(definition)

**Definition:**
The number of nodes in a *quadtree* region representation for a simple polygon (i.e. with nonintersecting edges and without holes) is O(p+q) for a 2^{q}× 2^{q} image with perimeter p measured in pixel widths. In most cases, q is negligible, and thus, the number of nodes is proportional to the perimeter. It also holds for three-dimensional data where the perimeter is replaced by surface area, and in general for d-dimensions where instead of perimeter we have the size of the (d-1)-dimensional interfaces between the d-dimensional objects.

*Note:
From Algorithms and Theory of Computation Handbook, page 18-24, Copyright © 1999 by CRC Press LLC. Appearing in the Dictionary of Computer Science, Engineering and Technology, Copyright © 2000 CRC Press LLC.*

Author: CRC-A

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Entry modified 17 December 2004.

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Cite this as:

Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "quadtree complexity theorem", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 17 December 2004. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/quadtreecplx.html