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| − | Below is the full source code to a tree that will form the basis for a future kd tree tutorial I'm working on. This code is released under the [[RWPCL]]. It is decently optimised—theoretically, anyway. I don't specialise in writing fast code, but the algorithms used are designed to ensure whole trees are ruled out as early as possible.
| + | There used to be a kd-tree on this page. It has since been packaged into a JAR file and is available at: [[File:Duyn_tutorial_kd_trees.jar]]. Its source is available under the [[RWPCL]]. |
| − | | |
| − | ===Exemplar.java===
| |
| − | Base class for items to be added into the tree. Sub-classes can carry useful data like guess factors.
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| − | | |
| − | <pre>
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| − | /**
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| − | * A sample point in multi-dimensional space. Needed because each sample
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| − | * may contain an arbitrary payload.
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| − | *
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| − | * Note: this class does not make allowance for a payload. Sub-class if
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| − | * you want to store something more than just data points.
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| − | *
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| − | * @author duyn
| |
| − | */
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| − | public class Exemplar {
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| − | public final double[] domain;
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| − | | |
| − | public Exemplar(final double[] coords) {
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| − | this.domain = coords;
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| − | }
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| − | | |
| − | public boolean domainEquals(final Exemplar other) {
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| − | return Arrays.equals(domain, other.domain);
| |
| − | }
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| − | | |
| − | @Override public String
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| − | toString() {
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| − | return Arrays.toString(domain);
| |
| − | }
| |
| − | }
| |
| − | </pre>
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| − | | |
| − | ===BucketKdTree.java===
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| − | Tree with k-nearest neighbour search. Does not support deletion or rebalancing—a re-build is required if you want to do either.
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| − | | |
| − | <pre>
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| − | import java.util.*;
| |
| − | | |
| − | /**
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| − | * A k-dimensional binary partitioning tree which splits space on the
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| − | * mean of the dimension with the largest variance. Points are held in
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| − | * buckets so we can pick a better split point than whatever comes first.
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| − | *
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| − | * Does not store tree depth. If you want balance, re-build the tree
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| − | * periodically.
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| − | *
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| − | * Optimisations in this tree assume distance metric is euclidian distance.
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| − | * May work if retrofitted with other metrics, but that is purely
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| − | * accidental.
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| − | *
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| − | * Note: results can become unpredictable if values are different but so
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| − | * close together that rounding errors in computing their mean result in
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| − | * all exemplars being on one side of the mean. Performance degrades when
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| − | * this occurs. Nearest neighbour search tested to work up to range
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| − | * [1, 1 + 5e-16).
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| − | *
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| − | * Ideas for path ordering and bounds come from:
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| − | * NEAL SAMPLE, MATTHEW HAINES, MARK ARNOLD, TIMOTHY PURCELL,
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| − | * 'Optimizing Search Strategies in k-d Trees'
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| − | * http://ilpubs.stanford.edu:8090/723/
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| − | *
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| − | * Computation of variance from:
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| − | * John Cook, 'Accurately computing running variance'
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| − | * http://www.johndcook.com/standard_deviation.html
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| − | *
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| − | * Terminology note: points are called Exemplars. They must all be
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| − | * descended from Exemplar class. Position in k-d space is stored in each
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| − | * exemplar's domain member. This is to avoid conflicting with already
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| − | * existing classes referring to geometric points.
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| − | *
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| − | * Terminology comes from:
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| − | * Andrew Moore, 'An intoductory tutorial on kd'
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| − | * http://www.autonlab.org/autonweb/14665
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| − | *
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| − | * @author duyn
| |
| − | */
| |
| − | public class BucketKdTree<T extends Exemplar> {
| |
| − | | |
| − | // Only leaf nodes contain points
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| − | private List<T> exemplars = new LinkedList<T>();
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| − | // These aren't initialised until add() is called.
