peclet-morton
A small, header-only C++17 library for Morton (Z-order) codes whose distinguishing feature is arithmetic directly in Morton space.
Most Morton libraries only encode (interleave coordinates into a code) and decode (the inverse). This one adds the operations you actually want when you navigate a Z-ordered grid:
- increment / decrement / add along a single axis,
- find a neighbour,
- step to the next or previous cell in Z-order,
- sweep an axis-aligned region,
…all without the usual decode → modify → re-encode round trip. Each is a
handful of branchless integer instructions.
#include "morton/morton.hpp"
using morton::Morton;
Morton<2, 32> m = Morton<2, 32>::encode(x, y); // interleave (PDEP)
m.inc(0); // x += 1, in place, leaving y's interleaved bits untouched
m.add(1, 10); // y += 10
auto n = m.neighbor(0, -1); // the cell at (x-1, y+10)
auto [xx, yy] = m.decode(); // back to coordinates (PEXT)
Why this is faster
Holding a Morton code and needing a neighbour is extremely common: stencils, finite-difference sweeps, octree/quadtree traversal, ray marching, spatial hashing. The conventional approach decodes to coordinates, changes one, and re-encodes. This library mutates the relevant axis' bits in place.
Measured on this machine (Intel x86-64 with BMI2, g++ 14, -O3), 2D 32-bit:
| operation | this library | conventional (decode+encode) | speedup |
|---|---|---|---|
| encode | 1398 Mops/s | 1400 Mops/s (libmorton) | parity |
| decode | 2040 Mops/s | 2012 Mops/s (libmorton) | parity |
| single neighbour (+1 on an axis) | 2721 Mops/s | 1110 Mops/s | 2.5× |
| 4-neighbour stencil (scattered) | 6718 Mops/s | 2245 Mops/s | 3.0× |
Encode/decode are at parity with libmorton
(both use the hardware PDEP/PEXT instructions). The win is in the arithmetic
operations, which libmorton and similar libraries do not provide.
Honest caveat (from profiling): for a dense, contiguous full-region sweep, simply re-encoding each
(x, y)from scratch is actually faster than the arithmetic walk, because independent encodes pipeline while the arithmetic walk has a loop-carried dependency. The arithmetic advantage is for scattered / data-dependent neighbour access, where you already hold a code and cannot batch. Seedocs/EVALUATION.md.
Design
- Header-only, C++17, no dependencies.
#include "morton/morton.hpp". Morton<Dim, Bits>interleavesDimcoordinates ofBitsbits each; the code is stored in the smallest unsigned integer that fits.Dim * Bits ≤ 64uses a built-in integer; 65–128 bits use__uint128_twhere available, so 3D 32-bit (Morton3D32, 96-bit) and 2D 64-bit (Morton2D64, 128-bit) work too.- Encode/decode use BMI2
PDEP/PEXTwhen compiled with-mbmi2, with a portable software fallback otherwise. Both paths are tested for agreement. constexpr: the software path runs at compile time (selected via__builtin_is_constant_evaluated()), so you can build lookup tables inconstexpr.- Arithmetic wraps modulo
2^Bitsper axis (branchless);add_sat/sub_sat/try_add/try_subclamp or refuse instead of wrapping.
Headers
| header | provides |
|---|---|
morton/morton.hpp |
Morton<Dim,Bits>: encode/decode, axis arithmetic, saturating ops, face_neighbors/all_neighbors, ancestor/child hierarchy, Z-order ++/-- |
morton/iterate.hpp |
for_each_in_box (row-major), for_each_in_box_zorder (Z-order), and the Tropf-Herzog range-search pair bigmin_in_box / litmax_in_box |
morton/batch.hpp |
vectorised bulk add/sub/step/encode over arrays (AVX2 auto-vectorised) |
morton/wide_uint.hpp |
fixed-width word-array unsigned backing codes wider than 128 bits (pulled in automatically) |
Convenience aliases: Morton2D32, Morton2D16, Morton3D21, Morton3D16,
Morton3D32, Morton2D64. Codes wider than 128 bits (up to MORTON_MAX_BITS,
default 256) work too, e.g. Morton<3,64> (192-bit), Morton<2,128> (256-bit).
