zorder / curve index conversions

This crate provides functions to convert N-dimensional¹ coordinates to Z-order curve indexes and back. Z-order curve, also known as Morton code, is a mapping of N-dimensional coordinates to 1D index which preserves locality. It is cache-efficient way of storing N-dimensional data in 1D array.

Examples

Software implementation

use zorder::{index_of, coord_of};

// 2D coordinates
let idx = index_of([1u16, 1u16]);
assert_eq!(idx, 3u32);

let coord = coord_of(idx);
assert_eq!(coord, [1u16, 1u16]);

// 3D coordinates
let idx = index_of([1u16, 1u16, 1u16]);
assert_eq!(idx, 7u64);

let coord = coord_of(idx);
assert_eq!(coord, [1u16, 1u16, 1u16]);

bmi2 implementation

This should be faster but requires x86 specific instruction set support.

use zorder::bmi2::{coord_of, coord_of_unchecked, HardwareSupportToken, index_of, index_of_unchecked};

// Safe interface with hardware support token.
let support_token = HardwareSupportToken::new();
if let Some(support_token) = support_token {
    let idx = index_of([1u16, 1u16], support_token);
    assert_eq!(idx, 3u32);

    let coord = coord_of(idx, support_token);
    assert_eq!(coord, [1u16, 1u16]);
}

// Unsafe interface with hardware support check.
// Only works on x86_64 CPUs.
if zorder::bmi2::has_hardware_support() {
    let idx = unsafe { index_of_unchecked([1u16, 1u16]) };
    assert_eq!(idx, 3u32);

    let coord = unsafe { coord_of_unchecked(idx) };
    assert_eq!(coord, [1u16, 1u16]);
}

You can validate that your CPU supports bmi2 with the provided example:

$ cargo run --example bmi2_support

no_std

zorder supports no_std targets but std feature is enabled by default so you need to disable it:

[dependencies]
zorder = { version = "<latest>", default-features = false }

std library is needed for checking BMI2 support.

Optimization

zorder can be around 500 times slower in dev mode compared to release mode. It is therefore strongly recommended that you enable package level optimizations for zorder. Due to limitations of the Rust compiler described here, you need to increase the dev mode optimization level for your package as well.

Add the following to your Cargo.toml manifest file:

# Small amount of optimization for your package in `dev` mode.
[profile.dev]
opt-level = 1

# Compile `zorder` with high optimization.
[profile.dev.package.zorder]
opt-level = 3

Benchmarks

Below are benchmark results using two different systems; PC with AMD Ryzen 9 7950X in Ubuntu WSL2 and Raspberry Pi 5 on Raspberry Pi OS. Standard release profile was used. All results are rounded up to three significant figures.

You can run cargo bench to see the results on your machine.

Raspberry Pi 5 has non-x86_64 architecture and doesn't support BMI2, thus there are no results for those benchmarks.

Function	Dimension	Index width (bits)	7950X (ns)	Raspberry Pi 5 (ns)
index_of	2	16 (2 x 8)	2.00	4.60
		32 (2 x 16)	1.50	5.90
		64 (2 x 32)	1.32	7.28
		128 (2 x 64)	6.34	7.28
	3	32 (3 x 8)	1.77	4.12
		64 (3 x 16)	2.23	5.37
		128 (3 x 32)	6.42	21.0
coord_of	2	16 (2 x 8)	1.59	3.04
		32 (2 x 16)	1.54	3.79
		64 (2 x 32)	1.86	4.54
		128 (2 x 64)	3.90	9.29
	3	32 (3 x 8)	1.93	3.79
		64 (3 x 16)	2.36	5.72
		128 (3 x 32)	6.11	12.2
bmi2::index_of	2	16 (2 x 8)	1.03	-
		32 (2 x 16)	0.935	-
		64 (2 x 32)	0.994	-
	3	32 (3 x 8)	1.07	-
		64 (3 x 16)	5.17	-
bmi2::coord_of	2	16 (2 x 8)	0.947	-
		32 (2 x 16)	0.938	-
		64 (2 x 32)	1.13	-
	3	32 (3 x 8)	1.14	-
		64 (3 x 16)	1.14	-

License

Licensed under either of

• Apache License, Version 2.0 (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
• MIT license (LICENSE-MIT or http://opensource.org/licenses/MIT)

at your option.

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.

Footnotes

1. Maximum number of dimensions is limited by the largest unsigned integer type, u128, which is able to store 16 8-bit coordinates. bmi2 based approach is limited to u64. ↩