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If I have a set of quad tree (say on a Hilbert curve) what would be a good way to approach finding the optimum (or good enough) set of ranges at particular depth.

For example, if I'm searching for points between the bounding box 0,0 and 1,3 then I can apply the following naive ranges:

  • Depth 1 - Range 0,0-1,0 (~33% search space)
  • Depth 2 - Ranges 0,0-1,0 and 1,0-0,1 (~13% search space)
  • Depth 3 - Ranges 0,0-1,0 and 1,3-0,3 (~9.8% search space)

Hilbert curves

Clearly depth 3 for this search is optimal but the reduced search space has only dropped a small amount compared to the drop from depth 1 to depth 2.

At (much) bigger depths, or with searches that cross boundaries, is there a good algorithm(s) for estimating the difference between various depths, or ideally picking a mix of ranges at different depths that ideally cover the bounding box.

I'm not interested in polygons specifically but bonus points if there is a solution that works for polygons.

sh1mmer
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    I don't get it at all. A quadtree will adapt to the topology of your points, so I can't see how you could have 3 different quadtrees for the same set of points. I also don't understand the coordinates system you are using in your 3 depth examples. Maybe you are refering to a technique so obvious that it does not need explaining, but in that case I would rather like to have a couple of keywords to feed Google with and try to educate myself. – kuroi neko Jan 23 '14 at 02:47
  • There are at least 8 differnet quad trees, so please speficy whether is it a point quadtree or a object (bounds) bases, or line bases quad? MX-quad, MX-cif, PM, PMR-quad? Is it a bucket based quad? Was the quad transformed to a linear array using the hilbert curve / index? – AlexWien Mar 28 '14 at 14:13

2 Answers2

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Althoug your question needs more details, some answers:

You can estimate the depth of a quad by log4(N).
(Take the logarithm of base 4 of the number of elements N.)

Depeending on the type of quadtree you can limit the maximum depth to that number.

The order of inserting the elements influences the structure of the quad. Pre sorting the data before inserting can improve a bit the quad structure. The type of pre sort depens on the quad. if you use a hilbert backed up quad, you could pre sort the data by hilbert index.

AlexWien
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When you use a hilbert curve it's a spatial index it's not a quadtree. A quadtree has also some limitations for example how many points you can store. So, on a hilbert curve it's better to use small tiles so the bounding box can fit nice.

Micromega
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  • It often is possible to transform a quad tree to a linear array using morton index or hilbert curve. – AlexWien Mar 28 '14 at 14:12