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| − | private double[] exMean;
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| − | private double[] exSumSqDev;
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| − | | |
| − | // Optimisation when tree contains large number of duplicates
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| − | private boolean exemplarsAreUniform = true;
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| − | | |
| − | private int bucketSize;
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| − | private BucketKdTree<T> left, right;
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| − | | |
| − | private int splitDim;
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| − | private double split;
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| − | | |
| − | private final int dimensions;
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| − | | |
| − | // Optimisation for searches. This lets us skip a node if its
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| − | // scope intersects with a search hypersphere but it doesn't contain
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| − | // any points that actually intersect.
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| − | private double[] maxBounds;
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| − | private double[] minBounds;
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| − | | |
| − | public BucketKdTree(int bucketSize, int dimensions) {
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| − | this.bucketSize = bucketSize;
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| − | this.dimensions = dimensions;
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| − | }
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| − | | |
| − | //
| |
| − | // PUBLIC METHODS
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| − | //
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| − | | |
| − | public void
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| − | add(T ex) {
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| − | BucketKdTree<T> tree = addNoSplit(this, ex);
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| − | if (shouldSplit(tree)) {
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| − | split(tree);
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| − | }
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| − | }
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| − | | |
| − | public void
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| − | addAll(Collection<T> exs) {
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| − | // Some spurious function calls. Optimised for readability over
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| − | // efficiency.
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| − | final Set<BucketKdTree<T>> modTrees =
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| − | new HashSet<BucketKdTree<T>>();
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| − | for(T ex : exs) {
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| − | modTrees.add(addNoSplit(this, ex));
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| − | }
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| − | | |
| − | for(BucketKdTree<T> tree : modTrees) {
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| − | if (shouldSplit(tree)) {
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| − | split(tree);
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| − | }
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| − | }
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| − | }
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| − | | |
| − | public SortedMap<Double, List<T>>
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| − | search(double[] query, int nMinResults) {
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| − | // Forward to a static method to avoid accidental reference to
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| − | // instance variables while descending the tree
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| − | return search(this, query, nMinResults);
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| − | }
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| − |
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| − | @Override public String
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| − | toString() {
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| − | return toString("");
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| − | }
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| − | | |
| − | //
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| − | // IMPLEMENTATION DETAILS
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| − | //
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| − | | |
| − | private boolean
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| − | isTree() { return left != null; }
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| − | | |
| − | // Addition
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| − | | |
| − | // Adds an exemplar without splitting overflowing leaves.
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| − | // Returns leaf to which exemplar was added.
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| − | private static <T extends Exemplar> BucketKdTree<T>
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| − | addNoSplit(BucketKdTree<T> tree, T ex) {
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| − | // Some spurious function calls. Optimised for readability over
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| − | // efficiency.
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| − | BucketKdTree<T> cursor = tree;
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| − | while (cursor != null) {
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| − | updateBounds(cursor, ex);
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| − | if (cursor.isTree()) {
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| − | // Sub-tree
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| − | cursor = ex.domain[cursor.splitDim] <= cursor.split
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| − | ? cursor.left : cursor.right;
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| − | } else {
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| − | // Leaf
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| − | cursor.exemplars.add(ex);
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| − | final int nExs = cursor.exemplars.size();
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| − | if (nExs == 1) {
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| − | cursor.exMean =
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| − | Arrays.copyOf(ex.domain, cursor.dimensions);
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| − | cursor.exSumSqDev = new double[cursor.dimensions];
