KmTree.cpp

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/* +------------------------------------------------------------------------+
   |                     Mobile Robot Programming Toolkit (MRPT)            |
   |                          https://www.mrpt.org/                         |
   |                                                                        |
   | Copyright (c) 2005-2020, Individual contributors, see AUTHORS file     |
   | See: https://www.mrpt.org/Authors - All rights reserved.               |
   | Released under BSD License. See: https://www.mrpt.org/License          |
   +------------------------------------------------------------------------+ */
// See KmTree.cpp
//
// Author: David Arthur (darthur@gmail.com), 2009

// Includes
#include "KmTree.h"
#include <cstdlib>
#include <iostream>
using namespace std;

KmTree::KmTree(int n, int d, Scalar* points) : n_(n), d_(d), points_(points)
{
    // Initialize memory
    int node_size = sizeof(Node) + d_ * 3 * sizeof(Scalar);
    node_data_ = (char*)malloc((2 * n - 1) * node_size);
    point_indices_ = (int*)malloc(n * sizeof(int));
    for (int i = 0; i < n; i++) point_indices_[i] = i;
    KM_ASSERT(node_data_ != nullptr && point_indices_ != nullptr);

    // Calculate the bounding box for the points
    Scalar* bound_v1 = PointAllocate(d_);
    Scalar* bound_v2 = PointAllocate(d_);
    KM_ASSERT(bound_v1 != nullptr && bound_v2 != nullptr);
    PointCopy(bound_v1, points, d_);
    PointCopy(bound_v2, points, d_);
    for (int i = 1; i < n; i++)
        for (int j = 0; j < d; j++)
        {
            if (bound_v1[j] > points[i * d_ + j])
                bound_v1[j] = points[i * d_ + j];
            if (bound_v2[j] < points[i * d_ + j])
                bound_v1[j] = points[i * d_ + j];
        }

    // Build the tree
    char* temp_node_data = node_data_;
    top_node_ = BuildNodes(points, 0, n - 1, &temp_node_data);

    // Cleanup
    PointFree(bound_v1);
    PointFree(bound_v2);
}

KmTree::~KmTree()
{
    free(point_indices_);
    free(node_data_);
}

Scalar KmTree::DoKMeansStep(int k, Scalar* centers, int* assignment) const
{
    // Create an invalid center for comparison purposes
    Scalar* bad_center = PointAllocate(d_);
    KM_ASSERT(bad_center != nullptr);
    memset(bad_center, 0xff, d_ * sizeof(Scalar));

    // Allocate data
    auto* sums = (Scalar*)calloc(k * d_, sizeof(Scalar));
    int* counts = (int*)calloc(k, sizeof(int));
    int num_candidates = 0;
    int* candidates = (int*)malloc(k * sizeof(int));
    KM_ASSERT(sums != nullptr && counts != nullptr && candidates != nullptr);
    for (int i = 0; i < k; i++)
        if (memcmp(centers + i * d_, bad_center, d_ * sizeof(Scalar)) != 0)
            candidates[num_candidates++] = i;

    // Find nodes
    Scalar result = DoKMeansStepAtNode(
        top_node_, num_candidates, candidates, centers, sums, counts,
        assignment);

    // Set the new centers
    for (int i = 0; i < k; i++)
    {
        if (counts[i] > 0)
        {
            PointScale(sums + i * d_, Scalar(1) / counts[i], d_);
            PointCopy(centers + i * d_, sums + i * d_, d_);
        }
        else
        {
            memcpy(centers + i * d_, bad_center, d_ * sizeof(Scalar));
        }
    }

    // Cleanup memory
    PointFree(bad_center);
    free(candidates);
    free(counts);
    free(sums);
    return result;
}

