There is a lot of literature on join operations using discrete global grid systems (DGGS). H3 is a widely used DGGS optimized for visualization.

If joins are a critical performance-sensitive operation, the most important property of a DGGS is congruency. H3 is not congruent it was optimized for visualization, where congruency doesn’t matter, rather than analytical computation. For example, the article talks about deduplication, which is not even necessary with a congruent DGGS. You can do joins with H3 but it is not recommended as a general rule unless the data is small such that you can afford to brute-force it to some extent.

H3 is great for doing point geometry aggregates. It shines at that. Not so much geospatial joins though. DGGS optimized for analytic computation (and joins by implication) exist, they just aren’t optimal for trivial visualization.

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0xfaded 14 hours ago | parent | next [-]

S2 has this property https://s2geometry.io

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jandrewrogers 14 hours ago | parent [-]

Yes, S2 is a congruent DGGS. Unfortunately, it kind of straddles the analytics and visualization property space, being not particularly good at either. It made some design choices, like being projective, that limit its generality as an analytic DGGS. In fairness, its objectives were considerably more limited when it was created. The potential use cases have changed since then.

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mdasen 14 hours ago | parent [-]

Is there a congruent DGGS that you would recommend?

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jandrewrogers 13 hours ago | parent [-]

None that are well-documented publicly. There are a multitude of DGGS, often obscure, and they are often designed to satisfy specific applications. Most don’t have a public specification but they are easy to design.

If the objective is to overfit for high-performance scalable analytics, including congruency, the most capable DGGS designs are constructed by embedding a 2-spheroid in a synthetic Euclidean 3-space. The metric for the synthetic 3-space is usually defined to be both binary and as a whole multiple of meters. The main objection is that it is not an “equal area” DGGS, so not good for a pretty graphic, but it is trivially projected into it as needed so it doesn’t matter that much. The main knobs you might care about is the spatial resolution and how far the 3-space extends e.g. it is common to include low-earth orbit in the addressable space.

I was working with a few countries on standardizing one such design but we never got it over the line. There is quite a bit of literature on this, but few people read it and most of it is focused on visualization rather than analytic applications.

	▲	srean 10 hours ago \| parent [-]
		Pointers to the literature please. I don't work in this space but love geometry.

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ajfriend 14 hours ago | parent | prev | next [-]

I agree that the lack of congruency in H3 hexagons can cause weird overlaps and gaps if you plot mixed resolutions naively, but there are some workarounds that work pretty well in practice. For example, if you have mixed resolutions from compacted H3 cells but a single “logical” target resolution underneath, you can plot the coarser cells not with their native geometry, but using the outline of their children. When you do that, there are no gaps. (Totally unrelated but fun: that shape is a fractal sometimes called a "flowsnake" or a "Gosper Island" (https://en.wikipedia.org/wiki/Gosper_curve), which predates H3 by decades.)

That said, this feels like an issue with rendering geometry rather than with the index itself. I’m curious to hear more about why you think the lack of congruency affects H3’s performance for spatial joins. Under the hood, it’s still a parent–child hierarchy very similar to S2’s — H3 children are topological rather than geometric children (even though they still mostly overlap).

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jandrewrogers 13 hours ago | parent [-]

In a highly optimized system, spatial joins are often bandwidth bound.

Congruency allows for much more efficient join schedules and maximizes selectivity. This minimizes data motion, which is particularly important as data becomes large. Congruent shards also tend to be more computationally efficient generally, which does add up.

The other important aspect not raised here, is that congruent DGGS have much more scalable performance when using them to build online indexes during ingestion. This follows from them being much more concurrency friendly.

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ajfriend 11 hours ago | parent [-]

I appreciate the reply! So, I might be wrong here, but I think we may be talking about two different layers. I’m also not very familiar with the literature, so I’d be interested if you could point me to relevant work or explain where my understanding is off.

To me, the big selling point of H3 is that once you’re "in the H3 system", many operations don’t need to worry about geometry at all. Everything is discrete. H3 cells are nodes in a tree with prefixes that can be exploited, and geometry or congruency never really enter the picture at this layer.

Where geometry and congruency do come in is when you translate continuous data (points, polygons, and so on) into H3. In that scenario, I can totally see congruency being a useful property for speed, and that H3 is probably slower than systems that are optimized for that conversion step.

However, in most applications I’ve seen, the continuous-to-H3 conversion happens upstream, or at least isn’t the bottleneck. The primary task is usually operating on already "hexagonified" data, such as joins or other set operations on discrete cell IDs.

Am I understanding the bottleneck correctly?

	▲	jandrewrogers an hour ago \| parent [-]
		A DGGS is a specialized spatial unit system that, depending on the design, allows you to elide expensive computation for a select set of operations with the tradeoff that other operations become much more expensive. H3 is optimized for equal-area point aggregates. Congruency does not matter for these aggregates because there is only a single resolution. To your point, in H3 these are implemented as simple scalar counting aggregates -- little computational geometry required. Optimized implementations can generate these aggregates more or less at the speed of memory bandwidth. Ideal for building heat maps! H3 works reasonably for sharding spatial joins if all of the cell resolutions have the same size and are therefore disjoint. The number of records per cell can be highly variable so this is still suboptimal; adjusting the cell size to get better distribution just moves the suboptimality around. There is also the complexity if polygon data is involved. The singular importance of congruence as a property is that it enables efficient and uniform sharding of spatial data for distributed indexes, regardless of data distribution or geometry size. The practical benefits follow from efficient and scalable computation over data stored in cells of different size, especially for non-point geometry. Some DGGS optimized for equal-area point aggregates are congruent, such as HEALPix[0]. However, that congruency comes at high computational cost and unreasonably difficult technical implementation. Not recommended for geospatial use cases. Congruence has an important challenge that most overlook: geometric relationships on a 2-spheroid can only be approximated on a discrete computer. If you are not careful, quantization to the discrete during computation can effectively create tiny gaps between cells or tiny slivers of overlap. I've seen bugs in the wild from when the rare point lands in one of these non-congruent slivers. Mitigating this can be costly. This is how we end up with DGGS that embed the 2-spheroid in a synthetic Euclidean 3-space. Quantization issues on the 2-spheroid become trivial in 3-space. People tend to hate two things about these DGGS designs though, neither of which is a technical critique. First, these are not equal area designs like H3; cell size does not indicate anything about the area on the 2-sphere. Since they are efficiently congruent, the resolution can be locally scaled as needed so there are no technical ramifications. It just isn't intuitive like tiling a map or globe. Second, if you do project the cell boundaries onto the 2-sphere and then project that geometry into something like Web Mercator for visualization, it looks like some kind of insane psychedelic hallucination. These cells are designed for analytic processing, not visualization; the data itself is usually WGS84 and can be displayed in exactly the same way you would if you were using PostGIS, the DGGS just doesn't act as a trivial built-in visualization framework. Taking data stored in a 3-space embedding and converting it to H3-ified data or aggregates on demand is simple, efficient, and highly scalable. I often do things this way even when the data will only ever be visualized in H3 because it scales better. [0] https://en.wikipedia.org/wiki/HEALPix

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TacticalCoder 12 hours ago | parent | prev [-]

> If joins are a critical performance-sensitive operation, the most important property of a DGGS is congruency.

Not familiar with geo stuff / DGGS. Is H3 not congruent because hexagons, unlike squares or triangles, do not tile the plane perfectly?

I mean: could a system using hexagons ever be congruent?

	▲	urschrei 8 hours ago \| parent [-]
		Hexagons do tile the Euclidean plane perfectly. They are the largest of the three n-gons that do so.