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| − | } else {
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| − | for(int d = 0; d < cursor.dimensions; d++) {
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| − | final double coord = ex.domain[d];
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| − | | |
| − | final double oldExMean = cursor.exMean[d];
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| − | final double newMean = cursor.exMean[d] =
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| − | oldExMean + (coord - oldExMean)/nExs;
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| − | | |
| − | final double oldSumSqDev = cursor.exSumSqDev[d];
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| − | cursor.exSumSqDev[d] = oldSumSqDev
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| − | + (coord - oldExMean)*(coord - newMean);
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| − | }
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| − | }
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| − | if (cursor.exemplarsAreUniform) {
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| − | final List<T> cExs = cursor.exemplars;
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| − | if (cExs.size() > 0 && !ex.domainEquals(cExs.get(0)))
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| − | cursor.exemplarsAreUniform = false;
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| − | }
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| − | return cursor;
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| − | }
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| − | }
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| − | throw new RuntimeException("Walked tree without adding anything");
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| − | }
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| − | | |
| − | private static <T extends Exemplar> void
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| − | updateBounds(BucketKdTree<T> tree, Exemplar ex) {
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| − | final int dims = tree.dimensions;
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| − | if (tree.maxBounds == null) {
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| − | tree.maxBounds = Arrays.copyOf(ex.domain, dims);
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| − | tree.minBounds = Arrays.copyOf(ex.domain, dims);
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| − | } else {
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| − | for(int d = 0; d < dims; d++) {
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| − | final double dimVal = ex.domain[d];
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| − | if (dimVal > tree.maxBounds[d])
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| − | tree.maxBounds[d] = dimVal;
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| − | else if (dimVal < tree.minBounds[d])
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| − | tree.minBounds[d] = dimVal;
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| − | }
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| − | }
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| − | }
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| − | | |
| − | // Splitting (internal operation)
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| − | | |
| − | private static <T extends Exemplar> boolean
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| − | shouldSplit(BucketKdTree<T> tree) {
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| − | return tree.exemplars.size() > tree.bucketSize
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| − | && !tree.exemplarsAreUniform;
| |
| − | }
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| − | | |
| − | @SuppressWarnings("unchecked") private static <T extends Exemplar> void
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| − | split(BucketKdTree<T> tree) {
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| − | assert !tree.exemplarsAreUniform;
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| − | // Find dimension with largest variance to split on
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| − | double largestVar = -1;
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| − | int splitDim = 0;
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| − | for(int d = 0; d < tree.dimensions; d++) {
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| − | final double var =
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| − | tree.exSumSqDev[d]/(tree.exemplars.size() - 1);
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| − | if (var > largestVar) {
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| − | largestVar = var;
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| − | splitDim = d;
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| − | }
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| − | }
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| − | | |
| − | // Find mean as position for our split
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| − | double splitValue = tree.exMean[splitDim];
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| − | | |
| − | // Check that our split actually splits our data. This also lets
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| − | // us bulk load exemplars into sub-trees, which is more likely
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| − | // to keep optimal balance.
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| − | final List<T> leftExs = new LinkedList<T>();
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| − | final List<T> rightExs = new LinkedList<T>();
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| − | for(T s : tree.exemplars) {
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| − | if (s.domain[splitDim] <= splitValue)
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| − | leftExs.add(s);
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| − | else
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| − | rightExs.add(s);
| |
| − | }
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| − | int leftSize = leftExs.size();
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| − | final int treeSize = tree.exemplars.size();
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| − | if (leftSize == treeSize || leftSize == 0) {
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| − | System.err.println(
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| − | "WARNING: Randomly splitting non-uniform tree");
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| − | // We know the exemplars aren't all the same, so try picking
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| − | // an exemplar and a dimension at random for our split point
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| − | | |
| − | // This might take several tries, so we copy our exemplars to