// Helper functions for constructor
// ================================

// Build a kd tree from the given set of points
KmTree::Node* KmTree::BuildNodes(
    Scalar* points, int first_index, int last_index, char** next_node_data)
{
    // Allocate the node
    Node* node = (Node*)(*next_node_data);
    (*next_node_data) += sizeof(Node);
    node->sum = (Scalar*)(*next_node_data);
    (*next_node_data) += sizeof(Scalar) * d_;
    node->median = (Scalar*)(*next_node_data);
    (*next_node_data) += sizeof(Scalar) * d_;
    node->radius = (Scalar*)(*next_node_data);
    (*next_node_data) += sizeof(Scalar) * d_;

    // Fill in basic info
    node->num_points = (last_index - first_index + 1);
    node->first_point_index = first_index;

    // Calculate the bounding box
    Scalar* first_point = points + point_indices_[first_index] * d_;
    Scalar* bound_p1 = PointAllocate(d_);
    Scalar* bound_p2 = PointAllocate(d_);
    KM_ASSERT(bound_p1 != nullptr && bound_p2 != nullptr);
    PointCopy(bound_p1, first_point, d_);
    PointCopy(bound_p2, first_point, d_);
    for (int i = first_index + 1; i <= last_index; i++)
        for (int j = 0; j < d_; j++)
        {
            Scalar c = points[point_indices_[i] * d_ + j];
            if (bound_p1[j] > c) bound_p1[j] = c;
            if (bound_p2[j] < c) bound_p2[j] = c;
        }

    // Calculate bounding box stats and delete the bounding box memory
    Scalar max_radius = -1;
    int split_d = -1;
    for (int j = 0; j < d_; j++)
    {
        node->median[j] = (bound_p1[j] + bound_p2[j]) / 2;
        node->radius[j] = (bound_p2[j] - bound_p1[j]) / 2;
        if (node->radius[j] > max_radius)
        {
            max_radius = node->radius[j];
            split_d = j;
        }
    }
    PointFree(bound_p2);
    PointFree(bound_p1);

    // If the max spread is 0, make this a leaf node
    if (max_radius == 0)
    {
        node->lower_node = node->upper_node = nullptr;
        PointCopy(node->sum, first_point, d_);
        if (last_index != first_index)
            PointScale(node->sum, Scalar(last_index - first_index + 1), d_);
        node->opt_cost = 0;
        return node;
    }

    // Partition the points around the midpoint in this dimension. The
    // partitioning is done in-place
    // by iterating from left-to-right and right-to-left in the same way that
    // partioning is done for
    // quicksort.
    Scalar split_pos = node->median[split_d];
    int i1 = first_index, i2 = last_index, size1 = 0;
    while (i1 <= i2)
    {
        bool is_i1_good =
            (points[point_indices_[i1] * d_ + split_d] < split_pos);
        bool is_i2_good =
            (points[point_indices_[i2] * d_ + split_d] >= split_pos);
        if (!is_i1_good && !is_i2_good)
        {
            int temp = point_indices_[i1];
            point_indices_[i1] = point_indices_[i2];
            point_indices_[i2] = temp;
            is_i1_good = is_i2_good = true;
        }
        if (is_i1_good)
        {
            i1++;
            size1++;
        }
        if (is_i2_good)
        {
            i2--;
        }
    }

    // Create the child nodes
    KM_ASSERT(size1 >= 1 && size1 <= last_index - first_index);
    node->lower_node = BuildNodes(
        points, first_index, first_index + size1 - 1, next_node_data);
    node->upper_node =
        BuildNodes(points, first_index + size1, last_index, next_node_data);

    // Calculate the new sum and opt cost
    PointCopy(node->sum, node->lower_node->sum, d_);
    PointAdd(node->sum, node->upper_node->sum, d_);
    Scalar* center = PointAllocate(d_);
    KM_ASSERT(center != nullptr);
    PointCopy(center, node->sum, d_);
    PointScale(center, Scalar(1) / node->num_points, d_);
    node->opt_cost = GetNodeCost(node->lower_node, center) +
                     GetNodeCost(node->upper_node, center);
    PointFree(center);
    return node;
}