GPU (Kokkos)
A portable Kokkos backend lives in
include/morton/kokkos.hpp (morton::kokkos).
Because the core marks its functions with MORTON_HD (which defers to
KOKKOS_FUNCTION), the device kernels run the same Morton<Dim,Bits> code as
the CPU — no separate implementation — on any Kokkos backend (CUDA / HIP /
OpenMP / Serial); the backend and architecture come from the Kokkos build, not
the source. It offers Kokkos::View-based bulk ops (encode2/3, decode2/3,
per-axis add/sub/step) plus *_host raw-pointer convenience wrappers. On
an RTX 5080 (CUDA backend), 2D-32 encode hits ~51,000 Mops/s when data is
resident on the GPU (~33× one CPU core); a one-shot call on host data is
PCIe-transfer-bound and no faster than the CPU, so the GPU pays off only when
codes live on-device across a pipeline. Opt in with -DMORTON_ENABLE_KOKKOS=ON
(see DEVELOPMENT.md).
Build, test, benchmark
Full per-platform setup (incl. Python wheels and the GPU build) is in DEVELOPMENT.md. Quick start:
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build -j
ctest --test-dir build --output-on-failure # or: ./build/tests/morton_tests
./build/benchmarks/morton_bench
As a header-only dependency in another CMake project — either vendored:
add_subdirectory(morton) # this repo, e.g. as a submodule
target_link_libraries(your_target PRIVATE morton::morton)
…or installed and found via find_package (an install exports a CMake
package config; Conan recipe in conanfile.py, vcpkg port in
packaging/vcpkg/morton/):
find_package(morton CONFIG REQUIRED)
target_link_libraries(your_target PRIVATE morton::morton)
The test suite cross-checks every encode/decode against both a bit-by-bit
reference and libmorton, validates all arithmetic exhaustively on small grids
and against decode/modify/encode on large random inputs, and validates the
region iterators against brute force (>1M assertions).
Python
A vectorised NumPy interface (pure ctypes, no pybind11/native build deps
beyond a C++ compiler), published on PyPI as peclet-morton and imported as
peclet.morton (part of the peclet suite namespace). Install as a wheel
with scikit-build-core:
pip install . # builds the extension and bundles it into the wheel
python -m pytest bindings/python/tests -q
import numpy as np, peclet.morton as ma
x = np.arange(1000, dtype=np.uint32); y = x * 2
codes = ma.encode(x, y, bits=32)
shifted = ma.shift(codes, axis=0, delta=+1, dims=2, bits=32) # +1 in x, no decode
print(ma.decode(shifted, dims=2, bits=32))
For a quick dev loop without installing, drop the .so next to the package and
use PYTHONPATH:
cmake --build build --target peclet_morton_c
PYTHONPATH=bindings/python python3 -m pytest bindings/python/tests -q
Every call runs over whole arrays in compiled code. The arithmetic shift is
~3× faster than decode+encode in Python too (see bindings/python/bench_python.py).
Repository layout
include/morton/ the library: morton.hpp, iterate.hpp, batch.hpp, simd.hpp,
wide_uint.hpp, kokkos.hpp (portable GPU backend)
tests/ doctest suite (encode/decode, arithmetic, constexpr, wide,
neighbours, batch, iterate)
benchmarks/ C++ micro-benchmarks (vs libmorton) + batch/SIMD benchmark
bindings/ morton_c.h + morton_c.cpp: the C ABI (libpeclet_morton_c);
python/: the ctypes + NumPy wrapper and its pytest tests
pyproject.toml scikit-build-core wheel build + cibuildwheel config
conanfile.py Conan recipe; packaging/vcpkg/ vcpkg port
cmake/ CMake package-config template (find_package(morton))
.github/workflows/ ci.yml (build matrix) + release.yml (PyPI wheels on tag)
docs/ EVALUATION.md, ROADMAP.md, HILBERT_GPU_NOTES.md, Doxyfile
third_party/ vendored doctest and libmorton (tests/benchmarks only)
The project's original prototype (an arbitrary-width BitArray, a wide-code
Morton, and an octree built on it) is no longer in the tree; it is reachable at
the pre-legacy-removal git tag.
License
MIT — see LICENSE.
1 maintainer
Metadata
1 maintainer
Release files for peclet-morton 1.0.2
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