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| − | // an array to speed up process of picking a random point
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| − | Object[] exs = tree.exemplars.toArray();
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| − | while (leftSize == treeSize || leftSize == 0) {
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| − | leftExs.clear();
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| − | rightExs.clear();
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| − | | |
| − | splitDim = (int)
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| − | Math.floor(Math.random()*tree.dimensions);
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| − | final int splitPtIdx = (int)
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| − | Math.floor(Math.random()*exs.length);
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| − | // Cast is inevitable consequence of java's inability to
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| − | // create a generic array
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| − | splitValue = ((T)exs[splitPtIdx]).domain[splitDim];
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| − | for(T s : tree.exemplars) {
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| − | if (s.domain[splitDim] <= splitValue)
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| − | leftExs.add(s);
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| − | else
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| − | rightExs.add(s);
| |
| − | }
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| − | leftSize = leftExs.size();
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| − | }
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| − | }
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| − | | |
| − | // We have found a valid split. Start building our sub-trees
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| − | final BucketKdTree<T> left =
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| − | new BucketKdTree<T>(tree.bucketSize, tree.dimensions);
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| − | final BucketKdTree<T> right =
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| − | new BucketKdTree<T>(tree.bucketSize, tree.dimensions);
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| − | left.addAll(leftExs);
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| − | right.addAll(rightExs);
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| − | | |
| − | // Finally, commit the split
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| − | tree.splitDim = splitDim;
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| − | tree.split = splitValue;
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| − | tree.left = left;
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| − | tree.right = right;
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| − | | |
| − | // Let go of exemplars (and their running stats) held in this leaf
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| − | tree.exemplars = null;
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| − | tree.exMean = tree.exSumSqDev = null;
| |
| − | }
| |
| − | | |
| − | // Searching
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| − | | |
| − | // May return more results than requested if multiple exemplars have
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| − | // same distance from target.
| |
| − | //
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| − | // Note: this function works with squared distances to avoid sqrt()
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| − | // operations
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| − | private static <T extends Exemplar> SortedMap<Double, List<T>>
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| − | search(BucketKdTree<T> tree, double[] query, int nMinResults) {
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| − | // distance => list of points that distance away from query
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| − | final NavigableMap<Double, List<T>> results =
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| − | new TreeMap<Double, List<T>>();
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| − | | |
| − | final SearchState state = new SearchState();
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| − | final Deque<SearchStackEntry<T>> stack =
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| − | new LinkedList<SearchStackEntry<T>>();
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| − | stack.addFirst(new SearchStackEntry<T>(state.maxDistance, tree));
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| − | while (!stack.isEmpty()) {
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| − | final SearchStackEntry<T> entry = stack.removeFirst();
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| − | final BucketKdTree<T> cur = entry.tree;
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| − | | |
| − | if (cur.isTree()) {
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| − | searchTree(query, nMinResults, cur, state, stack);
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| − | } else if (entry.minDFromQ <= state.maxDistance
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| − | || state.nResults < nMinResults)
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| − | {
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| − | searchLeaf(query, nMinResults, cur, state, results);
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| − | }
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| − | }
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| − | | |
| − | return results;
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| − | }
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| − | | |
| − | private static <T extends Exemplar> void
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| − | searchTree(double[] query, int nMinResults, BucketKdTree<T> tree,
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| − | SearchState searchState, Deque<SearchStackEntry<T>> stack)
| |
| − | {
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| − | // Left is presumed near. This is verified further down.
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| − | BucketKdTree<T> nearTree = tree.left, farTree = tree.right;
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| − | // These variables let us skip empty sub-trees
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| − | boolean nearEmpty = nearTree.minBounds == null;
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| − | boolean farEmpty = farTree.minBounds == null;
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| − | | |
| − | // Find distance from nearest possible point in each
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| − | // sub-tree to query. If that is greater than max distance,
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| − | // we can rule out that sub-tree.