// Returns the total contribution of all points in the given kd-tree node,
// assuming they are all
// assigned to a center at the given location. We need to return:
//
//   sum_{x \in node} ||x - center||^2.
//
// If c denotes the center of mass of the points in this node and n denotes the
// number of points in
// it, then this quantity is given by
//
//   n * ||c - center||^2 + sum_{x \in node} ||x - c||^2
//
// The sum is precomputed for each node as opt_cost. This formula follows from
// expanding both sides
// as dot products. See Kanungo/Mount for more info.
Scalar KmTree::GetNodeCost(const Node* node, Scalar* center) const
{
    Scalar dist_sq = 0;
    for (int i = 0; i < d_; i++)
    {
        Scalar x = (node->sum[i] / node->num_points) - center[i];
        dist_sq += x * x;
    }
    return node->opt_cost + node->num_points * dist_sq;
}

// Helper functions for DoKMeans step
// ==================================

// A recursive version of DoKMeansStep. This determines which clusters all
// points that are rooted
// node will be assigned to, and updates sums, counts and assignment (if not
// null) accordingly.
// candidates maintains the set of cluster indices which could possibly be the
// closest clusters
// for points in this subtree.
Scalar KmTree::DoKMeansStepAtNode(
    const Node* node, int k, int* candidates, Scalar* centers, Scalar* sums,
    int* counts, int* assignment) const
{
    // Determine which center the node center is closest to
    Scalar min_dist_sq =
        PointDistSq(node->median, centers + candidates[0] * d_, d_);
    int closest_i = candidates[0];
    for (int i = 1; i < k; i++)
    {
        Scalar dist_sq =
            PointDistSq(node->median, centers + candidates[i] * d_, d_);
        if (dist_sq < min_dist_sq)
        {
            min_dist_sq = dist_sq;
            closest_i = candidates[i];
        }
    }

    // If this is a non-leaf node, recurse if necessary
    if (node->lower_node != nullptr)
    {
        // Build the new list of candidates
        int new_k = 0;
        int* new_candidates = (int*)malloc(k * sizeof(int));
        KM_ASSERT(new_candidates != nullptr);
        for (int i = 0; i < k; i++)
            if (!ShouldBePruned(
                    node->median, node->radius, centers, closest_i,
                    candidates[i]))
                new_candidates[new_k++] = candidates[i];

        // Recurse if there's at least two
        if (new_k > 1)
        {
            Scalar result = DoKMeansStepAtNode(
                                node->lower_node, new_k, new_candidates,
                                centers, sums, counts, assignment) +
                            DoKMeansStepAtNode(
                                node->upper_node, new_k, new_candidates,
                                centers, sums, counts, assignment);
            free(new_candidates);
            return result;
        }
        else
        {
            free(new_candidates);
        }
    }

    // Assigns all points within this node to a single center
    PointAdd(sums + closest_i * d_, node->sum, d_);
    counts[closest_i] += node->num_points;
    if (assignment != nullptr)
    {
        for (int i = node->first_point_index;
             i < node->first_point_index + node->num_points; i++)
            assignment[point_indices_[i]] = closest_i;
    }
    return GetNodeCost(node, centers + closest_i * d_);
}

// Determines whether every point in the box is closer to centers[best_index]
// than to
// centers[test_index].
//
// If x is a point, c_0 = centers[best_index], c = centers[test_index], then:
//       (x-c).(x-c) < (x-c_0).(x-c_0)
//   <=> (c-c_0).(c-c_0) < 2(x-c_0).(c-c_0)
//
// The right-hand side is maximized for a vertex of the box where for each
// dimension, we choose
// the low or high value based on the sign of x-c_0 in that dimension.
bool KmTree::ShouldBePruned(
    Scalar* box_median, Scalar* box_radius, Scalar* centers, int best_index,
    int test_index) const
{
    if (best_index == test_index) return false;

