K-d tree

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Task
K-d tree
You are encouraged to solve this task according to the task description, using any language you may know.
This page uses content from Wikipedia. The original article was at K-d tree. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

A k-d tree (short for k-dimensional tree) is a space-partitioning data structure for organizing points in a k-dimensional space. k-d trees are a useful data structure for several applications, such as searches involving a multidimensional search key (e.g. range searches and nearest neighbor searches). k-d trees are a special case of binary space partitioning trees.

k-d trees are not suitable, however, for efficiently finding the nearest neighbor in high dimensional spaces. As a general rule, if the dimensionality is k, the number of points in the data, N, should be N ≫ 2k. Otherwise, when k-d trees are used with high-dimensional data, most of the points in the tree will be evaluated and the efficiency is no better than exhaustive search, and other methods such as approximate nearest-neighbor are used instead.

Task: Construct a k-d tree and perform a nearest neighbor search for two example data sets:

  1. The Wikipedia example data of [(2,3), (5,4), (9,6), (4,7), (8,1), (7,2)].
  2. 1000 3-d points uniformly distributed in a 3-d cube.

For the Wikipedia example, find the nearest neighbor to point (9, 2) For the random data, pick a random location and find the nearest neighbor.

In addition, instrument your code to count the number of nodes visited in the nearest neighbor search. Count a node as visited if any field of it is accessed.

Output should show the point searched for, the point found, the distance to the point, and the number of nodes visited.

There are variant algorithms for constructing the tree. You can use a simple median strategy or implement something more efficient. Variants of the nearest neighbor search include nearest N neighbors, approximate nearest neighbor, and range searches. You do not have to implement these. The requirement for this task is specifically the nearest single neighbor. Also there are algorithms for inserting, deleting, and balancing k-d trees. These are also not required for the task.

Contents

[edit] C

Using a Quickselectesque median algorithm. Compared to unbalanced trees (random insertion), it takes slightly longer (maybe half a second or so) to construct a million-node tree, though average look up visits about 1/3 fewer nodes.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <time.h>
 
#define MAX_DIM 3
struct kd_node_t{
double x[MAX_DIM];
struct kd_node_t *left, *right;
};
 
inline double
dist(struct kd_node_t *a, struct kd_node_t *b, int dim)
{
double t, d = 0;
while (dim--) {
t = a->x[dim] - b->x[dim];
d += t * t;
}
return d;
}
 
/* see quickselect method */
struct kd_node_t*
find_median(struct kd_node_t *start, struct kd_node_t *end, int idx)
{
if (end <= start) return NULL;
if (end == start + 1)
return start;
 
inline void swap(struct kd_node_t *x, struct kd_node_t *y) {
double tmp[MAX_DIM];
memcpy(tmp, x->x, sizeof(tmp));
memcpy(x->x, y->x, sizeof(tmp));
memcpy(y->x, tmp, sizeof(tmp));
}
 
struct kd_node_t *p, *store, *md = start + (end - start) / 2;
double pivot;
while (1) {
pivot = md->x[idx];
 
swap(md, end - 1);
for (store = p = start; p < end; p++) {
if (p->x[idx] < pivot) {
if (p != store)
swap(p, store);
store++;
}
}
swap(store, end - 1);
 
/* median has duplicate values */
if (store->x[idx] == md->x[idx])
return md;
 
if (store > md) end = store;
else start = store;
}
}
 
struct kd_node_t*
make_tree(struct kd_node_t *t, int len, int i, int dim)
{
struct kd_node_t *n;
 
if (!len) return 0;
 
if ((n = find_median(t, t + len, i))) {
i = (i + 1) % dim;
n->left = make_tree(t, n - t, i, dim);
n->right = make_tree(n + 1, t + len - (n + 1), i, dim);
}
return n;
}
 
/* global variable, so sue me */
int visited;
 
void nearest(struct kd_node_t *root, struct kd_node_t *nd, int i, int dim,
struct kd_node_t **best, double *best_dist)
{
double d, dx, dx2;
 
if (!root) return;
d = dist(root, nd, dim);
dx = root->x[i] - nd->x[i];
dx2 = dx * dx;
 
visited ++;
 
if (!*best || d < *best_dist) {
*best_dist = d;
*best = root;
}
 
/* if chance of exact match is high */
if (!*best_dist) return;
 
if (++i >= dim) i = 0;
 
nearest(dx > 0 ? root->left : root->right, nd, i, dim, best, best_dist);
if (dx2 >= *best_dist) return;
nearest(dx > 0 ? root->right : root->left, nd, i, dim, best, best_dist);
}
 
#define N 1000000
#define rand1() (rand() / (double)RAND_MAX)
#define rand_pt(v) { v.x[0] = rand1(); v.x[1] = rand1(); v.x[2] = rand1(); }
int main(void)
{
int i;
struct kd_node_t wp[] = {
{{2, 3}}, {{5, 4}}, {{9, 6}}, {{4, 7}}, {{8, 1}}, {{7, 2}}
};
struct kd_node_t this = {{9, 2}};
struct kd_node_t *root, *found, *million;
double best_dist;
 
root = make_tree(wp, sizeof(wp) / sizeof(wp[1]), 0, 2);
 
visited = 0;
found = 0;
nearest(root, &this, 0, 2, &found, &best_dist);
 
printf(">> WP tree\nsearching for (%g, %g)\n"
"found (%g, %g) dist %g\nseen %d nodes\n\n",
this.x[0], this.x[1],
found->x[0], found->x[1], sqrt(best_dist), visited);
 
million = calloc(N, sizeof(struct kd_node_t));
srand(time(0));
for (i = 0; i < N; i++) rand_pt(million[i]);
 
root = make_tree(million, N, 0, 3);
rand_pt(this);
 
visited = 0;
found = 0;
nearest(root, &this, 0, 3, &found, &best_dist);
 
printf(">> Million tree\nsearching for (%g, %g, %g)\n"
"found (%g, %g, %g) dist %g\nseen %d nodes\n",
this.x[0], this.x[1], this.x[2],
found->x[0], found->x[1], found->x[2],
sqrt(best_dist), visited);
 
/* search many random points in million tree to see average behavior.
tree size vs avg nodes visited:
10 ~ 7
100 ~ 16.5
1000 ~ 25.5
10000 ~ 32.8
100000 ~ 38.3
1000000 ~ 42.6
10000000 ~ 46.7 */

int sum = 0, test_runs = 100000;
for (i = 0; i < test_runs; i++) {
found = 0;
visited = 0;
rand_pt(this);
nearest(root, &this, 0, 3, &found, &best_dist);
sum += visited;
}
printf("\n>> Million tree\n"
"visited %d nodes for %d random findings (%f per lookup)\n",
sum, test_runs, sum/(double)test_runs);
 
// free(million);
 
return 0;
}
output
>> WP tree

searching for (9, 2) found (8, 1) dist 1.41421 seen 3 nodes

>> Million tree searching for (0.29514, 0.897237, 0.941998) found (0.296093, 0.896173, 0.948082) dist 0.00624896 seen 44 nodes

>> Million tree visited 4271442 nodes for 100000 random findings (42.714420 per lookup)

[edit] D

Translation of: Go

Points are values, the code is templated on the the dimensionality of the points and the floating point type of the coordinate. Instead of sorting it uses the faster topN, that partitions the points array in two halves around their median.

