60 lines
2.3 KiB
Markdown
60 lines
2.3 KiB
Markdown
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---
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id: 5900f48b1000cf542c50ff9e
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title: 'Problem 287: Quadtree encoding (a simple compression algorithm)'
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challengeType: 5
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forumTopicId: 301938
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dashedName: problem-287-quadtree-encoding-a-simple-compression-algorithm
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---
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# --description--
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The 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:
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- the first bit deals with the complete $2^N×2^N$ region;
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- "0" denotes a split:
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- the current $2^n×2^n$ region is divided into 4 sub-regions of dimension $2^{n - 1}×2^{n - 1}$,
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- the next bits contains the description of the top left, top right, bottom left and bottom right sub-regions - in that order;
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- "10" indicates that the current region contains only black pixels;
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- "11" indicates that the current region contains only white pixels.
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Consider the following 4×4 image (colored marks denote places where a split can occur):
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<img class="img-responsive center-block" alt="4x4 image with colored marks denoting place where split can occur" src="https://cdn.freecodecamp.org/curriculum/project-euler/quadtree-encoding-a-simple-compression-algorithm.gif" style="background-color: white; padding: 10px;" />
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This image can be described by several sequences, for example : "<strong><span style="color: red">0</span></strong><strong><span style="color: blue">0</span></strong>10101010<strong><span style="color: green">0</span></strong>1011111011<strong><span style="color: orange">0</span></strong>10101010", of length 30, or "<strong><span style="color: red">0</span></strong>10<strong><span style="color: green">0</span></strong>101111101110", of length 16, which is the minimal sequence for this image.
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For a positive integer $N$, define $D_N$ as the $2^N×2^N$ image with the following coloring scheme:
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- the pixel with coordinates $x = 0$, $y = 0$ corresponds to the bottom left pixel,
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- if ${(x - 2^{N - 1})}^2 + {(y - 2^{N - 1})}^2 ≤ 2^{2N - 2}$ then the pixel is black,
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- otherwise the pixel is white.
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What is the length of the minimal sequence describing $D_{24}$?
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# --hints--
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`quadtreeEncoding()` should return `313135496`.
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```js
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assert.strictEqual(quadtreeEncoding(), 313135496);
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```
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# --seed--
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## --seed-contents--
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```js
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function quadtreeEncoding() {
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return true;
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}
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quadtreeEncoding();
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```
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# --solutions--
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```js
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// solution required
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```
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