    Scalar* best = centers + best_index * d_;
    Scalar* test = centers + test_index * d_;
    Scalar lhs = 0, rhs = 0;
    for (int i = 0; i < d_; i++)
    {
        Scalar component = test[i] - best[i];
        lhs += component * component;
        if (component > 0)
            rhs += (box_median[i] + box_radius[i] - best[i]) * component;
        else
            rhs += (box_median[i] - box_radius[i] - best[i]) * component;
    }
    return (lhs >= 2 * rhs);
}

Scalar KmTree::SeedKMeansPlusPlus(int k, Scalar* centers) const
{
    auto* dist_sq = (Scalar*)malloc(n_ * sizeof(Scalar));
    KM_ASSERT(dist_sq != nullptr);

    // Choose an initial center uniformly at random
    SeedKmppSetClusterIndex(top_node_, 0);
    int i = GetRandom(n_);
    memcpy(centers, points_ + point_indices_[i] * d_, d_ * sizeof(Scalar));
    Scalar total_cost = 0;
    for (int j = 0; j < n_; j++)
    {
        dist_sq[j] = PointDistSq(points_ + point_indices_[j] * d_, centers, d_);
        total_cost += dist_sq[j];
    }

    // Repeatedly choose more centers
    for (int new_cluster = 1; new_cluster < k; new_cluster++)
    {
        while (true)
        {
            Scalar cutoff = (rand() / Scalar(RAND_MAX)) * total_cost;
            Scalar cur_cost = 0;
            for (i = 0; i < n_; i++)
            {
                cur_cost += dist_sq[i];
                if (cur_cost >= cutoff) break;
            }
            if (i < n_) break;
        }
        memcpy(
            centers + new_cluster * d_, points_ + point_indices_[i] * d_,
            d_ * sizeof(Scalar));
        total_cost =
            SeedKmppUpdateAssignment(top_node_, new_cluster, centers, dist_sq);
    }

    // Clean up and return
    free(dist_sq);
    return total_cost;
}

// Helper functions for SeedKMeansPlusPlus
// =======================================

// Sets kmpp_cluster_index to 0 for all nodes
void KmTree::SeedKmppSetClusterIndex(const Node* node, int value) const
{
    node->kmpp_cluster_index = value;
    if (node->lower_node != nullptr)
    {
        SeedKmppSetClusterIndex(node->lower_node, value);
        SeedKmppSetClusterIndex(node->upper_node, value);
    }
}

Scalar KmTree::SeedKmppUpdateAssignment(
    const Node* node, int new_cluster, Scalar* centers, Scalar* dist_sq) const
{
    // See if we can assign all points in this node to one cluster
    if (node->kmpp_cluster_index >= 0)
    {
        if (ShouldBePruned(
                node->median, node->radius, centers, node->kmpp_cluster_index,
                new_cluster))
            return GetNodeCost(node, centers + node->kmpp_cluster_index * d_);
        if (ShouldBePruned(
                node->median, node->radius, centers, new_cluster,
                node->kmpp_cluster_index))
        {
            SeedKmppSetClusterIndex(node, new_cluster);
            for (int i = node->first_point_index;
                 i < node->first_point_index + node->num_points; i++)
                dist_sq[i] = PointDistSq(
                    points_ + point_indices_[i] * d_,
                    centers + new_cluster * d_, d_);
            return GetNodeCost(node, centers + new_cluster * d_);
        }

        // It may be that the a leaf-node point is equidistant from the new
        // center or old
        if (node->lower_node == nullptr)
            return GetNodeCost(node, centers + node->kmpp_cluster_index * d_);
    }

    // Recurse
    Scalar cost = SeedKmppUpdateAssignment(
                      node->lower_node, new_cluster, centers, dist_sq) +
                  SeedKmppUpdateAssignment(
                      node->upper_node, new_cluster, centers, dist_sq);
    int i1 = node->lower_node->kmpp_cluster_index,
        i2 = node->upper_node->kmpp_cluster_index;
    if (i1 == i2 && i1 != -1)
        node->kmpp_cluster_index = i1;
    else
        node->kmpp_cluster_index = -1;
    return cost;
}