// Implmentation following pseudocode from
// "An introductory tutorial on kd-trees" by Andrew W. Moore,
// Carnegie Mellon University, PDF accessed from:
// http://www.autonlab.org/autonweb/14665
 
import std.typecons, std.math, std.algorithm, std.random, std.range,
std.traits, core.memory;
 
/// k-dimensional point.
struct Point(size_t k, F) if (isFloatingPoint!F) {
F[k] data;
alias data this; // Kills DMD std.algorithm.swap inlining.
// Define opIndexAssign and opIndex for dmd.
enum size_t length = k;
 
/// Square of the euclidean distance.
double sqd(in ref Point!(k, F) q) const pure nothrow @nogc {
double sum = 0;
foreach (immutable dim, immutable pCoord; data)
sum += (pCoord - q[dim]) ^^ 2;
return sum;
}
}
 
// Following field names in the paper.
// rangeElt would be whatever data is associated with the Point.
// We don't bother with it for this example.
struct KdNode(size_t k, F) {
Point!(k, F) domElt;
immutable int split;
typeof(this)* left, right;
}
 
struct Orthotope(size_t k, F) { /// k-dimensional rectangle.
Point!(k, F) min, max;
}
 
struct KdTree(size_t k, F) {
KdNode!(k, F)* n;
Orthotope!(k, F) bounds;
 
// Constructs a KdTree from a list of points, also associating the
// bounds of the tree. The bounds could be computed of course, but
// in this example we know them already. The algorithm is table
// 6.3 in the paper.
this(Point!(k, F)[] pts, in Orthotope!(k, F) bounds_) pure {
static KdNode!(k, F)* nk2(size_t split)(Point!(k, F)[] exset)
pure {
if (exset.empty)
return null;
if (exset.length == 1)
return new KdNode!(k, F)(exset[0], split, null, null);
 
// Pivot choosing procedure. We find median, then find
// largest index of points with median value. This
// satisfies the inequalities of steps 6 and 7 in the
// algorithm.
auto m = exset.length / 2;
topN!((p, q) => p[split] < q[split])(exset, m);
immutable d = exset[m];
while (m+1 < exset.length && exset[m+1][split] == d[split])
m++;
 
enum nextSplit = (split + 1) % d.length;//cycle coordinates
return new KdNode!(k, F)(d, split,
nk2!nextSplit(exset[0 .. m]),
nk2!nextSplit(exset[m + 1 .. $]));
}
 
this.n = nk2!0(pts);
this.bounds = bounds_;
}
}
 
/**
Find nearest neighbor. Return values are:
nearest neighbor--the ooint within the tree that is nearest p.
square of the distance to that point.
a count of the nodes visited in the search.
*/

auto findNearest(size_t k, F)(KdTree!(k, F) t, in Point!(k, F) p)
pure nothrow @nogc {
// Algorithm is table 6.4 from the paper, with the addition of
// counting the number nodes visited.
static Tuple!(Point!(k, F), "nearest",
F, "distSqd",
int, "nodesVisited")
nn(KdNode!(k, F)* kd, in Point!(k, F) target,
Orthotope!(k, F) hr, F maxDistSqd) pure nothrow @nogc {
if (kd == null)
return typeof(return)(Point!(k, F)(), F.infinity, 0);
 
int nodesVisited = 1;
immutable s = kd.split;
auto pivot = kd.domElt;
auto leftHr = hr;
auto rightHr = hr;
leftHr.max[s] = pivot[s];
rightHr.min[s] = pivot[s];
 
KdNode!(k, F)* nearerKd, furtherKd;
Orthotope!(k, F) nearerHr, furtherHr;
if (target[s] <= pivot[s]) {
//nearerKd, nearerHr = kd.left, leftHr;
//furtherKd, furtherHr = kd.right, rightHr;
nearerKd = kd.left;
nearerHr = leftHr;
furtherKd = kd.right;
furtherHr = rightHr;
} else {
//nearerKd, nearerHr = kd.right, rightHr;
//furtherKd, furtherHr = kd.left, leftHr;
nearerKd = kd.right;
nearerHr = rightHr;
furtherKd = kd.left;
furtherHr = leftHr;
}
 
auto n1 = nn(nearerKd, target, nearerHr, maxDistSqd);
auto nearest = n1.nearest;
auto distSqd = n1.distSqd;
nodesVisited += n1.nodesVisited;
 
if (distSqd < maxDistSqd)
maxDistSqd = distSqd;
auto d = (pivot[s] - target[s]) ^^ 2;
if (d > maxDistSqd)
return typeof(return)(nearest, distSqd, nodesVisited);
d = pivot.sqd(target);
if (d < distSqd) {
nearest = pivot;
distSqd = d;
maxDistSqd = distSqd;
}
 
immutable n2 = nn(furtherKd, target, furtherHr, maxDistSqd);
nodesVisited += n2.nodesVisited;
if (n2.distSqd < distSqd) {
nearest = n2.nearest;
distSqd = n2.distSqd;
}
 
return typeof(return)(nearest, distSqd, nodesVisited);
}
 
return nn(t.n, p, t.bounds, F.infinity);
}
 
void showNearest(size_t k, F)(in string heading, KdTree!(k, F) kd,
in Point!(k, F) p) {
import std.stdio: writeln;
writeln(heading, ":");
writeln("Point: ", p);
immutable n = kd.findNearest(p);
writeln("Nearest neighbor: ", n.nearest);
writeln("Distance: ", sqrt(n.distSqd));
writeln("Nodes visited: ", n.nodesVisited, "\n");
}
 
void main() {
static Point!(k, F) randomPoint(size_t k, F)() {
typeof(return) result;
foreach (immutable i; 0 .. k)
result[i] = uniform(F(0), F(1));
return result;
}
 
static Point!(k, F)[] randomPoints(size_t k, F)(in size_t n) {
return n.iota.map!(_ => randomPoint!(k, F)).array;
}
 
import std.stdio, std.conv, std.datetime, std.typetuple;
rndGen.seed(1); // For repeatable outputs.
 
alias D2 = TypeTuple!(2, double);
alias P = Point!D2;
auto kd1 = KdTree!D2([P([2, 3]), P([5, 4]), P([9, 6]),
P([4, 7]), P([8, 1]), P([7, 2])],
Orthotope!D2(P([0, 0]), P([10, 10])));
showNearest("Wikipedia example data", kd1, P([9, 2]));
 
enum int N = 400_000;
alias F3 = TypeTuple!(3, float);
alias Q = Point!F3;
StopWatch sw;
GC.disable;
sw.start;
auto kd2 = KdTree!F3(randomPoints!F3(N),
Orthotope!F3(Q([0, 0, 0]), Q([1, 1, 1])));
sw.stop;
GC.enable;
showNearest(text("k-d tree with ", N,
" random 3D ", F3[1].stringof,
" points (construction time: ",
sw.peek.msecs, " ms)"), kd2, randomPoint!F3);
 
sw.reset;
sw.start;
enum int M = 10_000;
size_t visited = 0;
foreach (immutable _; 0 .. M) {
immutable n = kd2.findNearest(randomPoint!F3);
visited += n.nodesVisited;
}
sw.stop;
 
writefln("Visited an average of %0.2f nodes on %d searches " ~
"in %d ms.", visited / double(M), M, sw.peek.msecs);
}
Output, using the ldc2 compiler:
Wikipedia example data:
Point:            [9, 2]
Nearest neighbor: [8, 1]
Distance:         1.41421
Nodes visited:    3

k-d tree with 400000 random 3D float points (construction time: 250 ms):
Point:            [0.22012, 0.984514, 0.698782]
Nearest neighbor: [0.225766, 0.978981, 0.69885]
Distance:         0.00790531
Nodes visited:    54

Visited an average of 43.10 nodes on 10000 searches in 33 ms.

