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<!DOCTYPE html>
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<head>
<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Sortedness Demo - About</title>
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<div class="nav-l">
<a class="nav-brand" href="index.html"><span class="m">S</span>Sortedness</a>
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<div class="wrap">
<div class="container">
<div class="row" style="text-align: center; margin-top: 100px;">
<div class="col-lg-12">
<h1>Data sortedness and how do we quantify it?</h1>
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<div class="row" style="align-items: start;">
<div class="col-md-6 hdr">
<p>
Data sortedness captures the difference between the indexed order
and the arrival order of the indexed attribute.
A simple way to quantify sortedness is to count the number of entries
that are out of order - this metric is widely known in literature as the
number of <b><b><em>inversions</em></b></b>. <br><br>
Meanwhile, the <b><b><em>K-L-sortedness</em></b></b> metric quantifies sortedness with the
help
of two parameters: <b><b>K</b></b> that counts the number of entries in incorrect order and
<b><b>L</b></b> that measures the maximum displacement of any out-of-order entry.
Additionally, the K-L-sortedness metric also uses the distribution of
sortedness defined by a (α, β)-distribution to model the variability in the
displacement of out-of-order entries. <br><br>
</p>
</div>
<div class="col-md-5 text-center" style="margin-top: 100px;margin-left: 50px;">
<img src="img/k-l-metric-visual.jpg" alt="K-L metric visualization" class="img-fluid"
style="max-height:400px; object-fit:contain;" />
<!-- <p class="small mt-2">Figure: K-L metric visualization</p> -->
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<section id="sortedness-designspace">
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<h1>Sortedness Design Space in Indexes</h1>
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<div class="row" style="align-items: start;">
<div class="col-md-5 text-center" style="margin-top: 75px;margin-right: 50px;">
<img src="img/read-write-tradeoff-intro.jpg" alt="K-L metric visualization"
class="img-fluid" style="max-height:400px; object-fit:contain;" />
<!-- <p class="small mt-2">Figure: K-L metric visualization</p> -->
</div>
<div class="col-md-6 hdr">
<p>
Every data reorganization technique, including indexing data
structures, exhibit a fundamental tradeoff between
its read and write costs, along the
<b><b>thick black line</b></b>.
To achieve <em>logarithmic search time</em>, an index must
insert entries in the correct order
(i.e., <b><b><em>in-place insertion</em></b></b>, on the bottom
right corner).
On the other hand, if queries are infrequent and scanning is
acceptable, entries can simply be appended (top-left).
Indexes like the B+tree and B-epsilon tree navigate this
read vs. write tradeoff
to offer better read or write performance.
</p><br>
<p>
A <b><b><em>sortedness-adaptive</em></b></b> index achieves better
read-write tradeoffs as well as improved space utilization by
<b><b><em></em>harnessing intrinsic data order</b></b>.
This allows the index to move closer to
the origin of the green axis (bottom left corner) when ingesting
near-sorted data.
</p>
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<h1><br><br><b>TAIL B+tree</b></h1>
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