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Note: for updates please also check the fork by the original PH-Tree developer.

PH-Tree C++

The PH-Tree is an ordered index on an n-dimensional space (quad-/oct-/2^n-tree) where each dimension is (by default) indexed by a 64bit integer. The index order follows z-order / Morton order. The default implementation is effectively a 'map', i.e. each key is associated with at most one value. Keys are points or boxes in n-dimensional space.

Two strengths of PH-Trees are fast insert/removal operations and scalability with large datasets. It also provides fast window queries and k-nearest neighbor queries, and it scales well with higher dimensions. The default implementation is limited to 63 dimensions.

The API ist mostly analogous to STL's std::map, see function descriptions for details.

Theoretical background is listed here.

More information about PH-Trees (including a Java implementation) is available here.

User Guide

API Usage

Key Types

Basic operations



Custom Key Types


Troubleshooting / FAQ


When to use a PH-Tree

Optimising Performance

Compiling / Building

Build system & dependencies



Further Resources


API Usage

Key Types

The PH-Tree Map supports out of the box five types:

  • PhTreeD uses PhPointD keys, which are vectors/points of 64 bit double.
  • PhTreeF uses PhPointF keys, which are vectors/points of 32 bit float.
  • PhTreeBoxD uses PhBoxD keys, which consist of two PhPointD that define an axis-aligned rectangle/box.
  • PhTreeBoxF uses PhBoxF keys, which consist of two PhPointF that define an axis-aligned rectangle/box.
  • PhTree uses PhPoint keys, which are vectors/points of std::int64

The PH-Tree MultiMap supports out of the box three types:

  • PhTreeMultiMapD uses PhPointD keys, which are vectors/points of 64 bit double.
  • PhTreeMultiMapBoxD uses PhBoxD keys, which consist of two PhPointD that define an axis-aligned rectangle/box.
  • PhTreeMultiMap uses PhPoint keys, which are vectors/points of std::int64

Additional tree types can be defined easily analogous to the types above, please refer to the declaration of the tree types for an example. Support for custom key classes (points and boxes) as well as custom coordinate mappings can be implemented using custom Converter classes, see below. The PhTreeMultiMap is by default backed by std::unordered_set but this can be changed via a template parameter.

The PhTree and PhTreeMultiMap types are available from phtree.h and phtree_multimap.h.

Basic Operations

class MyData { ... };
MyData my_data; 

// Create a 3D point tree with floating point coordinates and a value type of `MyData`.
auto tree = PhTreeD<3, MyData>();

// Create coordinate
PhPointD<3> p{1.1, 1.0, 10.};

// Some operations
tree.emplace(p, my_data);
tree.emplace_hint(hint, p, my_data);
tree.insert(p, my_data);
tree[p] = my_data;

// Multi-map only
tree.relocate(p_old, p_new, value);


  • For-each over all elements: tree.fore_each(callback);
  • Iterator over all elements: auto iterator = tree.begin();
  • For-each with box shaped window queries: tree.fore_each(PhBoxD(min, max), callback);
  • Iterator for box shaped window queries: auto q = tree.begin_query(PhBoxD(min, max));
  • Iterator for k nearest neighbor queries: auto q = tree.begin_knn_query(k, center_point, distance_function);
  • Custom query shapes, such as spheres: tree.for_each(callback, FilterSphere(center, radius, tree.converter()));

For-each example
// Callback for counting entries
struct Counter {
    void operator()(PhPointD<3> key, T& t) {
    size_t n_ = 0;

// Count entries inside of an axis aligned box defined by the two points (1,1,1) and (3,3,3)
Counter callback;
tree.for_each({{1, 1, 1}, {3, 3, 3}}, callback);
// callback.n_ is now the number of entries in the box.

Iterator examples
// Iterate over all entries
for (auto it : tree) {

// Iterate over all entries inside of an axis aligned box defined by the two points (1,1,1) and (3,3,3)    
for (auto it = tree.begin_query({{1, 1, 1}, {3, 3, 3}}); it != tree.end(); ++it) {

// Find 5 nearest neighbors of (1,1,1)    
for (auto it = tree.begin_knn_query(5, {1, 1, 1}); it != tree.end(); ++it) {


All queries allow specifying an additional filter. The filter is called for every key/value pair that would normally be returned (subject to query constraints) and to every node in the tree that the query decides to traverse (also subject to query constraints). Returning true in the filter does not change query behaviour, returning false means that the current value or child node is not returned or traversed. An example of a geometric filter can be found in phtree/common/filter.h in FilterAABB.