[edit] Faster Alternative Version

Translation of: C

This version performs less lookups. Compiled with DMD this version is two times slower than the C version. Compiled with ldc2 it's a little faster than the C version compiled with gcc.

import std.stdio, std.algorithm, std.math, std.random, std.typetuple;
 
enum maxDim = 3;
 
template Iota(int stop) {
static if (stop <= 0)
alias Iota = TypeTuple!();
else
alias Iota = TypeTuple!(Iota!(stop - 1), stop - 1);
}
 
struct KdNode {
double[maxDim] x;
KdNode* left, right;
}
 
// See QuickSelect method.
KdNode* findMedian(size_t idx)(KdNode[] nodes) pure nothrow @nogc {
auto start = nodes.ptr;
auto end = &nodes[$ - 1] + 1;
 
if (end <= start)
return null;
if (end == start + 1)
return start;
 
KdNode* md = start + (end - start) / 2;
 
while (true) {
immutable double pivot = md.x[idx];
 
swap(md.x, (end - 1).x); // Swaps the whole arrays x.
auto store = start;
foreach (p; start .. end) {
if (p.x[idx] < pivot) {
if (p != store)
swap(p.x, store.x);
store++;
}
}
swap(store.x, (end - 1).x);
 
// Median has duplicate values.
if (store.x[idx] == md.x[idx])
return md;
 
if (store > md)
end = store;
else
start = store;
}
}
 
KdNode* makeTree(size_t dim, size_t i)(KdNode[] nodes)
pure nothrow @nogc {
if (!nodes.length)
return null;
 
auto n = findMedian!i(nodes);
if (n != null) {
enum i2 = (i + 1) % dim;
immutable size_t nPos = n - nodes.ptr;
n.left = makeTree!(dim, i2)(nodes[0 .. nPos]);
n.right = makeTree!(dim, i2)(nodes[nPos + 1 .. $]);
}
 
return n;
}
 
void nearest(size_t dim)(in KdNode* root,
in ref KdNode nd,
in size_t i,
ref const(KdNode)* best,
ref double bestDist,
ref size_t nVisited) pure nothrow @nogc {
static double dist(in ref KdNode a, in ref KdNode b)
pure nothrow @nogc {
typeof(KdNode.x[0]) result = 0;
foreach (i; Iota!dim)
result += (a.x[i] - b.x[i]) ^^ 2;
return result;
}
 
if (root == null)
return;
 
immutable double d = dist(*root, nd);
immutable double dx = root.x[i] - nd.x[i];
immutable double dx2 = dx ^^ 2;
nVisited++;
 
if (!best || d < bestDist) {
bestDist = d;
best = root;
}
 
// If chance of exact match is high.
if (!bestDist)
return;
 
immutable i2 = (i + 1 >= dim) ? 0 : i + 1;
 
nearest!dim(dx > 0 ? root.left : root.right,
nd, i2, best, bestDist, nVisited);
if (dx2 >= bestDist)
return;
nearest!dim(dx > 0 ? root.right : root.left,
nd, i2, best, bestDist, nVisited);
}
 
void randPt(size_t dim=3)(ref KdNode v, ref Xorshift rng)
nothrow @nogc {
foreach (immutable i; Iota!dim)
v.x[i] = rng.uniform01;
}
 
void smallTest() {
KdNode[] wp = [{[2, 3]}, {[5, 4]}, {[9, 6]},
{[4, 7]}, {[8, 1]}, {[7, 2]}];
KdNode thisPt = {[9, 2]};
 
KdNode* root = makeTree!(2, 0)(wp);
 
const(KdNode)* found = null;
double bestDist = 0;
size_t nVisited = 0;
nearest!2(root, thisPt, 0, found, bestDist, nVisited);
 
writefln("WP tree:\n Searching for %s\n" ~
" Found %s, dist = %g\n Seen %d nodes.\n",
thisPt.x[0..2], found.x[0..2], sqrt(bestDist), nVisited);
}
 
void bigTest() {
enum N = 1_000_000;
enum testRuns = 100_000;
 
auto bigTree = new KdNode[N];
auto rng = Xorshift(1);
foreach (ref node; bigTree)
randPt(node, rng);
 
KdNode* root = makeTree!(3, 0)(bigTree);
KdNode thisPt;
randPt(thisPt, rng);
 
const(KdNode)* found = null;
double bestDist = 0;
size_t nVisited = 0;
nearest!3(root, thisPt, 0, found, bestDist, nVisited);
 
writefln("Big tree (%d nodes):\n Searching for %s\n"~
" Found %s, dist = %g\n Seen %d nodes.",
N, thisPt.x, found.x, sqrt(bestDist), nVisited);
 
size_t sum = 0;
foreach (immutable _; 0 .. testRuns) {
found = null;
nVisited = 0;
randPt(thisPt, rng);
nearest!3(root, thisPt, 0, found, bestDist, nVisited);
sum += nVisited;
}
writefln("\nBig tree:\n Visited %d nodes for %d random "~
"searches (%.2f per lookup).",
sum, testRuns, sum / double(testRuns));
}
 
void main() {
smallTest;
bigTest;
}
Output:
WP tree:
  Searching for [9, 2]
  Found [8, 1], dist = 1.41421
  Seen 3 nodes.

Big tree (1000000 nodes):
  Searching for [0.225893, 0.725471, 0.486279]
  Found [0.220761, 0.729613, 0.489134], dist = 0.00718703
  Seen 35 nodes.

Big tree:
  Visited 4267592 nodes for 100000 random searches (42.68 per lookup).

[edit] Go

// Implmentation following pseudocode from "An intoductory tutorial on kd-trees"
// by Andrew W. Moore, Carnegie Mellon University, PDF accessed from
// http://www.autonlab.org/autonweb/14665
package main
 
import (
"fmt"
"math"
"math/rand"
"sort"
"time"
)
 
// point is a k-dimensional point.
type point []float64
 
// sqd returns the square of the euclidean distance.
func (p point) sqd(q point) float64 {
var sum float64
for dim, pCoord := range p {
d := pCoord - q[dim]
sum += d * d
}
return sum
}
 
// kdNode following field names in the paper.
// rangeElt would be whatever data is associated with the point. we don't
// bother with it for this example.
type kdNode struct {
domElt point
split int
left, right *kdNode
}
 
type kdTree struct {
n *kdNode
bounds hyperRect
}
 
type hyperRect struct {
min, max point
}
 
// Go slices are reference objects. The data must be copied if you want
// to modify one without modifying the original.
func (hr hyperRect) copy() hyperRect {
return hyperRect{append(point{}, hr.min...), append(point{}, hr.max...)}
}
 
// newKd constructs a kdTree from a list of points, also associating the
// bounds of the tree. The bounds could be computed of course, but in this
// example we know them already. The algorithm is table 6.3 in the paper.
func newKd(pts []point, bounds hyperRect) kdTree {
var nk2 func([]point, int) *kdNode
nk2 = func(exset []point, split int) *kdNode {
if len(exset) == 0 {
return nil
}
// pivot choosing procedure. we find median, then find largest
// index of points with median value. this satisfies the
// inequalities of steps 6 and 7 in the algorithm.
sort.Sort(part{exset, split})
m := len(exset) / 2
d := exset[m]
for m+1 < len(exset) && exset[m+1][split] == d[split] {
m++
}
// next split
s2 := split + 1
if s2 == len(d) {
s2 = 0
}
return &kdNode{d, split, nk2(exset[:m], s2), nk2(exset[m+1:], s2)}
}
return kdTree{nk2(pts, 0), bounds}
}
 
// a container type used for sorting. it holds the points to sort and
// the dimension to use for the sort key.
type part struct {
pts []point
dPart int
}
 
// satisfy sort.Interface
func (p part) Len() int { return len(p.pts) }
func (p part) Less(i, j int) bool {
return p.pts[i][p.dPart] < p.pts[j][p.dPart]
}
func (p part) Swap(i, j int) { p.pts[i], p.pts[j] = p.pts[j], p.pts[i] }
 
// nearest. find nearest neighbor. return values are:
// nearest neighbor--the point within the tree that is nearest p.
// square of the distance to that point.
// a count of the nodes visited in the search.
func (t kdTree) nearest(p point) (best point, bestSqd float64, nv int) {
return nn(t.n, p, t.bounds, math.Inf(1))
}
 