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| − | double nearD = nearEmpty ? Double.POSITIVE_INFINITY
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| − | : minDistanceSqFrom(query,
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| − | nearTree.minBounds, nearTree.maxBounds);
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| − | double farD = farEmpty ? Double.POSITIVE_INFINITY
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| − | : minDistanceSqFrom(query,
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| − | farTree.minBounds, farTree.maxBounds);
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| − | | |
| − | // Swap near and far if they're incorrect
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| − | if (farD < nearD) {
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| − | final double tmpD = nearD;
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| − | final BucketKdTree<T> tmpTree = nearTree;
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| − | final boolean tmpEmpty = nearEmpty;
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| − | | |
| − | nearD = farD;
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| − | nearTree = farTree;
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| − | nearEmpty = farEmpty;
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| − | | |
| − | farD = tmpD;
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| − | farTree = tmpTree;
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| − | farEmpty = tmpEmpty;
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| − | }
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| − | | |
| − | // Add nearest sub-tree to stack later so we descend it
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| − | // first. This is likely to constrict our max distance
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| − | // sooner, resulting in less visited nodes
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| − | if (!farEmpty && (farD <= searchState.maxDistance
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| − | || searchState.nResults < nMinResults))
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| − | {
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| − | stack.addFirst(new SearchStackEntry<T>(farD, farTree));
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| − | }
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| − | | |
| − | if (!nearEmpty && (nearD <= searchState.maxDistance
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| − | || searchState.nResults < nMinResults))
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| − | {
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| − | stack.addFirst(new SearchStackEntry<T>(nearD, nearTree));
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| − | }
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| − | }
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| − | | |
| − | private static <T extends Exemplar> void
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| − | searchLeaf(double[] query, int nMinResults,
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| − | BucketKdTree<T> leaf, SearchState searchState,
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| − | NavigableMap<Double, List<T>> results)
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| − | {
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| − | // Keep track of elements at max distance so we know
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| − | // whether we can just drop entire list of furthest
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| − | // exemplars
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| − | final int nMinResultsBeforeAddition = nMinResults - 1;
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| − | for(T ex : leaf.exemplars) {
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| − | final double exD = distanceSqFrom(query, ex.domain);
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| − | if (searchState.nResults < nMinResults
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| − | || exD == searchState.maxDistance)
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| − | {
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| − | // Blindly add this exemplar
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| − | List<T> exsAtD = getOrElseInit(results, exD);
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| − | if (leaf.exemplarsAreUniform) {
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| − | // No need to go through every one if all
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| − | // exemplars are the same
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| − | exsAtD.addAll(leaf.exemplars);
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| − | searchState.nResults += leaf.exemplars.size();
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| − | } else {
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| − | exsAtD.add(ex);
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| − | searchState.nResults++;
| |
| − | }
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| − | | |
| − | // Update information about furthest exemplars
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| − | if (exD > searchState.maxDistance
