mlpack

Octree

The Octree class implements the generalized octree, a hierarchical tree structure. When built on data in two dimensions, it is also called a ‘quadtree’. The generalized octree is most useful on data in only two or three dimensions, as its number of children is exponential in the data dimension: so, e.g., an octree node in two dimensions has up to four children; in three dimensions has up to eight; and so on.

mlpack’s Octree implementation differs from many textbook descriptions of quadtrees and octrees in that mlpack allows the bounding box for an Octree node to be a general hyperrectangle instead of a hypercube (e.g. each dimension may have a different width). This allows the Octree to fit the data points with tighter bounds for each node—very important for pruning in distance-based tasks.

For higher-dimensional use cases (e.g. data dimension of 5 or more), the KDTree, other BinarySpaceTree variants, or virtually any other type of tree will perform better.

mlpack’s Octree implementation supports three template parameters for configurable behavior, and implements all the functionality required by the TreeType API, plus some additional functionality specific to octrees.

🔗 See also

🔗 Template parameters

In accordance with the TreeType API (see also this more detailed section), the Octree class takes three template parameters:

Octree<DistanceType, StatisticType, MatType>

If no template parameters are explicitly specified, then defaults are used:

Octree<> = Octree<EuclideanDistance, EmptyStatistic, arma::mat>

🔗 Constructors

Octrees are efficiently constructed by permuting points in a dataset in a quicksort-like algorithm. However, this means that the ordering of points in the tree’s dataset (accessed with node.Dataset()) after construction may be different.

Notes:

🔗 Constructor parameters:

name type description default
data arma::mat Column-major matrix to build the tree on. Pass with std::move(data) to avoid copying the matrix. (N/A)
maxLeafSize size_t Maximum number of points to store in each leaf. 20
oldFromNew std::vector<size_t> Mappings from points in node.Dataset() to points in data. (N/A)
newFromOld std::vector<size_t> Mappings from points in data to points in node.Dataset(). (N/A)

🔗 Basic tree properties

Once an Octree object is constructed, various properties of the tree can be accessed or inspected. Many of these functions are required by the TreeType API.

🔗 Accessing members of a tree

See also the developer documentation for basic tree functionality in mlpack.

🔗 Accessing data held in a tree

🔗 Accessing computed bound quantities of a tree

The following quantities are cached for each node in an Octree, and so accessing them does not require any computation.

Notes:

🔗 Other functionality

🔗 Bounding distances with the tree

The primary use of trees in mlpack is bounding distances to points or other tree nodes. The following functions can be used for these tasks.

🔗 Tree traversals

Like every mlpack tree, the Octree class provides a single-tree and dual-tree traversal that can be paired with a RuleType class to implement a single-tree or dual-tree algorithm.

🔗 Example usage

Build an Octree on the iris dataset and print basic statistics about the tree.

// See https://datasets.mlpack.org/iris.csv.
arma::mat dataset;
mlpack::data::Load("iris.csv", dataset, true);

// Build the octree with a leaf size of 10.  (This means that nodes are split
// until they contain 10 or fewer points.)
//
// The std::move() means that `dataset` will be empty after this call, and no
// data will be copied during tree building.
//
// Note that the '<>' isn't necessary if C++20 is being used (e.g.
// `mlpack::Octree tree(...)` will work fine in C++20 or newer).
mlpack::Octree<> tree(std::move(dataset));

// Print the bounding box of the root node.
std::cout << "Bounding box of root node:" << std::endl;
for (size_t i = 0; i < tree.Bound().Dim(); ++i)
{
  std::cout << " - Dimension " << i << ": [" << tree.Bound()[i].Lo() << ", "
      << tree.Bound()[i].Hi() << "]." << std::endl;
}
std::cout << std::endl;

// Print the number of descendant points of the root, and of each of its
// children.
std::cout << "Descendant points of root:        "
    << tree.NumDescendants() << "." << std::endl;
for (size_t i = 0; i < tree.NumChildren(); ++i)
{
  std::cout << "Descendant points of child " << i << ": "
      << tree.Child(i).NumDescendants() << "." << std::endl;
}
std::cout << std::endl;

// Compute the center of the octree.
arma::vec center;
tree.Center(center);
std::cout << "Center of octree: " << center.t();

Build two Octrees on subsets of the tiny LCDM dataset (3-dimensional astronomical data) and compute minimum and maximum distances between different nodes in the tree.

// See https://datasets.mlpack.org/lcdm_tiny.csv.
arma::mat dataset;
mlpack::data::Load("lcdm_tiny.csv", dataset, true);

// Build octrees on the first half and the second half of points.
mlpack::Octree<> tree1(dataset.cols(0, dataset.n_cols / 2));
mlpack::Octree<> tree2(dataset.cols(dataset.n_cols / 2 + 1,
    dataset.n_cols - 1));

// Compute the maximum distance between the trees.
std::cout << "Maximum distance between tree root nodes: "
    << tree1.MaxDistance(tree2) << "." << std::endl;

// Get the leftmost grandchild of the first tree's root---if it exists.
if (!tree1.IsLeaf() && !tree1.Child(0).IsLeaf())
{
  mlpack::Octree<>& node1 = tree1.Child(0).Child(0);