// algorithm is table 6.4 from the paper, with the addition of counting
// the number nodes visited.
func nn(kd *kdNode, target point, hr hyperRect,
maxDistSqd float64) (nearest point, distSqd float64, nodesVisited int) {
if kd == nil {
return nil, math.Inf(1), 0
}
nodesVisited++
s := kd.split
pivot := kd.domElt
leftHr := hr.copy()
rightHr := hr.copy()
leftHr.max[s] = pivot[s]
rightHr.min[s] = pivot[s]
targetInLeft := target[s] <= pivot[s]
var nearerKd, furtherKd *kdNode
var nearerHr, furtherHr hyperRect
if targetInLeft {
nearerKd, nearerHr = kd.left, leftHr
furtherKd, furtherHr = kd.right, rightHr
} else {
nearerKd, nearerHr = kd.right, rightHr
furtherKd, furtherHr = kd.left, leftHr
}
var nv int
nearest, distSqd, nv = nn(nearerKd, target, nearerHr, maxDistSqd)
nodesVisited += nv
if distSqd < maxDistSqd {
maxDistSqd = distSqd
}
d := pivot[s] - target[s]
d *= d
if d > maxDistSqd {
return
}
if d = pivot.sqd(target); d < distSqd {
nearest = pivot
distSqd = d
maxDistSqd = distSqd
}
tempNearest, tempSqd, nv := nn(furtherKd, target, furtherHr, maxDistSqd)
nodesVisited += nv
if tempSqd < distSqd {
nearest = tempNearest
distSqd = tempSqd
}
return
}
 
func main() {
rand.Seed(time.Now().Unix())
kd := newKd([]point{{2, 3}, {5, 4}, {9, 6}, {4, 7}, {8, 1}, {7, 2}},
hyperRect{point{0, 0}, point{10, 10}})
showNearest("WP example data", kd, point{9, 2})
kd = newKd(randomPts(3, 1000), hyperRect{point{0, 0, 0}, point{1, 1, 1}})
showNearest("1000 random 3d points", kd, randomPt(3))
}
 
func randomPt(dim int) point {
p := make(point, dim)
for d := range p {
p[d] = rand.Float64()
}
return p
}
 
func randomPts(dim, n int) []point {
p := make([]point, n)
for i := range p {
p[i] = randomPt(dim)
}
return p
}
 
func showNearest(heading string, kd kdTree, p point) {
fmt.Println()
fmt.Println(heading)
fmt.Println("point: ", p)
nn, ssq, nv := kd.nearest(p)
fmt.Println("nearest neighbor:", nn)
fmt.Println("distance: ", math.Sqrt(ssq))
fmt.Println("nodes visited: ", nv)
}
Output:
WP example data
point:            [9 2]
nearest neighbor: [8 1]
distance:         1.4142135623730951
nodes visited:    3

1000 random 3d points
point:            [0.314731890562714 0.5908890147906868 0.2657722255021785]
nearest neighbor: [0.2541611609533609 0.5781168738628141 0.27829000365095274]
distance:         0.06315564613771865
nodes visited:    25

[edit] Haskell

There is a space leak when creating the trees which will lead to terrible performance on massive trees. This can probably be quelled with strictness annotations.

import System.Random
import Data.List (sortBy, genericLength, minimumBy)
import Data.Ord (comparing)
 
-- A finite list of dimensional accessors tell a KDTree how to get a
-- Euclidean dimensional value 'b' out of an arbitrary datum 'a'.
type DimensionalAccessors a b = [a -> b]
 
-- A binary tree structure of 'a'.
data Tree a = Node a (Tree a) (Tree a)
| Empty
 
instance Show a => Show (Tree a) where
show Empty = "Empty"
show (Node value left right) =
"(" ++ show value ++ " " ++ show left ++ " " ++ show right ++ ")"
 
-- A k-d tree structure of 'a' with Euclidean dimensions of 'b'.
data KDTree a b = KDTree (DimensionalAccessors a b) (Tree a)
 
instance Show a => Show (KDTree a b) where
show (KDTree _ tree) = "KDTree " ++ show tree
 
-- The squared Euclidean distance formula.
sqrDist :: Num b => DimensionalAccessors a b -> a -> a -> b
sqrDist dims a b = sum $ map square $ zipWith (-) a' b'
where
a' = map ($ a) dims
b'
= map ($ b) dims
 
square :: Num a => a -> a
square = (^ 2)
 
-- Insert a value into a k-d tree.
insert :: Ord b => KDTree a b -> a -> KDTree a b
insert (KDTree dims tree) value = KDTree dims $ ins (cycle dims) tree
where
ins _ Empty = Node value Empty Empty
ins (d:ds) (Node split left right) =
if d value < d split
then Node split (ins ds left) right
else Node split left (ins ds right)
 
-- Produce an empty k-d tree.
empty :: DimensionalAccessors a b -> KDTree a b
empty dims = KDTree dims Empty
 
-- Produce a k-d tree with one value.
singleton :: Ord b => DimensionalAccessors a b -> a -> KDTree a b
singleton dims value = insert (empty dims) value
 
-- Create a k-d tree from a list of values using the median-finding algorithm.
fromList :: Ord b => DimensionalAccessors a b -> [a] -> KDTree a b
fromList dims values = KDTree dims $ fList (cycle dims) values
where
fList _ [] = Empty
fList (d:ds) values =
let sorted = sortBy (comparing d) values
(lower, higher) = splitAt (genericLength sorted `div` 2) sorted
in case higher of
[] -> Empty
median:rest -> Node median (fList ds lower) (fList ds rest)
 
-- Create a k-d tree from a list of values by repeatedly inserting the values
-- into a tree. Faster than median-finding, but can create unbalanced trees.
fromListLinear :: Ord b => DimensionalAccessors a b -> [a] -> KDTree a b
fromListLinear dims values = foldl insert (empty dims) values
 
-- Given a k-d tree, find the nearest value to a given value.
-- Also report how many nodes were visited.
nearest :: (Ord b, Num b, Integral c) => KDTree a b -> a -> (Maybe a, c)
nearest (KDTree dims tree) value = near (cycle dims) tree
where
dist = sqrDist dims
-- If we have an empty tree, then return nothing.
near _ Empty = (Nothing, 1)
-- We hit a leaf node, so it is the current best.
near _ (Node split Empty Empty) = (Just split, 1)
near (d:ds) (Node split left right) =
-- Move down the tree in the fashion of insertion.
let dimdist x y = square (d x - d y)
splitDist = dist value split
hyperPlaneDist = dimdist value split
bestLeft = near ds left
bestRight = near ds right
-- maybeThisBest is the node of the side of the split where the value
-- resides, and maybeOtherBest is the node on the other side of the split.
((maybeThisBest, thisCount), (maybeOtherBest, otherCount)) =
if d value < d split
then (bestLeft, bestRight)
else (bestRight, bestLeft)
in case maybeThisBest of
Nothing ->
-- From the search point (in this case), the hypersphere radius to
-- the split node is always >= the hyperplane distance, so we will
-- always check the other side.
let count = 1 + thisCount + otherCount
in case maybeOtherBest of
-- We are currently at a leaf node, so this is the only choice.
-- It is not strictly necessary to take care of this case
-- because of the above pattern matching in near.
Nothing -> (Just split, count)
-- We have a node on the other side, so compare
-- it to the split point to see which is closer.
Just otherBest ->
if dist value otherBest < splitDist
then (maybeOtherBest, count)
else (Just split, count)
 
Just thisBest ->
let thisBestDist = dist value thisBest
best =
-- Determine which is the closer node of this side.
if splitDist < thisBestDist
then split
else thisBest
bestDist = dist value best
in
if bestDist < hyperPlaneDist
-- If the distance to the best node is less than the distance
-- to the splitting hyperplane, then the current best node is the
-- only choice.
then (Just best, 1 + thisCount)
-- There is a chance that a node on the other side is closer
-- than the current best.
else
let count = 1 + thisCount + otherCount
in case maybeOtherBest of
Nothing -> (Just best, count)
Just otherBest ->
if bestDist < dist value otherBest
then (Just best, count)
else (maybeOtherBest, count)
 
-- Dimensional accessors for a 2-tuple
tuple2D :: [(a, a) -> a]
tuple2D = [fst, snd]
 
