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erasure_code_demo.html
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<title>Erasure codes</title>
</head>
<body>
<script src="https://unpkg.com/[email protected]"></script>
<script src="https://cdn.rawgit.com/jasondavies/jsbn/v1.4/jsbn.js"></script>
<script src="https://cdn.rawgit.com/jasondavies/jsbn/v1.4/jsbn2.js"></script>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.9.0/katex.min.css" integrity="sha384-TEMocfGvRuD1rIAacqrknm5BQZ7W7uWitoih+jMNFXQIbNl16bO8OZmylH/Vi/Ei" crossorigin="anonymous">
<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.9.0/katex.min.js" integrity="sha384-jmxIlussZWB7qCuB+PgKG1uLjjxbVVIayPJwi6cG6Zb4YKq0JIw+OMnkkEC7kYCq" crossorigin="anonymous"></script>
<style>
.katex { font-size: 1.0em; }
</style>
<script src="arithmetic.js"></script>
<script src="math.js"></script>
<script src="field_257.js"></script>
<script src="carryless.js"></script>
<script src="field_256.js"></script>
<script src="rational.js"></script>
<script src="matrix.js"></script>
<script src="cauchy_erasure_code.js"></script>
<script src="demo_common.js"></script>
<script src="erasure_code_demo_common.js"></script>
<script src="cauchy_matrix_demo.js"></script>
<script src="row_reduce.js"></script>
<script src="compute_parity_demo.js"></script>
<script src="reconstruct_data_demo.js"></script>
<style>
input:invalid { color: red; }
</style>
<div>
<h3>Example 1: <code>ComputeParity</code></h3>
Let
<span style="white-space: nowrap;">
<var>d</var> = [ da, db, 0d ]
</span>
be the input data bytes and let
<span style="white-space: nowrap;">
<var>m</var> = 2
</span>
be the desired parity byte count. Then the output parity bytes
are
<span style="white-space: nowrap;">
<var>p</var> = [ <span style="color: red;">52</span>, <span style="color: red;">0c</span> ].
</span>
</div>
<div id="computeParityDemo">
</div>
<script>
'use strict';
(function() {
const { h, render } = window.preact;
const root = document.getElementById('computeParityDemo');
render(h(ComputeParityDemo, {
initialD: 'da, db, 0d', initialM: '2',
name: 'computeParityDemo',
detailed: false,
header: h('h3', null, 'Interactive Example 1: ', h('code', null, 'ComputeParity')),
resultColor: 'red',
}), root.parent, root);
})();
</script>
<div>
<h3>Example 2: <code>ComputeParity</code> in detail</h3>
Let
<span style="white-space: nowrap;">
<var>d</var> = [ da, db, 0d ]
</span>
be the input data bytes and let
<span style="white-space: nowrap;">
<var>m</var> = 2
</span>
be the desired parity byte count. Then, with the input byte
count
<span style="white-space: nowrap;">
<var>n</var> = 3,
</span>
the
<span style="white-space: nowrap;">
<var>m</var> × <var>n</var>
</span>
Cauchy parity matrix computed using
<span style="white-space: nowrap;">
<var>x</var><sub>i</sub> = <var>n</var> + <var>i</var>
</span>
and
<span style="white-space: nowrap;">
<var>y</var><sub>i</sub> = <var>i</var>
</span>
is
<pre>/ f6 8d 01 \
\ cb 52 7b /.</pre>
Therefore, the parity bytes are computed as
<pre> _ _ _ _
/ f6 8d 01 \ | da | | 52 |
\ cb 52 7b / * | db | = |_ 0c _|,
|_ 0d _|</pre>
and thus the output parity bytes are
<span style="white-space: nowrap;">
<var>p</var> = [ <span style="color: red;">52</span>, <span style="color: red;">0c</span> ].