template <dimension_t DIM, typename T>
struct FilterByValueId {
    [[nodiscard]] constexpr bool IsEntryValid(const PhPoint<DIM>& key, const T& value) const {
        // Arbitrary example: Only allow values with even values of id_
        return value.id_ % 2 == 0;
    [[nodiscard]] constexpr bool IsNodeValid(const PhPoint<DIM>& prefix, int bits_to_ignore) const {
        // Allow all nodes
        return true;

// Iterate over all entries inside of an axis aligned box defined by the two points (1,1,1) and (3,3,3).
// Return only entries that suffice the filter condition.    
for (auto it = tree.begin_query({1, 1, 1}, {3, 3, 3}, FilterByValueId<3, T>())); it != tree.end(); ++it) {

Distance function

Nearest neighbor queries can also use custom distance metrics, such as L1 distance. Note that this returns a special iterator that provides a function to get the distance of the current entry:

#include "phtree/phtree.h"

// Find 5 nearest neighbors of (1,1,1) using L1 distance    
for (auto it = tree.begin_knn_query(5, {1, 1, 1}, DistanceL1<3>())); it != tree.end(); ++it) {
    std::cout << "distance = " << it.distance() << std::endl;


The PH-Tree can internally only process integer keys. In order to use floating point coordinates, the floating point coordinates must be converted to integer coordinates. The PhTreeD and PhTreeBoxD use by default the PreprocessIEEE & PostProcessIEEE functions. The IEEE processor is a loss-less converter (in terms of numeric precision) that simply takes the 64bits of a double value and treats them as if they were a 64bit integer (it is slightly more complicated than that, see discussion in the papers referenced above). In other words, it treats the IEEE 754 representation of the double value as integer, hence the name IEEE converter.

The IEEE conversion is fast and reversible without loss of precision. However, it has been shown that other converters can result in indexes that are up to 20% faster. One useful alternative is a Multiply converter that convert floating point to integer by multiplication and casting:

double my_float = ...;
// Convert to int 
std::int64_t my_int = (std::int64_t) my_float * 1000000.;

// Convert back
double resultung_float = ((double)my_int) / 1000000.;  

It is obvious that this approach leads to a loss of numerical precision. Moreover, the loss of precision depends on the actual range of the double values and the constant. The chosen constant should probably be as large as possible but small enough such that converted values do not exceed the 64bit limit of std::int64_t. Note that the PH-Tree provides several ConverterMultiply implementations for point/box and double/float.

template <dimension_t DIM>
struct MyConverterMultiply : public ConverterPointBase<DIM, double, scalar_64_t> {
    explicit MyConverterMultiply(double multiplier)
    : multiplier_{multiplier}, divider_{1. / multiplier} {}

    [[nodiscard]] PhPoint<DIM> pre(const PhPointD<DIM>& point) const {
        PhPoint<DIM> out;
        for (dimension_t i = 0; i < DIM; ++i) {
            out[i] = point[i] * multiplier_;
        return out;

    [[nodiscard]] PhPointD<DIM> post(const PhPoint<DIM>& in) const {
        PhPointD<DIM> out;
        for (dimension_t i = 0; i < DIM; ++i) {
            out[i] = ((double)in[i]) * divider_;
        return out;

    [[nodiscard]] auto pre_query(const PhBoxD<DIM>& query_box) const {
        return PhBox{pre(query_box.min()), pre(query_box.max())};

    const double multiplier_;
    const double divider_;

template <dimension_t DIM, typename T>
using MyTree = PhTreeD<DIM, T, MyConverterMultiply<DIM>>;

void test() {
    MyConverterMultiply<3> converter{1000000};
    MyTree<3, MyData> tree(converter);
    ...  // use the tree

It is also worth trying out constants that are 1 or 2 orders of magnitude smaller or larger than this maximum value. Experience shows that this may affect query performance by up to 10%. This is due to a more compact structure of the resulting index tree.

