## Quadtree encoding (a simple compression algorithm)

### Problem 287

Published on Saturday, 10th April 2010, 09:00 am; Solved by 829The quadtree encoding allows us to describe a 2^{N}×2^{N} black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:

- the first bit deals with the complete 2
^{N}×2^{N}region; - "0" denotes a split:

the current 2^{n}×2^{n}region is divided into 4 sub-regions of dimension 2^{n-1}×2^{n-1},

the next bits contains the description of the top left, top right, bottom left and bottom right sub-regions - in that order; - "10" indicates that the current region contains only black pixels;
- "11" indicates that the current region contains only white pixels.

Consider the following 4×4 image (colored marks denote places where a split can occur):

This image can be described by several sequences, for example :**0****0**10101010**0**1011111011**0**10101010", of length 30, or

"**0**10**0**101111101110", of length 16, which is the minimal sequence for this image.

For a positive integer `N`, define `D _{N}` as the 2

^{N}×2

^{N}image with the following coloring scheme:

- the pixel with coordinates
`x`= 0,`y`= 0 corresponds to the bottom left pixel, - if (
`x`- 2^{N-1})^{2}+ (`y`- 2^{N-1})^{2}≤ 2^{2N-2}then the pixel is black, - otherwise the pixel is white.

What is the length of the minimal sequence describing `D`_{24} ?