-- Dimensional accessors for a 3-tuple
tuple3D :: [(a, a, a) -> a]
tuple3D = [d1, d2, d3]
where
d1 (a, _, _) = a
d2 (_, b, _) = b
d3 (_, _, c) = c
 
-- Random 3-tuple generation
instance (Random a, Random b, Random c) => Random (a, b, c) where
random gen =
let (vA, genA) = random gen
(vB, genB) = random genA
(vC, genC) = random genB
in ((vA, vB, vC), genC)
 
randomR ((lA, lB, lC), (hA, hB, hC)) gen =
let (vA, genA) = randomR (lA, hA) gen
(vB, genB) = randomR (lB, hB) genA
(vC, genC) = randomR (lC, hC) genB
in ((vA, vB, vC), genC)
 
printResults :: (Show a, Show b, Show c, Floating c) =>
a -> (Maybe a, b) -> DimensionalAccessors a c -> IO ()
printResults point result dims = do
let (nearest, visited) = result
case nearest of
Nothing -> putStrLn "Could not find nearest."
Just value -> do
let dist = sqrt $ sqrDist dims point value
putStrLn $ "Point: " ++ show point
putStrLn $ "Nearest: " ++ show value
putStrLn $ "Distance: " ++ show dist
putStrLn $ "Visited: " ++ show visited
putStrLn ""
 
-- Naive nearest search used to confirm results.
linearNearest :: (Ord b, Num b) => DimensionalAccessors a b -> a -> [a] -> Maybe a
linearNearest _ _ [] = Nothing
linearNearest dims value xs = Just $ minimumBy (comparing $ sqrDist dims value) xs
 
main :: IO ()
main = do
let wikiValues :: [(Double, Double)]
wikiValues = [(2, 3), (5, 4), (9, 6), (4, 7), (8, 1), (7, 2)]
wikiTree = fromList tuple2D wikiValues
wikiSearch = (9, 2)
wikiNearest = nearest wikiTree wikiSearch
putStrLn "Wikipedia example:"
printResults wikiSearch wikiNearest tuple2D
 
let stdGen = mkStdGen 0
randRange :: ((Double, Double, Double), (Double, Double, Double))
randRange = ((0, 0, 0), (1000, 1000, 1000))
(randSearch, stdGenB) = randomR randRange stdGen
randValues = take 1000 $ randomRs randRange stdGenB
randTree = fromList tuple3D randValues
randNearest = nearest randTree randSearch
randNearestLinear = linearNearest tuple3D randSearch randValues
putStrLn "1000 random 3D points on the range of [0, 1000):"
printResults randSearch randNearest tuple3D
putStrLn "Confirm naive nearest:"
print randNearestLinear
Output:
Wikipedia example:
Point:    (9.0,2.0)
Nearest:  (8.0,1.0)
Distance: 1.4142135623730951
Visited:  3

1000 random 3D points on the range of [0, 1000):
Point:    (992.9251340102518,993.3624439225405,464.8305261946105)
Nearest:  (939.1965739740829,980.2876583283734,452.4829965078272)
Distance: 56.658359235505955
Visited:  23

Confirm naive nearest:
Just (939.1965739740829,980.2876583283734,452.4829965078272)

[edit] J

As a general rule, tree algorithms are a bad idea in J. That said, here's an implementation:

coclass 'kdnode'
create=:3 :0
Axis=: ({:$y)|<.2^.#y
Mask=: Axis~:i.{:$y
if. 3>#y do.
Leaf=:1
Points=: y
else.
Leaf=:0
data=. y /: Axis|."1 y
n=. <.-:#data
Points=: ,:n{data
Left=: conew&'kdnode' n{.data
Right=: conew&'kdnode' (1+n)}.data
end.
)
 
distance=: +/&.:*:@:-"1
 
nearest=:3 :0
_ 0 nearest y
 :
n=.' ',~":N_base_=:N_base_+1
dists=. Points distance y
ndx=. (i. <./) dists
nearest=. ndx { Points
range=. ndx { dists
if. Leaf do.
range;nearest return.
else.
d0=. x <. range
p0=. nearest
if. d0=0 do. 0;y return. end.
if. 0={./:Axis|."1 y,Points do.
'dist pnt'=.d0 nearest__Left y
if. dist > d0 do.
d0;p0 return.
end.
if. dist < d0 do.
if. dist > (Mask#pnt) distance Mask#,Points do.
'dist2 pnt2'=. d0 nearest__Right y
if. dist2 < dist do. dist2;pnt2 return. end.
end.
end.
else.
'dist pnt'=. d0 nearest__Right y
if. dist > d0 do.
d0;p0 return.
end.
if. dist < d0 do.
if. dist > (Mask#pnt) distance Mask#,Points do.
'dist2 pnt2'=. d0 nearest__Left y
if. dist2 < dist do. dist2;pnt2 return. end.
end.
end.
end.
end.
dist;pnt return.
)
 
coclass 'kdtree'
create=:3 :0
root=: conew&'kdnode' y
)
nearest=:3 :0
N_base_=:0
'dist point'=. nearest__root y
dist;N_base_;point
)

And here's example use:

   tree=:conew&'kdtree' (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
nearest__tree 9 2
┌───────┬─┬───┐
1.4142148 1
└───────┴─┴───┘

The first box is distance from argument point to selected point. The second box is the number of nodes visited. The third box is the selected point.

Here's the bigger problem:

   tree=:conew&'kdtree' dataset=:?1000 3$0
nearest__tree pnt
┌─────────┬──┬──────────────────────────┐
0.0387914120.978082 0.767632 0.392523
└─────────┴──┴──────────────────────────┘

So, why are trees "generally a bad idea in J"?

First off, that's a lot of code, it took time to write. Let's assume that that time was free. Let's also assume that the time taken to build the tree structure was free. We're going to use this tree billions of times. Now what?

Well, let's compare the above implementation to a brute force implementation for time. Here's a "visit all nodes" implementation. It should give us the same kinds of results but we will claim that each candidate point is a node so we'll be visiting a lot more "nodes":

build0=:3 :0
data=: y
)
 
distance=: +/&.:*:@:-"1
 
nearest0=:3 :0
nearest=. data {~ (i. <./) |data distance y
(nearest distance y);(#data);nearest
)

Here's the numbers we get:

   build0 (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
nearest0 9 2
┌───────┬─┬───┐
1.4142168 1
└───────┴─┴───┘
build0 dataset
nearest0 pnt
┌─────────┬────┬──────────────────────────┐
0.038791410000.978082 0.767632 0.392523
└─────────┴────┴──────────────────────────┘

But what about timing?

   tree=:conew&'kdtree' (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
timespacex 'nearest__tree 9 2'
0.000487181 19328
build0 (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
timespacex 'nearest0 9 2'
3.62419e_5 6016

The kdtree implementation is over ten times slower than the brute force implementation for this small dataset. How about the bigger dataset?

   tree=:conew&'kdtree' dataset
timespacex 'nearest__tree pnt'
0.00141408 45312
build0 dataset
timespacex 'nearest0 pnt'
0.00140702 22144

On the bigger dataset, the kdtree implementation is about the same speed as the brute force implementation.

For a more practical approach to this kind of problem, see https://github.com/locklin/j-nearest-neighbor (that is: link to a high performance implementation).

