[hex12648430]

The big advantage of Z-order curves is that the addressing computation is very cheap, which is why it's used a lot in computer graphics.

[leni536]

While I agree that Z-order curves are simpler, but it's fast to calculate Hilbert curves on modern CPUs too. Just to self plug:

https://github.com/leni536/fast_hilbert_curve

I only implemented the index->XY calculation yet. It compiles to 36 instructions without any branches and takes up 86 bytes.

https://github.com/leni536/fast_hilbert_curve/wiki/How-effic...

I think I can apply the same tricks for the inverse function too.

[m1el]

But using the same set of instructions, z-order encoding and decoding is 8 instructions (5 if you exclude size conversion and return):

    zorder64_inv:
        movabsq $0x5555555555555555, %rax
        pextq   %rax, %rcx, %rdx
        shrq    %rcx
        pextq   %rax, %rcx, %rcx
        shlq    $32, %rcx
        movl    %edx, %eax
        orq     %rcx, %rax
        retq

    zorder64:
        movl    %ecx, %eax
        movabsq $0x5555555555555555, %r8
        pdepq   %r8, %rax, %rcx
        movl    %edx, %eax
        pdepq   %r8, %rax, %rax
        addq    %rax, %rax
        orq     %rcx, %rax
        retq

[leni536]

Nice! Now I wonder when 36 vs 8 machine instructions become a bottleneck. I have seen applications of space-filling curves in quasi Monte Carlo integration, it could be potentially significant there.

[oppositelock]

Hilbert curves are used in a lot of graphics too. Heck, the old SGI Octane with Vpro graphics used a recursive Hilbert curve rasterizer. They show up a lot today in geospatial big-data since hilbert addresses make good shard keys.

[jacobolus]

I suspect that most production applications of Hilbert curve ordering would work just as well with Z order (a.k.a. Morton order), with the additional benefit of being simpler to reason about (just interleave/de-interleave the bits).

I haven’t ever seen any convincing benchmarks or other analysis where the Hilbert curve created any notable performance advantage vs. Z order; the only time you really need it is if moving along the linearized coordinate must never have jumps in the multidimensional coordinates, but I’m not convinced there are many if any real-world cases where that is important (note that in either case small movements in the multidimensional coordinates are associated with large jumps in the linearized coordinate). If the only goal is to minimize memory fetches, etc. then the Z ordering works just fine.

(If you know any good comparisons where the Hilbert curve comes out ahead, I’d be curious to read them.)

[leni536]

There is an error-diffusion dithering algorithm based on the Hilbert-curve called Riemersma dither.

https://www.compuphase.com/riemer.htm

I suspect that error diffusion along the long jumps of a z-order curve could create strange undesired artifacts.