  // Get the rightmost grandchild of the second tree's root---if it exists.
  if (!tree2.IsLeaf() && !tree2.Child(0).IsLeaf())
  {
    mlpack::Octree<>& node2 = tree2.Child(0).Child(0);

    // Print the minimum and maximum distance between the nodes.
    mlpack::Range dists = node1.RangeDistance(node2);
    std::cout << "Possible distances between two grandchild nodes: ["
        << dists.Lo() << ", " << dists.Hi() << "]." << std::endl;

    // Print the minimum distance between the first node and the first
    // descendant point of the second node.
    const size_t descendantIndex = node2.Descendant(0);
    const double descendantMinDist =
        node1.MinDistance(node2.Dataset().col(descendantIndex));
    std::cout << "Minimum distance between grandchild node and descendant "
        << "point: " << descendantMinDist << "." << std::endl;

    // Which child of node2 is closer to node1?
    const size_t closerIndex = node2.GetNearestChild(node1);
    if (closerIndex == 0)
      std::cout << "The left child of node2 is closer to node1." << std::endl;
    else if (closerIndex == 1)
      std::cout << "The right child of node2 is closer to node1." << std::endl;
    else // closerIndex == 2 in this case.
      std::cout << "Both children of node2 are equally close to node1."
          << std::endl;

    // And which child of node1 is further from node2?
    const size_t furtherIndex = node1.GetFurthestChild(node2);
    if (furtherIndex == 0)
      std::cout << "The left child of node1 is further from node2."
          << std::endl;
    else if (furtherIndex == 1)
      std::cout << "The right child of node1 is further from node2."
          << std::endl;
    else // furtherIndex == 2 in this case.
      std::cout << "Both children of node1 are equally far from node2."
          << std::endl;
  }
}

Build an Octree on 32-bit floating point data and save it to disk.

// See https://datasets.mlpack.org/iris.csv.
arma::fmat dataset;
mlpack::data::Load("iris.csv", dataset);

// Build the Octree using 32-bit floating point data as the matrix type.
// We will still use the default EmptyStatistic and EuclideanDistance
// parameters.  A leaf size of 100 is used here.
mlpack::Octree<mlpack::EuclideanDistance,
               mlpack::EmptyStatistic,
               arma::fmat> tree(std::move(dataset), 100);

// Save the Octree to disk with the name 'tree'.
mlpack::data::Save("tree.bin", "tree", tree);

std::cout << "Saved tree with " << tree.Dataset().n_cols << " points to "
    << "'tree.bin'." << std::endl;

Load a 32-bit floating point Octree from disk, then traverse it manually and find the number of leaf nodes with fewer than 10 points.

// This assumes the tree has already been saved to 'tree.bin' (as in the example
// above).

// This convenient typedef saves us a long type name!
using TreeType = mlpack::Octree<mlpack::EuclideanDistance,
                                mlpack::EmptyStatistic,
                                arma::fmat>;

TreeType tree;
mlpack::data::Load("tree.bin", "tree", tree);
std::cout << "Tree loaded with " << tree.NumDescendants() << " points."
    << std::endl;

// Recurse in a depth-first manner.  Count both the total number of leaves, and
// the number of leaves with fewer than 10 points.
size_t leafCount = 0;
size_t totalLeafCount = 0;
std::stack<TreeType*> stack;
stack.push(&tree);
while (!stack.empty())
{
  TreeType* node = stack.top();
  stack.pop();

  if (node->NumPoints() < 10)
    ++leafCount;
  ++totalLeafCount;

  if (!node->IsLeaf())
  {
    for (size_t i = 0; i < node->NumChildren(); ++i)
      stack.push(&node->Child(i));
  }
}

// Note that it would be possible to use TreeType::SingleTreeTraverser to
// perform the recursion above, but that is more well-suited for more complex
// tasks that require pruning and other non-trivial behavior; so using a simple
// stack is the better option here.

// Print the results.
std::cout << leafCount << " out of " << totalLeafCount << " leaves have fewer "
  << "than 10 points." << std::endl;

Build an Octree and map between original points and new points.

// See https://datasets.mlpack.org/lcdm_tiny.csv.
arma::mat dataset;
mlpack::data::Load("lcdm_tiny.csv", dataset, true);

// Build the tree.
std::vector<size_t> oldFromNew, newFromOld;
mlpack::Octree<> tree(dataset, oldFromNew, newFromOld);

// oldFromNew and newFromOld will be set to the same size as the dataset.
std::cout << "Number of points in dataset: " << dataset.n_cols << "."
    << std::endl;
std::cout << "Size of oldFromNew: " << oldFromNew.size() << "." << std::endl;
std::cout << "Size of newFromOld: " << newFromOld.size() << "." << std::endl;
std::cout << std::endl;

// See where point 42 in the tree's dataset came from.
std::cout << "Point 42 in the permuted tree's dataset:" << std::endl;
std::cout << "  " << tree.Dataset().col(42).t();
std::cout << "Was originally point " << oldFromNew[42] << ":" << std::endl;
std::cout << "  " << dataset.col(oldFromNew[42]).t();
std::cout << std::endl;

// See where point 7 in the original dataset was mapped.
std::cout << "Point 7 in original dataset:" << std::endl;
std::cout << "  " << dataset.col(7).t();
std::cout << "Mapped to point " << newFromOld[7] << ":" << std::endl;
std::cout << "  " << tree.Dataset().col(newFromOld[7]).t();