See also: wp:KISS_principle

[edit] Perl 6

Translation of: Python
class Kd_node {
has $.d;
has $.split;
has $.left;
has $.right;
}
 
class Orthotope {
has $.min;
has $.max;
}
 
class Kd_tree {
has $.n;
has $.bounds;
method new($pts, $bounds) { self.bless(n => nk2(0,$pts), bounds => $bounds) }
 
sub nk2($split, @e) {
return () unless @e;
my @exset = @e.sort(*.[$split]);
my $m = +@exset div 2;
my @d = @exset[$m][];
while $m+1 < @exset and @exset[$m+1][$split] eqv @d[$split] {
++$m;
}
 
my $s2 = ($split + 1) % @d; # cycle coordinates
Kd_node.new: :@d, :$split,
left => nk2($s2, @exset[0 ..^ $m]),
right => nk2($s2, @exset[$m ^.. *]);
}
}
 
class T3 {
has $.nearest;
has $.dist_sqd = Inf;
has $.nodes_visited = 0;
}
 
sub find_nearest($k, $t, @p) {
return nn($t.n, @p, $t.bounds, Inf);
 
sub nn($kd, @target, $hr, $max_dist_sqd is copy) {
return T3.new(:nearest([0.0 xx $k])) unless $kd;
 
my $nodes_visited = 1;
my $s = $kd.split;
my $pivot = $kd.d;
my $left_hr = $hr.clone;
my $right_hr = $hr.clone;
$left_hr.max[$s] = $pivot[$s];
$right_hr.min[$s] = $pivot[$s];
 
my $nearer_kd;
my $further_kd;
my $nearer_hr;
my $further_hr;
if @target[$s] <= $pivot[$s] {
($nearer_kd, $nearer_hr) = $kd.left, $left_hr;
($further_kd, $further_hr) = $kd.right, $right_hr;
}
else {
($nearer_kd, $nearer_hr) = $kd.right, $right_hr;
($further_kd, $further_hr) = $kd.left, $left_hr;
}
 
my $n1 = nn($nearer_kd, @target, $nearer_hr, $max_dist_sqd);
my $nearest = $n1.nearest;
my $dist_sqd = $n1.dist_sqd;
$nodes_visited += $n1.nodes_visited;
 
if $dist_sqd < $max_dist_sqd {
$max_dist_sqd = $dist_sqd;
}
my $d = ($pivot[$s] - @target[$s]) ** 2;
if $d > $max_dist_sqd {
return T3.new(:$nearest, :$dist_sqd, :$nodes_visited);
}
$d = [+] (@$pivot Z- @target) X** 2;
if $d < $dist_sqd {
$nearest = $pivot;
$dist_sqd = $d;
$max_dist_sqd = $dist_sqd;
}
 
my $n2 = nn($further_kd, @target, $further_hr, $max_dist_sqd);
$nodes_visited += $n2.nodes_visited;
if $n2.dist_sqd < $dist_sqd {
$nearest = $n2.nearest;
$dist_sqd = $n2.dist_sqd;
}
 
T3.new(:$nearest, :$dist_sqd, :$nodes_visited);
}
}
 
sub show_nearest($k, $heading, $kd, @p) {
print qq:to/END/;
$heading:
Point: [@p.join(',')]
END
my $n = find_nearest($k, $kd, @p);
print qq:to/END/;
Nearest neighbor: [$n.nearest.join(',')]
Distance: &sqrt($n.dist_sqd)
Nodes visited: $n.nodes_visited()
 
END
}
 
sub random_point($k) { [rand xx $k] }
sub random_points($k, $n) { [random_point($k) xx $n] }
 
sub MAIN {
my $kd1 = Kd_tree.new([[2, 3],[5, 4],[9, 6],[4, 7],[8, 1],[7, 2]],
Orthotope.new(:min([0, 0]), :max([10, 10])));
show_nearest(2, "Wikipedia example data", $kd1, [9, 2]);
 
my $N = 1000;
my $t0 = now;
my $kd2 = Kd_tree.new(random_points(3, $N), Orthotope.new(:min([0,0,0]), :max([1,1,1])));
my $t1 = now;
show_nearest(2,
"k-d tree with $N random 3D points (generation time: {$t1 - $t0}s)",
$kd2, random_point(3));
}
Output:
Wikipedia example data:
Point:            [9,2]
Nearest neighbor: [8,1]
Distance:         1.4142135623731
Nodes visited:    3

k-d tree with 1000 random 3D points (generation time: 67.0934954s):
Point:            [0.765565651400664,0.223251226280109,0.00536717765240979]
Nearest neighbor: [0.758919336088656,0.228895111242011,0.0383284709862686]
Distance:         0.0340950700678338
Nodes visited:    23

[edit] Python

Translation of: D
from random import seed, random
from time import clock
from operator import itemgetter
from collections import namedtuple
from math import sqrt
from copy import deepcopy
 
 
def sqd(p1, p2):
return sum((c1 - c2) ** 2 for c1, c2 in zip(p1, p2))
 
 
class KdNode(object):
__slots__ = ("dom_elt", "split", "left", "right")
 
def __init__(self, dom_elt, split, left, right):
self.dom_elt = dom_elt
self.split = split
self.left = left
self.right = right
 
 
class Orthotope(object):
__slots__ = ("min", "max")
 
def __init__(self, mi, ma):
self.min, self.max = mi, ma
 
 
class KdTree(object):
__slots__ = ("n", "bounds")
 
def __init__(self, pts, bounds):
def nk2(split, exset):
if not exset:
return None
exset.sort(key=itemgetter(split))
m = len(exset) // 2
d = exset[m]
while m + 1 < len(exset) and exset[m + 1][split] == d[split]:
m += 1
 
s2 = (split + 1) % len(d) # cycle coordinates
return KdNode(d, split, nk2(s2, exset[:m]),
nk2(s2, exset[m + 1:]))
self.n = nk2(0, pts)
self.bounds = bounds
 
T3 = namedtuple("T3", "nearest dist_sqd nodes_visited")
 
 
def find_nearest(k, t, p):
def nn(kd, target, hr, max_dist_sqd):
if kd is None:
return T3([0.0] * k, float("inf"), 0)
 
nodes_visited = 1
s = kd.split
pivot = kd.dom_elt
left_hr = deepcopy(hr)
right_hr = deepcopy(hr)
left_hr.max[s] = pivot[s]
right_hr.min[s] = pivot[s]
 
if target[s] <= pivot[s]:
nearer_kd, nearer_hr = kd.left, left_hr
further_kd, further_hr = kd.right, right_hr
else:
nearer_kd, nearer_hr = kd.right, right_hr
further_kd, further_hr = kd.left, left_hr
 
n1 = nn(nearer_kd, target, nearer_hr, max_dist_sqd)
nearest = n1.nearest
dist_sqd = n1.dist_sqd
nodes_visited += n1.nodes_visited
 
if dist_sqd < max_dist_sqd:
max_dist_sqd = dist_sqd
d = (pivot[s] - target[s]) ** 2
if d > max_dist_sqd:
return T3(nearest, dist_sqd, nodes_visited)
d = sqd(pivot, target)
if d < dist_sqd:
nearest = pivot
dist_sqd = d
max_dist_sqd = dist_sqd
 
n2 = nn(further_kd, target, further_hr, max_dist_sqd)
nodes_visited += n2.nodes_visited
if n2.dist_sqd < dist_sqd:
nearest = n2.nearest
dist_sqd = n2.dist_sqd
 
return T3(nearest, dist_sqd, nodes_visited)
 
return nn(t.n, p, t.bounds, float("inf"))
 
 
def show_nearest(k, heading, kd, p):
print(heading + ":")
print("Point: ", p)
n = find_nearest(k, kd, p)
print("Nearest neighbor:", n.nearest)
print("Distance: ", sqrt(n.dist_sqd))
print("Nodes visited: ", n.nodes_visited, "\n")
 
 
def random_point(k):
return [random() for _ in range(k)]
 
 
def random_points(k, n):
return [random_point(k) for _ in range(n)]
 
if __name__ == "__main__":
seed(1)
P = lambda *coords: list(coords)
kd1 = KdTree([P(2, 3), P(5, 4), P(9, 6), P(4, 7), P(8, 1), P(7, 2)],
Orthotope(P(0, 0), P(10, 10)))
show_nearest(2, "Wikipedia example data", kd1, P(9, 2))
 
N = 400000
t0 = clock()
kd2 = KdTree(random_points(3, N), Orthotope(P(0, 0, 0), P(1, 1, 1)))
t1 = clock()
text = lambda *parts: "".join(map(str, parts))
show_nearest(2, text("k-d tree with ", N,
" random 3D points (generation time: ",
t1-t0, "s)"),
kd2, random_point(3))
Output:
Wikipedia example data:
Point:            [9, 2]
Nearest neighbor: [8, 1]
Distance:         1.41421356237
Nodes visited:    3

k-d tree with 400000 random 3D points (generation time: 14.8755565302s):
Point:            [0.066694022911324868, 0.13692213852082813, 0.94939167224227283]
Nearest neighbor: [0.067027753280507252, 0.14407354836507069, 0.94543775920177597]
Distance:         0.00817847583914
Nodes visited:    33

[edit] Racket

The following code is optimized for readability.

