounded aspect ratio.
//
// A constant FS_ASPECT_RATIO is defined. Given a box, those sides
// which can be split so that the ratio of the longest to shortest
// side does not exceed ASPECT_RATIO are identified. Among these
// sides, we select the one in which the points have the largest
// spread. We then split the points in a manner which most evenly
// distributes the points on either side of the splitting plane,
// subject to maintaining the bound on the ratio of long to short
// sides. To determine that the aspect ratio will be preserved,
// we determine the longest side (other than this side), and
// determine how narrowly we can cut this side, without causing the
// aspect ratio bound to be exceeded (small_piece).
//
// This procedure is more robust than either kd_split or midpt_split,
// but is more complicated as well. When point distribution is
// extremely skewed, this degenerates to midpt_split (actually
// 1/3 point split), and when the points are most evenly distributed,
// this degenerates to kd-split.
//----------------------------------------------------------------------
void fair_split(
ANNpointArray pa, // point array
ANNidxArray pidx, // point indices (permuted on return)
const ANNorthRect &bnds, // bounding rectangle for cell
int n, // number of points
int dim, // dimension of space
int &cut_dim, // cutting dimension (returned)
ANNcoord &cut_val, // cutting value (returned)
int &n_lo) // num of points on low side (returned)
{
int d;
ANNcoord max_length = bnds.hi[0] - bnds.lo[0];
cut_dim = 0;
for (d = 1; d < dim; d++) { // find length of longest box side
ANNcoord length = bnds.hi[d] - bnds.lo[d];
if (length > max_length) {
max_length = length;
cut_dim = d;
}
}
ANNcoord max_spread = 0; // find legal cut with max spread
cut_dim = 0;
for (d = 0; d < dim; d++) {
ANNcoord length = bnds.hi[d] - bnds.lo[d];
// is this side midpoint splitable
// without violating aspect ratio?
if (((double) max_length)*2.0/((double) length) <= FS_ASPECT_RATIO) {
// compute spread along this dim
ANNcoord spr = annSpread(pa, pidx, n, d);
if (spr > max_spread) { // best spread so far
max_spread = spr;
cut_dim = d; // this is dimension to cut
}
}
}
max_length = 0; // find longest side other than cut_dim
for (d = 0; d < dim; d++) {
ANNcoord length = bnds.hi[d] - bnds.lo[d];
if (d != cut_dim && length > max_length)
max_length = length;
}
// consider most extreme splits
ANNcoord small_piece = max_length / FS_ASPECT_RATIO;
ANNcoord lo_cut = bnds.lo[cut_dim] + small_piece;// lowest legal cut
ANNcoord hi_cut = bnds.hi[cut_dim] - small_piece;// highest legal cut
int br1, br2;
// is median below lo_cut ?
if (annSplitBalance(pa, pidx, n, cut_dim, lo_cut) >= 0) {
cut_val = lo_cut; // cut at lo_cut
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
n_lo = br1;
}
// is median above hi_cut?
else if (annSplitBalance(pa, pidx, n, cut_dim, hi_cut) <= 0) {
cut_val = hi_cut; // cut at hi_cut
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
n_lo = br2;
}
else { // median cut preserves asp ratio
n_lo = n/2; // split about median
annMedianSplit(pa, pidx, n, cut_dim, cut_val, n_lo);
}
}
//----------------------------------------------------------------------
// sl_fair_split - sliding fair split splitting rule
//
// Sliding fair split is a splitting rule that combines the
// strengths of both fair split with sliding midpoint split.
// Fair split tends to produce balanced splits when the points
// are roughly uniformly distributed, but it can produce many
// trivial splits when points are highly clustered. Sliding
// midpoint never produces trivial splits, and shrinks boxes
// nicely if points are highly clustered, but it may produce
// rather unbalanced splits when points are unclustered but not
// quite uniform.
