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diff --git a/geom_bottleneck/bottleneck/src/ann/kd_split.cpp b/geom_bottleneck/bottleneck/src/ann/kd_split.cpp new file mode 100644 index 0000000..7979aaa --- /dev/null +++ b/geom_bottleneck/bottleneck/src/ann/kd_split.cpp @@ -0,0 +1,632 @@ +//---------------------------------------------------------------------- +// 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 + +namespace geom_bt { +//---------------------------------------------------------------------- +// 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. + +//---------------------------------------------------------------------- +// NOTE: Virtually all point indexing is done through an index (i.e. +// permutation) array pidx. Consequently, a reference to the d-th +// coordinate of the i-th point is pa[pidx[i]][d]. The macro PA(i,d) +// is a shorthand for this. +//---------------------------------------------------------------------- + // standard 2-d indirect indexing +#define PA(i,d) (pa[pidx[(i)]][(d)]) + // accessing a single point +#define PP(i) (pa[pidx[(i)]]) + + +//---------------------------------------------------------------------- +// 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); + } +} + + +///////////////////////////////////////////////////////////////////////////////// +// for kd-trees with deletion +// +//---------------------------------------------------------------------- +// 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_wd( + 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) + int &cut_pt_idx) // index of cutting point (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); + cut_pt_idx = n_lo; + cut_val = PA(cut_pt_idx, cut_dim); +} + +//---------------------------------------------------------------------- +// 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_wd( + 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 &cut_pt_idx) // index of cutting point (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; + + cut_pt_idx = n_lo; + cut_val = PA(cut_pt_idx, cut_dim); + +} + +//---------------------------------------------------------------------- +// 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_wd( + 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 &cut_pt_idx) // index of cutting point (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; +} +} |