 
#lang racket
 
; A tree consists of a point, a left and a right subtree.
(struct tree (p l r) #:transparent)
; If the node is in depth d, then the points in l has
; the (d mod k)'th coordinate less than the same coordinate in p.
 
(define (kdtree d k ps)
(cond [(empty? ps) #f] ; #f represents an empty subtree
[else (define-values (p l r) (split-points ps (modulo d k)))
(tree p (kdtree (+ d 1) k l) (kdtree (+ d 1) k r))]))
 
(define (split-points ps d)
(define (ref p) (vector-ref p d))
(define sorted-ps (sort ps < #:key ref))
(define mid (quotient (+ (length ps)) 2))
(define median (ref (list-ref sorted-ps mid)))
(define-values (l r) (partition(λ(x)(< (ref x) median))sorted-ps))
(values (first r) l (rest r)))
 
; The bounding box of a subtree:
(struct bb (mins maxs) #:transparent)
 
(define (infinite-bb k)
(bb (make-vector k -inf.0) (make-vector k +inf.0)))
 
(define/match (copy-bb h)
[((bb mins maxs))
(bb (vector-copy mins) (vector-copy maxs))])
 
(define (dist v w) (for/sum ([x v] [y w]) (sqr (- x y))))
(define (intersects? g r hr) (<= (dist (closest-in-hr g hr) g) r))
(define (closest-in-hr g hr)
(for/vector ([gi g] [mini (bb-mins hr)] [maxi (bb-maxs hr)])
(cond [(<= gi mini) mini]
[(< mini gi maxi) gi]
[else maxi])))
 
(define (split-bb hr d x)
(define left (copy-bb hr))
(define right (copy-bb hr))
(vector-set! (bb-maxs left) d x)
(vector-set! (bb-mins right) d x)
(values left right))
 
(define visits 0) ; for statistics only
(define (visit) (set! visits (+ visits 1)))
(define (reset-visits) (set! visits 0))
(define (regret-visit) (set! visits (- visits 1)))
 
(define (nearest-neighbor g t k)
(define (nearer? p q) (< (dist p g) (dist q g)))
(define (nearest p q) (if (nearer? p q) p q))
(define (nn d t bb) (visit)
(define (ref p) (vector-ref p (modulo d k)))
(match t
[#f (regret-visit) #(+inf.0 +inf.0 +inf.0)]
[(tree p l r)
(define-values (lbb rbb) (split-bb bb (modulo d k) (ref p)))
(define-values (near near-bb far far-bb)
(if (< (ref g) (ref p))
(values l lbb r rbb)
(values r rbb l lbb)))
(define n (nearest p (nn (+ d 1) near near-bb)))
(if (intersects? g (dist n g) far-bb)
(nearest n (nn (+ d 1) far far-bb))
n)]))
(nn 0 t (infinite-bb k)))
 

Tests:

 
(define (wikipedia-test)
(define t (kdtree 0 2 '(#(2 3) #(5 4) #(9 6) #(4 7) #(8 1) #(7 2))))
(reset-visits)
(define n (nearest-neighbor #(9 2) t 2))
(displayln "Wikipedia Test")
(displayln (~a "Nearest neighbour to (9,2) is: " n))
(displayln (~a "Distance: " (dist n #(9 2))))
(displayln (~a "Visits: " visits "\n")))
 
(define (test k n)
(define (random!) (for/vector ([_ k]) (random)))
(define points (for/list ([_ n]) (random!)))
(define t (kdtree 0 k points))
(reset-visits)
(define target (for/vector ([_ k]) 0.75))
(define nb (nearest-neighbor target t k))
(define nb-control (argmin (λ (p) (dist p target)) points))
(displayln (~a n " points in R^3 test"))
(displayln (~a "Nearest neighbour to " target " is: \n\t\t" nb))
(displayln (~a "Control: \t" nb-control))
(displayln (~a "Distance: \t" (dist nb target)))
(displayln (~a "Control: \t" (dist nb-control target)))
(displayln (~a "Visits: \t" visits)))
 
(wikipedia-test)
(test 3 1000)
(test 3 1000)
 

Output:

 
Wikipedia Test
Nearest neighbour to (9,2) is: #(8 1)
Distance: 2
Visits: 3
 
1000 points in R^3 test
Nearest neighbour to #(0.75 0.75 0.75) is:
#(0.8092534479975508 0.7507095851813429 0.7706494651024903)
Control: #(0.8092534479975508 0.7507095851813429 0.7706494651024903)
Distance: 0.003937875019747008
Control: 0.003937875019747008
Visits: 83
 
1000 points in R^3 test
Nearest neighbour to #(0.75 0.75 0.75) is:
#(0.7775581478448806 0.7806612633582072 0.7396664367640902)
Control: #(0.7775581478448806 0.7806612633582072 0.7396664367640902)
Distance: 0.0018063471125121851
Control: 0.0018063471125121851
Visits: 39
 

[edit] Scala

This example works for sequences of Int, Double, etc, so it is non-minimal due to its type-safe Numeric parameterisation.

object KDTree {
import Numeric._
 
// Task 1A. Build tree of KDNodes. Translated from Wikipedia.
def apply[T](points: Seq[Seq[T]], depth: Int = 0)(implicit num: Numeric[T]): Option[KDNode[T]] = {
val dim = points.headOption.map(_.size) getOrElse 0
if (points.isEmpty || dim < 1) None
else {
val axis = depth % dim
val sorted = points.sortBy(_(axis))
val median = sorted(sorted.size / 2)(axis)
val (left, right) = sorted.partition(v => num.lt(v(axis), median))
Some(KDNode(right.head, apply(left, depth + 1), apply(right.tail, depth + 1), axis))
}
}
 
// Task 1B. Find the nearest node in this subtree. Translated from Wikipedia.
case class KDNode[T](value: Seq[T], left: Option[KDNode[T]], right: Option[KDNode[T]], axis: Int)(implicit num: Numeric[T]) {
def nearest(to: Seq[T]): Nearest[T] = {
val default = Nearest(value, to, Set(this))
compare(to, value) match {
case 0 => default // exact match
case t =>
lazy val bestL = left.map(_ nearest to).getOrElse(default)
lazy val bestR = right.map(_ nearest to).getOrElse(default)
val branch1 = if (t < 0) bestL else bestR
val best = if (num.lt(branch1.distsq, default.distsq)) branch1 else default
val splitDist = num.minus(to(axis), value(axis))
if (num.lt(num.times(splitDist, splitDist), best.distsq)) {
val branch2 = if (t < 0) bestR else bestL
val visited = branch2.visited ++ best.visited + this
if (num.lt(branch2.distsq, best.distsq))
branch2.copy(visited = visited)
else best.copy(visited = visited)
} else best.copy(visited = best.visited + this)
}
}
}
 
// Keep track of nodes visited, as per task. Pretty-printable.
case class Nearest[T](value: Seq[T], to: Seq[T], visited: Set[KDNode[T]] = Set[KDNode[T]]())(implicit num: Numeric[T]) {
lazy val distsq = KDTree.distsq(value, to)
override def toString = f"Searched for=${to} found=${value} distance=${math.sqrt(num.toDouble(distsq))}%.4f visited=${visited.size}"
}
 