//
// Sliding fair split is based on the theory that there are two
// types of splits that are "good": balanced splits that produce
// fat boxes, and unbalanced splits provided the cell with fewer
// points is fat.
//
// This splitting rule operates by first computing the longest
// side of the current bounding box. Then it asks which sides
// could be split (at the midpoint) and still satisfy the aspect
// ratio bound with respect to this side. Among these, it selects
// the side with the largest spread (as fair split would). It
// then considers the most extreme cuts that would be allowed by
// the aspect ratio bound. This is done by dividing the longest
// side of the box by the aspect ratio bound. If the median cut
// lies between these extreme cuts, then we use the median cut.
// If not, then consider the extreme cut that is closer to the
// median. If all the points lie to one side of this cut, then
// we slide the cut until it hits the first point. This may
// violate the aspect ratio bound, but will never generate empty
// cells. However the sibling of every such skinny cell is fat,
// and hence packing arguments still apply.
//
//----------------------------------------------------------------------
void sl_fair_split(
ANNpointArray pa, // point array
ANNidxArray pidx, // point indices (permuted on return)
const ANNorthRect &bnds, // bounding rectangle for cell
int n, // number of points
int dim, // dimension of space
int &cut_dim, // cutting dimension (returned)
ANNcoord &cut_val, // cutting value (returned)
int &n_lo) // num of points on low side (returned)
{
int d;
ANNcoord min, max; // min/max coordinates
int br1, br2; // split break points
ANNcoord max_length = bnds.hi[0] - bnds.lo[0];
cut_dim = 0;
for (d = 1; d < dim; d++) { // find length of longest box side
ANNcoord length = bnds.hi[d] - bnds.lo[d];
if (length > max_length) {
max_length = length;
cut_dim = d;
}
}
ANNcoord max_spread = 0; // find legal cut with max spread
cut_dim = 0;
for (d = 0; d < dim; d++) {
ANNcoord length = bnds.hi[d] - bnds.lo[d];
// is this side midpoint splitable
// without violating aspect ratio?
if (((double) max_length)*2.0/((double) length) <= FS_ASPECT_RATIO) {
// compute spread along this dim
ANNcoord spr = annSpread(pa, pidx, n, d);
if (spr > max_spread) { // best spread so far
max_spread = spr;
cut_dim = d; // this is dimension to cut
}
}
}
max_length = 0; // find longest side other than cut_dim
for (d = 0; d < dim; d++) {
ANNcoord length = bnds.hi[d] - bnds.lo[d];
if (d != cut_dim && length > max_length)
max_length = length;
}
// consider most extreme splits
ANNcoord small_piece = max_length / FS_ASPECT_RATIO;
ANNcoord lo_cut = bnds.lo[cut_dim] + small_piece;// lowest legal cut
ANNcoord hi_cut = bnds.hi[cut_dim] - small_piece;// highest legal cut
// find min and max along cut_dim
annMinMax(pa, pidx, n, cut_dim, min, max);
// is median below lo_cut?
if (annSplitBalance(pa, pidx, n, cut_dim, lo_cut) >= 0) {
if (max > lo_cut) { // are any points above lo_cut?
cut_val = lo_cut; // cut at lo_cut
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
n_lo = br1; // balance if there are ties
}
else { // all points below lo_cut
cut_val = max; // cut at max value
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
n_lo = n-1;
}
}
// is median above hi_cut?
else if (annSplitBalance(pa, pidx, n, cut_dim, hi_cut) <= 0) {
if (min < hi_cut) { // are any points below hi_cut?
cut_val = hi_cut; // cut at hi_cut
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
n_lo = br2; // balance if there are ties
}
else { // all points above hi_cut
cut_val = min; // cut at min value
annPlaneSplit(pa, pidx, n, cut_dim, cut_val, br1, br2);
n_lo = 1;
}
}
else { // median cut is good enough
n_lo = n/2; // split about median
annMedianSplit(pa, pidx, n, cut_dim, cut_val, n_lo);
}
}