</span>
</div>
<div id="computeParityDetailDemo">
</div>
<script>
'use strict';
(function() {
const { h, render } = window.preact;
const root = document.getElementById('computeParityDetailDemo');
render(h(ComputeParityDemo, {
initialD: 'da, db, 0d', initialM: '2',
name: 'computeParityDemo',
detailed: true,
header: h('h3', null, 'Interactive Example 2: ', h('code', null, 'ComputeParity'), ' in detail'),
resultColor: 'red',
}), root.parent, root);
})();
</script>
<div>
<h3>Example 3: <code>ReconstructData</code></h3>
Let
<span style="white-space: nowrap;">
<var>d</var><sub>partial</sub> = [ ??, db, ?? ]
</span>
be the input partial data bytes and
<span style="white-space: nowrap;">
<var>p</var><sub>partial</sub> = [ 52, 0c ]
</span>
be the input partial parity bytes. Then the output data bytes are
<span style="white-space: nowrap;">
<var>d</var> = [ <span style="color: red;">da</span>, <span style="color: red;">db</span>, <span style="color: red;">0d</span> ].
</span>
</div>
<div id="reconstructDataDemo">
</div>
<script>
'use strict';
(function() {
const { h, render } = window.preact;
const root = document.getElementById('reconstructDataDemo');
render(h(ReconstructDataDemo, {
initialPartialD: '??, db, ??', initialPartialP: '52, 0c',
name: 'reconstructDataDemo',
detailed: false,
header: h('h3', null, 'Interactive Example 3: ', h('code', null, 'ReconstructData')),
resultColor: 'red',
}), root.parent, root);
})();
</script>
<div>
<h3>Example 4: <code>ReconstructData</code> in detail</h3>
Let
<span style="white-space: nowrap;">
<var>d</var><sub>partial</sub> = [ ??, db, ?? ]
</span>
be the input partial data bytes and
<span style="white-space: nowrap;">
<var>p</var><sub>partial</sub> = [ 52, 0c ]
</span>
be the input partial parity bytes. Then, with the data byte
count
<span style="white-space: nowrap;">
<var>n</var> = 3
</span>
and the parity byte count
<span style="white-space: nowrap;">
<var>m</var> = 2,
</span>
and appending the rows of the
<span style="white-space: nowrap;">
<var>m</var> × <var>n</var>
</span>
Cauchy parity matrix to the
<span style="white-space: nowrap;">
<var>n</var> × <var>n</var>
</span>
identity matrix, we get
<pre>/ X01X X00X X00X \
| 00 01 00 |
| X00X X00X X01X |
| f6 8d 01 |
\ cb 52 7b /,</pre>
where the rows corresponding to the unknown data and parity
bytes are crossed out. Taking the first <var>n</var> rows that
aren’t crossed out, we get the square matrix
<pre>/ 00 01 00 \
| f6 8d 01 |
\ cb 52 7b /</pre>
which has inverse
<pre>/ 01 d0 d6 \
| 01 00 00 |
\ 7b b8 bb /.</pre>
Therefore, the data bytes are reconstructed from the first
<var>n</var> known data and parity bytes as
<pre> _ _ _ _
/ 01 d0 d6 \ | db | | da |
| 01 00 00 | * | 52 | = | db |
\ 7b b8 bb / |_ 0c _| |_ 0d _|,</pre>
and thus the output data bytes are
<span style="white-space: nowrap;">
<var>d</var> = [ <span style="color: red;">da</span>, <span style="color: red;">db</span>, <span style="color: red;">0d</span> ].
</span>
</div>
<div id="reconstructDataDetailDemo">
</div>
<script>
'use strict';
(function() {
const { h, render } = window.preact;
const root = document.getElementById('reconstructDataDetailDemo');
render(h(ReconstructDataDemo, {
initialPartialD: '??, db, ??', initialPartialP: '52, 0c',
name: 'reconstructDataDemo',
detailed: true,
header: h('h3', null, 'Interactive Example 4: ', h('code', null, 'ReconstructData'), ' in detail'),
resultColor: 'red',
}), root.parent, root);
})();
</script>
</body>
</html>