// Numeric utilities
def distsq[T](a: Seq[T], b: Seq[T])(implicit num: Numeric[T]) =
a.zip(b).map(c => num.times(num.minus(c._1, c._2), num.minus(c._1, c._2))).sum
def compare[T](a: Seq[T], b: Seq[T])(implicit num: Numeric[T]): Int =
a.zip(b).find(c => num.compare(c._1, c._2) != 0).map(c => num.compare(c._1, c._2)).getOrElse(0)
}

Task test:

object KDTreeTest extends App {
def test[T](haystack: Seq[Seq[T]], needles: Seq[T]*)(implicit num: Numeric[T]) = {
println
val tree = KDTree(haystack)
if (haystack.size < 20) tree.foreach(println)
for (kd <- tree; needle <- needles; nearest = kd nearest needle) {
println(nearest)
// Brute force proof
val better = haystack
.map(KDTree.Nearest(_, needle))
.filter(n => num.lt(n.distsq, nearest.distsq))
.sortBy(_.distsq)
assert(better.isEmpty, s"Found ${better.size} closer than ${nearest.value} e.g. ${better.head}")
}
}
 
// Results 1
val wikitest = List(List(2,3), List(5,4), List(9,6), List(4,7), List(8,1), List(7,2))
test(wikitest, List(9,2))
 
// Results 2 (1000 points uniformly distributed in 3-d cube coordinates, sides 2 to 20)
val uniform = for(x <- 1 to 10; y <- 1 to 10; z <- 1 to 10) yield List(x*2, y*2, z*2)
assume(uniform.size == 1000)
test(uniform, List(0, 0, 0), List(2, 2, 20), List(9, 10, 11))
 
// Results 3 (1000 points randomly distributed in 3-d cube coordinates, sides -1.0 to 1.0)
scala.util.Random.setSeed(0)
def random(n: Int) = (1 to n).map(_ => (scala.util.Random.nextDouble - 0.5)* 2)
test((1 to 1000).map(_ => random(3)), random(3))
 
// Results 4 (27 points uniformly distributed in 3-d cube coordinates, sides 3...9)
val small = for(x <- 1 to 3; y <- 1 to 3; z <- 1 to 3) yield List(x*3, y*3, z*3)
assume(small.size == 27)
test(small, List(0, 0, 0), List(4, 5, 6))
}
Output:
KDNode(List(7, 2),Some(KDNode(List(5, 4),Some(KDNode(List(2, 3),None,None,0)),Some(KDNode(List(4, 7),None,None,0)),1)),Some(KDNode(List(9, 6),Some(KDNode(List(8, 1),None,None,0)),None,1)),0)
Searched for=List(9, 2) found=List(8, 1) distance=1.4142 visited=3

Searched for=List(0, 0, 0) found=List(2, 2, 2) distance=3.4641 visited=10
Searched for=List(2, 2, 20) found=List(2, 2, 20) distance=0.0000 visited=9
Searched for=List(9, 10, 11) found=List(8, 10, 12) distance=1.4142 visited=134

Searched for=Vector(0.19269603520919643, -0.25958512078298535, -0.2572864045762784) found=Vector(0.07811099409527977, -0.2477618820196814, -0.20252227622550611) distance=0.1275 visited=25

Searched for=List(0, 0, 0) found=List(3, 3, 3) distance=5.1962 visited=4
Searched for=List(4, 5, 6) found=List(3, 6, 6) distance=1.4142 visited=6

[edit] Tcl

Translation of: Python
Library: TclOO
package require TclOO
 
oo::class create KDTree {
variable t dim
constructor {points} {
set t [my Build 0 $points 0 end]
set dim [llength [lindex $points 0]]
}
method Build {split exset from to} {
set exset [lsort -index $split -real [lrange $exset $from $to]]
if {![llength $exset]} {return 0}
set m [expr {[llength $exset] / 2}]
set d [lindex $exset $m]
while {[set mm $m;incr mm] < [llength $exset] && \
[lindex $exset $mm $split] == [lindex $d $split]} {
set m $mm
}
set s [expr {($split + 1) % [llength $d]}]
list 1 $d $split [my Build $s $exset 0 [expr {$m-1}]] \
[my Build $s $exset [expr {$m+1}] end]
}
 
method findNearest {p} {
lassign [my FN $t $p inf] p d2 count
return [list $p [expr {sqrt($d2)}] $count]
}
method FN {kd target maxDist2} {
if {[lindex $kd 0] == 0} {
return [list [lrepeat $dim 0.0] inf 0]
}
 
set nodesVisited 1
lassign $kd -> pivot s
 
if {[lindex $target $s] <= [lindex $pivot $s]} {
set nearerKD [lindex $kd 3]
set furtherKD [lindex $kd 4]
} else {
set nearerKD [lindex $kd 4]
set furtherKD [lindex $kd 3]
}
 
lassign [my FN $nearerKD $target $maxDist2] nearest dist2 count
incr nodesVisited $count
 
if {$dist2 < $maxDist2} {
set maxDist2 $dist2
}
set d2 [expr {([lindex $pivot $s]-[lindex $target $s])**2}]
if {$d2 > $maxDist2} {
return [list $nearest $dist2 $nodesVisited]
}
set d2 0.0
foreach pp $pivot tp $target {set d2 [expr {$d2+($pp-$tp)**2}]}
if {$d2 < $dist2} {
set nearest $pivot
set maxDist2 [set dist2 $d2]
}
 
lassign [my FN $furtherKD $target $maxDist2] fNearest fDist2 count
incr nodesVisited $count
if {$fDist2 < $dist2} {
set nearest $fNearest
set dist2 $fDist2
}
 
return [list $nearest $dist2 $nodesVisited]
}
}

Demonstration code:

proc showNearest {heading tree point} {
puts ${heading}:
puts "Point: \[[join $point ,]\]"
lassign [$tree findNearest $point] nearest distance count
puts "Nearest neighbor: \[[join $nearest ,]\]"
puts "Distance: $distance"
puts "Nodes visited: $count"
}
proc randomPoint k {
for {set j 0} {$j < $k} {incr j} {lappend p [::tcl::mathfunc::rand]}
return $p
}
proc randomPoints {k n} {
for {set i 0} {$i < $n} {incr i} {
set p {}
for {set j 0} {$j < $k} {incr j} {
lappend p [::tcl::mathfunc::rand]
}
lappend ps $p
}
return $ps
}
 
KDTree create kd1 {{2 3} {5 4} {9 6} {4 7} {8 1} {7 2}}
showNearest "Wikipedia example data" kd1 {9 2}
puts ""
 
set N 1000
set t [time {KDTree create kd2 [randomPoints 3 $N]}]
showNearest "k-d tree with $N random 3D points (generation time: [lindex $t 0] us)" kd2 [randomPoint 3]
kd2 destroy
puts ""
 
set N 1000000
set t [time {KDTree create kd2 [randomPoints 3 $N]}]
showNearest "k-d tree with $N random 3D points (generation time: [lindex $t 0] us)" kd2 [randomPoint 3]
puts "Search time: [time {kd2 findNearest [randomPoint 3]} 10000]"
Output:
Wikipedia example data:
Point:            [9,2]
Nearest neighbor: [8,1]
Distance:         1.4142135623730951
Nodes visited:    3

k-d tree with 1000 random 3D points (generation time: 11908 us):
Point:            [0.8480196329057308,0.6659702466176685,0.961934903153188]
Nearest neighbor: [0.8774737389187672,0.7011300077201472,0.8920397525150514]
Distance:         0.0836007490668497
Nodes visited:    29

k-d tree with 1000000 random 3D points (generation time: 19643366 us):
Point:            [0.10923849936073576,0.9714587558859301,0.30731017482807405]
Nearest neighbor: [0.10596616664247875,0.9733627601402638,0.3079096774141815]
Distance:         0.0038331184393709545
Nodes visited:    22
Search time:      289.894755 microseconds per iteration
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