(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

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 17 December 2004.

HTML page formatted Mon Feb 2 13:10:40 2015.

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: http://www.nist.gov/dads/HTML/quadtreecplx.html