## Quadtree encoding (a simple compression algorithm)

### Problem 287

Published on Saturday, 10th April 2010, 09:00 am; Solved by 786

The quadtree encoding allows us to describe a 2N2N 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 2N2N region;
- "0" denotes a split:
  the current 2n2n region is divided into 4 sub-regions of dimension 2n-12n-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 44 image (colored marks denote places where a split can occur):

This image can be described by several sequences, for example : "0 01010101001011111011010101010", of length 30, or
"0100101111101110", of length 16, which is the minimal sequence for this image.

For a positive integer N, define DN as the 2N2N image with the following coloring scheme:

- the pixel with coordinates x= 0, y= 0 corresponds to the bottom left pixel,
- if (x- 2N-1)2+ (y- 2N-1)2 22N-2 then the pixel is black,
- otherwise the pixel is white.

What is the length of the minimal sequence describing D24?