Custom key types

With custom converters it is also possible to use your own custom classes as keys (instead of PhPointD or PhBoxF). The following example defined custom MyPoint and MyBox types and a converter that allows using them with a PhTree:

struct MyPoint {
    double x_;
    double y_;
    double z_;

using MyBox = std::pair<MyPoint, MyPoint>;

class MyConverterMultiply : public ConverterBase<3, 3, double, scalar_64_t, MyPoint, MyBox> {
    using BASE = ConverterPointBase<3, double, scalar_64_t>;
    using PointInternal = typename BASE::KeyInternal;
    using QueryBoxInternal = typename BASE::QueryBoxInternal;

    explicit MyConverterMultiply(double multiplier = 1000000)
    : multiplier_{multiplier}, divider_{1. / multiplier} {}

    [[nodiscard]] PointInternal pre(const MyPoint& point) const {
        return {static_cast<long>(point.x_ * multiplier_),
                static_cast<long>(point.y_ * multiplier_),
                static_cast<long>(point.z_ * multiplier_)};

    [[nodiscard]] MyPoint post(const PointInternal& in) const {
        return {in[0] * divider_, in[1] * divider_, in[2] * divider_};

    [[nodiscard]] QueryBoxInternal pre_query(const MyBox& box) const {
        return {pre(box.first), pre(box.second)};

    const double multiplier_;
    const double divider_;

void test() {
    MyConverterMultiply tm;
    PhTree<3, Id, MyConverterMultiply> tree(tm);
    ... // use the tree


  • C++: Supports value types of T and T*, but not T&
  • C++: Return types of find(), emplace(), ... differ slightly from std::map, they have function first() , second() instead of fields of the same name.
  • General: PH-Trees are maps, i.e. each coordinate can hold only one entry. In order to hold multiple values per coordinate please use the PhTreeMultiMap implementations.
  • General: PH-Trees order entries internally in z-order (Morton order). However, the order is based on the ( unsigned) bit representation of keys, so negative coordinates are returned after positive coordinates.
  • General: The current implementation support between 2 and 63 dimensions.
  • Differences to std::map: There are several differences to std::map. Most notably for the iterators:
    • begin()/end() are not comparable with < or >. Only it == tree.end() and it != tree.end() is supported.
    • Value of end(): The tree has no linear memory layout, so there is no useful definition of a pointer pointing _ after_ the last entry or any entry. This should be irrelevant for normal usage.

Troubleshooting / FAQ

Problem: The PH-Tree appears to be losing updates/insertions.

Solution: Remember that the PH-Tree is a map, keys will not be inserted if an identical key already exists. The easiest solution is to use one of the PhTreeMultiMap implementations. Alternatively, this can be solved by turning the PH-Tree into a multi-map, for example by using something like std::map or std::set as member type: PhTree<3, std::set<MyDataClass>>. The set instances can then be used to handle key conflicts by storing multiple entries for the same key. The logic to handle conflicts must currently be implemented manually by the user.


When to use a PH-Tree

The PH-Tree is a multi-dimensional index or spatial index. This section gives a rough overview how the PH-Tree compares to other spatial indexes, such as kD-trees, R-trees/BV-hierarchies or quadtrees.

Disclaimer: This overview cannot be comprehensive (there are 100s of spatial indexes out there) and performance depends heavily on the actual dataset, usage patterns, hardware, ... .

Generally, the PH-Tree tends to have the following advantages:

  • Fast insertion/removal times. While some indexes, such as k-D-trees, trees can be build from scratch very fast, they tend to be be much slower when removing entries or when indexing large datasets. Also, most indexes require rebalancing which may result in unpredictable latency (R-trees) or may result in index degradation if delayed (kD-trees).

  • Competitive query performance. Query performance is generally comparable to other index structures. The PH-Tree is fast at looking up coordinates but requires more traversal than other indexes. This means it is especially efficient if the query results are 'small', e.g. up to 100 results per query.

  • Scalability with large datasets. The PH-Tree's insert/remove/query performance tends to scale well to large datasets with millions of entries.

  • Scalability with the number of dimensions. The PH-Tree has been shown to deal "well" with high dimensional data ( 1000k+ dimensions). What does "well" mean?

    • It works very well for up to 30 (sometimes 50) dimensions. Please note that the C++ implementation has not been optimised nearly as much as the Java implementation.
    • For more dimensions (Java was tested with 1000+ dimensions) the PH-Tree still has excellent insertion/deletion performance. However, the query performance cannot compete with specialised high-dim indexes such as cover-trees or pyramid-trees (these tend to be very slow on insertion/deletion though).
  • Modification operations (insert/delete) in a PH-Tree are guaranteed to modify only one Node (potentially creating/deleting a second one). This guarantee can have advantages for concurrent implementations or when serializing the index. Please note that this advantage is somewhat theoretical because this guarantee is not exploited by the current implementation (it doesn't support concurrency or serialization).

PH-Tree disadvantages:

  • A PH-Tree is a map, not a multi-map. This project also provides PhTreeMultiMap implementations that store a hash-set at each coordinate. In practice, the overhead of storing sets appears to be usually small enough to not matter much.

