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