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| − | || searchState.maxDistance == Double.POSITIVE_INFINITY)
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| − | {
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| − | searchState.maxDistance = exD;
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| − | }
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| − | | |
| − | if (searchState.maxDistance == exD)
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| − | searchState.nExsAtMaxD = exsAtD.size();
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| − | | |
| − | } else if (exD < searchState.maxDistance) {
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| − | // Point closer than furthest neighbour
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| − | | |
| − | if (searchState.nResults - searchState.nExsAtMaxD
| |
| − | >= nMinResultsBeforeAddition)
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| − | {
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| − | // Dropping furthest exemplars won't leave
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| − | // us with too little to meet return
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| − | // minimum
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| − | results.remove(searchState.maxDistance);
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| − | searchState.nResults -= searchState.nExsAtMaxD;
| |
| − | }
| |
| − | | |
| − | // Add new exemplar
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| − | List<T> exsAtD = getOrElseInit(results, exD);
| |
| − | if (leaf.exemplarsAreUniform) {
| |
| − | // No need to go through every one if all
| |
| − | // exemplars are the same
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| − | exsAtD.addAll(leaf.exemplars);
| |
| − | searchState.nResults += leaf.exemplars.size();
| |
| − | } else {
| |
| − | exsAtD.add(ex);
| |
| − | searchState.nResults++;
| |
| − | }
| |
| − | | |
| − | // Update information about furthest exemplars
| |
| − | Map.Entry<Double, List<T>> lastEnt = results.lastEntry();
| |
| − | searchState.maxDistance = lastEnt.getKey();
| |
| − | searchState.nExsAtMaxD = lastEnt.getValue().size();
| |
| − | } // exD < maxDistance
| |
| − | | |
| − | // No need to keep going if all exemplars are
| |
| − | // the same
| |
| − | if (leaf.exemplarsAreUniform) break;
| |
| − | } // for(T ex : cur.exemplars)
| |
| − | }
| |
| − | | |
| − | private static <T extends Exemplar> List<T>
| |
| − | getOrElseInit(Map<Double, List<T>> results, double dst) {
| |
| − | List<T> lst = results.get(dst);
| |
| − | if (lst == null) {
| |
| − | lst = new LinkedList<T>();
| |
| − | results.put(dst, lst);
| |
| − | }
| |
| − | return lst;
| |
| − | }
| |
| − | | |
| − | // Dumping to string (debug)
| |
| − | | |
| − | private String
| |
| − | toString(String indent) {
| |
| − | if (isTree()){
| |
| − | return String.format("%s{|%d|%f|\n%s\n%s} ", indent,
| |
| − | splitDim, split, left.toString(indent + "\t"),
| |
| − | right.toString(indent + "\t"));
| |
| − | } else {
| |
| − | return String.format("%sL%s", indent,
| |
| − | Arrays.toString(exemplars.toArray()));
| |
| − | }
| |
| − | }
| |
| − | | |
| − | // Distance calculations
| |
| − | | |
| − | // Gets distance from target of nearest point on hyper-rect defined
| |
| − | // by supplied min and max bounds
| |
| − | private static double
| |
| − | minDistanceSqFrom(double[] target, double[] min, double[] max) {
| |
| − | // Note: profiling shows this is called lots of times, so it pays
| |
| − | // to be well optimised
| |
| − | double distanceSq = 0;
| |
| − | for(int d = 0; d < target.length; d++) {
| |
| − | final double coord = target[d];
| |
| − | double nearCoord;
| |
| − | if (((nearCoord = min[d]) > coord)
| |
| − | || ((nearCoord = max[d]) < coord))
| |
| − | {
| |
| − | final double dst = nearCoord - coord;
| |
| − | distanceSq += dst*dst;
| |
| − | }
| |
| − | }
| |
| − | return distanceSq;
| |
| − | }
| |
| − | | |
| − | // Accessible to testing
| |
| − | static double
| |
| − | distanceSqFrom(double[] p1, double[] p2) {
| |
| − | // Note: profiling shows this is called lots of times, so it pays
| |
| − | // to be well optimised
| |
| − | double dSq = 0;
| |
| − | for(int d = 0; d < p1.length; d++) {
| |
| − | final double dst = p1[d] - p2[d];
| |
| − | if (dst != 0)
| |
| − | dSq += dst*dst;
| |
| − | }
| |
| − | return dSq;
| |
| − | }
| |
| − | | |
| − | //
| |
| − | // class SearchStackEntry
| |
| − | //
| |
| − | // Stores a precomputed distance so we don't have to do it again
| |
| − | // when we pop the tree off the search stack.
| |
| − | | |
| − | private static class SearchStackEntry<T extends Exemplar> {
| |
| − | public final double minDFromQ;
| |
| − | public final BucketKdTree<T> tree;
| |
| − | | |
| − | public SearchStackEntry(double minDFromQ, BucketKdTree<T> tree) {
| |
| − | this.minDFromQ = minDFromQ;
| |
| − | this.tree = tree;
| |
| − | }
| |
| − | }
| |
| − | | |
| − | //
| |
| − | // class SearchState
| |
| − | //
| |
| − | // Holds data about current state of the search. Used for live updating
| |
| − | // of pruning distance.
| |
| − | | |
| − | private static class SearchState {
| |
| − | int nResults = 0;
| |
| − | double maxDistance = Double.POSITIVE_INFINITY;
| |
| − | int nExsAtMaxD = 0;
| |
| − | }
| |
| − | }
| |
| − | </pre>
| |