  • PH-Trees are not very efficient in scenarios where queries tend to return large result sets in the order of 1000 or more.

Optimising Performance

There are numerous ways to improve performance. The following list gives an overview over the possibilities.

  1. Use for_each instead of iterators. This should improve performance of queries by 5%-10%.

  2. Use emplace_hint if possible. When updating the position of an entry, the naive way is to use erase() /emplace(). With emplace_hint, insertion can avoid navigation to the target node if the insertion coordinate is close to the removal coordinate.

    auto iter = tree.find(old_position);
    tree.emplace_hint(iter, new_position, value);
  3. Store pointers instead of large data objects. For example, use PhTree<3, MyLargeClass*> instead of PhTree<3, MyLargeClass> if MyLargeClass is large.

    • This prevents the PH-Tree from storing the values inside the tree. This should improve cache-locality and thus performance when operating on the tree.
    • Using pointers is also useful if construction/destruction of values is expensive. The reason is that the tree has to construct and destruct objects internally. This may be avoidable but is currently still happening.
  4. Use non-box query shapes. Depending on the use case it may be more suitable to use a custom filter for queries. For example:

    tree.for_each(callback, FilterSphere(center, radius, tree.converter()));

  5. Use a different data converter. The default converter of the PH-Tree results in a reasonably fast index. Its biggest advantage is that it provides lossless conversion from floating point coordinates to PH-Tree coordinates (integers) and back to floating point coordinates.

    • The ConverterMultiply is a lossy converter but it tends to improve performance by 10% or more. This is not caused by faster operation in the converter itself but by a more compact tree shape. The example shows how to use a converter that multiplies coordinates by 100'000, thus preserving roughly 5 fractional digits:

      PhTreeD<DIM, T, ConverterMultiply<3, 100 * 1000, 1>>

  6. Use custom key types. By default, the PH-Tree accepts only coordinates in the form of its own key types, such as PhPointD, PhBoxF or similar. To avoid conversion from custom types to PH-Tree key types, custom classes can often be adapted to be accepted directly by the PH-Tree without conversion. This requires implementing a custom converter as described in the section about Custom Key Types.

  7. Advanced: Adapt internal Node representation. Depending on the dimensionality DIM, the PH-Tree uses internally in Nodes different container types to hold entries. By default, it uses an array for DIM<=3, a vector for DIM<=8 and an ordered map for DIM>8. Adapting these thresholds can have strong effects on performance as well as memory usage. One example: Changing the threshold to use vector for DIM==3 reduced performance of the update_d benchmark by 40%-50% but improved performance of query_d by 15%-20%. The threshold is currently hardcoded.
    The effects are not always easy to predict but here are some guidelines:

    • "array" is the fastest solution for insert/update/remove type operations. Query performance is "ok". Memory consumption is O(DIM^2) for every node regardless of number of entries in the node.
    • "vector" is the fastest for queries but has for large nodes worst case O(DIM^2) insert/update/remove performance.
    • "map" scales well with DIM but is for low values of DIM generally slower than "array" or "vector".

Compiling the PH-Tree

This section will guide you through the initial build system and IDE you need to go through in order to build and run custom versions of the PH-Tree on your machine.

Build system & dependencies

PH-Tree can be built with cmake 3.14 or Bazel as build system. All code is written in C++ targeting the C++17 standard. The code has been verified to compile on Linux with Clang 9, 10, 11, 12, and GCC 9, 10, 11, and on Windows with Visual Studio 2019.

Ubuntu Linux

sudo add-apt-repository ppa:hnakamur/libarchive
sudo add-apt-repository ppa:hnakamur/libzstd
sudo add-apt-repository ppa:hnakamur/cmake
sudo apt update
sudo apt install cmake


To build on Windows, you'll need to have a version of Visual Studio 2019 installed (likely Professional), in addition to Bazel or cmake.


Once you have set up your dependencies, you should be able to build the PH-Tree repository by running:

bazel build ...

Similarly, you can run all unit tests with:

bazel test ...


mkdir build
cd build
cmake ..
cmake --build .

Further Resources


The PH-Tree is discussed in the following publications and reports:

  • T. Zaeschke, C. Zimmerli, M.C. Norrie: "The PH-Tree -- A Space-Efficient Storage Structure and Multi-Dimensional Index", (SIGMOD 2014)
  • T. Zaeschke: "The PH-Tree Revisited", (2015)
  • T. Zaeschke, M.C. Norrie: "Efficient Z-Ordered Traversal of Hypercube Indexes" (BTW 2017).