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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<meta property="og:title" content="patchmap"/>
<meta itemprop="thumbnailUrl" content="https://ssl.gstatic.com/atari/images/atari-logo.svg"/>
<meta itemprop="image" content="https://ssl.gstatic.com/atari/images/atari-logo.svg"/>
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<title>Hash table performance and memory efficiency</title>
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<body>
<section>
<nav>
<div class="sidebar">
<h3> <tt> sumpflager::patchmap </tt> </h3>
<p style="text-align: center;">Wolfgang Brehm</p>
<ul>
<li>
<a href="#The best hash table">
The best hash table
</a>
</li>
<br>
<li>
<a href="#The best space-time trade-off">
The best space-time trade-off
</a>
</li>
<br>
<li>
<a href="#The patchmap">
The <tt>patchmap</tt>
</a>
</li>
<br>
<li>
<a href="#Memory efficient associations">
Memory efficient associations
</a>
</li>
<br>
<li>
<a href="#Performance comparison">
Performance comparison
</a>
</li>
<br>
<li>
<a href="#The algorithm">
The algorithm
</a>
</li>
<br>
<li>
<a href="#Competing with google::sparse_hash_map">
Competing with <tt>google::sparse_hash_map</tt>
</a>
</li>
<br>
<li>
<a href="#Tombstones and zombies">Tombstones and zombies</a>
</li>
<br>
<li>
<a href="#The backstory">The backstory</a>
</li>
<br>
<li>
<a href="#Availability">Availability</a>
</li>
<br>
<li>
<a href="#Publication">Publication</a>
</li>
<br>
</ul>
</div>
<div class="content">
<h2 style="text-align: center;">Memory Efficient Hash Tables<br>
and Pseudorandom Ordering</h2>
<h2><a name="The best hash table">What makes a hash table good?</a></h3>
<p>
A hash table is a data struture commonly used to associate keys with values
to implement dictionaries or to test if a key is part of a set of keys to
implement set operations.
The best hash table is the one that enables these operations at the lowest
cost.
Cost in computing essentially is two-dimensionsional, time and space.
Keys or key-value pairs are stored at a pseudorandom position determined
first by the hash of the key and then either storing all the entries that map
to the same position in a secondary data structure
(this method is called chaining) or by a probing strategy that finds
another free position.
The first step creats a cache miss.
A cache miss is when data, that cannot be found in any of the caches of the
CPU, is requested.
The time this takes is around 300 CPU cycles and typically determines the
performance of a hash table.
It cannot be avoided due to the pseudorandom nature of hashing.
The second step however occurs more frequently the fuller the table becomes,
it is a function of the load factor α.
The load factor is defined as the number of elements stored in the table
divided by the total number of positions available in the table. Therefore,
for a given hash function and and collision resolution scheme, the larger
table is also faster because it has to resolve the less collisions,
and therefore less cache misses.
</p>
<h3><a name="Memory efficient associations">
The most memory efficient datastructure for associations
</a></h3>
<p>
The hash table with the best memory efficiency is simply the one with the
highest load factor, (it can even exceed 100% memory efficiency by using key
compression with <a href="https://arxiv.org/pdf/1905.00163.pdf">compact hashing</a> ).
A hash table like that does still provide O(1) lookups, just very slow.
</p>
<h3><a name="The best space-time trade-off">The best trade-off between memory usage and speed</a></h3>
<p>
I argue that the best hash table is not the fastest or the most memory
efficient, but the one with the best trade-off between space and time.
Asymptotically the best trade-off is the one that has the lowest product of
the the two.
Imagine building a big server that does nothing but searching entries in a
hash table.
Say the hash table is very large so that the offset of the price for the CPU
and the other hardware will be dwarfed by the cost of the RAM.
The server will have a finite life-time, therefore the cost of a single
lookup into the hash table will be directly proportional to the product of the
time it takes to access an element and the amount of space the hash table
needs per element.
Let's call this concept the space-time.
</p>
<p>
If the space-time seems too abstract let's take a common task we all probably
do from time to time:
Finding hash collisions by brute force.
By first computing as many hashes as fit in memory and then - using a hash
table - comparing to all of them at the same time with a single
lookup for each newly computed hash.
Say we can neglect the time to compute a hash, then the performance can be
expressed as the number of comparisons that can be done per unit of time.
That number is, not incidentally, proportional to the reciprocal of the
space-time.
</p>
<h2><a name="The patchmap">The <tt>patchmap</tt> </a></h2>
<p>
The <tt>patchmap</tt> is my attempt at achieving this magical smallest
space-time. Recently I was frustrated with the performance of
<tt>std::unordered_map</tt> and before I could even test other hash tables
I had come up with an idea of how to save space that I needed to try.
Let's first look at how it compares, then at the algorithm, more about the
motivation later.
</p>
<h3><a name="Performance comparison">Performance compared to other hash tables</a></h3>
<p>
I designed a small benchmark to be as fair and comprehensive as possible.
Only random keys can produce comparable memory access patterns across
different hash table implementations using different hash functions and
probing strategies, keep in mind however that this masks bad hash functions.
For the following benchmark 2²⁶ random 64 bit keys were associated with
arbitrary 64 bit values. The average memory consumption and time for
insertion, lookup and deletion per key-value pair can be found in the plots
below for the different hash tables.
</p>
<a name="Figure 1">
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<path stroke='black' d='M218.7,232.8 L218.7,222.3 M218.7,21.1 L218.7,31.6 '/> <g transform="translate(218.7,258.4)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" > 0.4</tspan></text>
</g>
</g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<path stroke='black' d='M286.1,232.8 L286.1,222.3 M286.1,21.1 L286.1,31.6 '/> <g transform="translate(286.1,258.4)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" > 0.6</tspan></text>
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</g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<path stroke='black' d='M353.4,232.8 L353.4,222.3 M353.4,21.1 L353.4,31.6 '/> <g transform="translate(353.4,258.4)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" > 0.8</tspan></text>
</g>
</g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<path stroke='black' d='M420.8,232.8 L420.8,222.3 M420.8,21.1 L420.8,31.6 '/> <g transform="translate(420.8,258.4)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" > 1</tspan></text>
</g>
</g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
</g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<path stroke='black' d='M84.0,21.1 L84.0,232.8 L420.8,232.8 L420.8,21.1 L84.0,21.1 Z '/></g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<g transform="translate(29.5,127.0) rotate(270)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" >time for successful lookup [ns]</tspan></text>
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</g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<g transform="translate(252.4,289.9)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" >memory efficiency</tspan></text>
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</g>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
</g>
<g id="gnuplot_plot_1" ><title>gnuplot_plot_1</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt3' transform='translate(325.5,132.9) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_2" ><title>gnuplot_plot_2</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt5' transform='translate(381.2,73.6) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_3" ><title>gnuplot_plot_3</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<g transform="translate(205.5,206.0)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" >⬟</tspan></text>
</g>
</g>
</g>
<g id="gnuplot_plot_4" ><title>gnuplot_plot_4</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<g transform="translate(331.8,143.1)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" >★</tspan></text>
</g>
<g transform="translate(331.8,132.4)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" >★</tspan></text>
</g>
</g>
</g>
<g id="gnuplot_plot_5" ><title>gnuplot_plot_5</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt7' transform='translate(165.0,201.3) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_6" ><title>gnuplot_plot_6</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt11' transform='translate(207.9,148.7) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_7" ><title>gnuplot_plot_7</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt6' transform='translate(285.2,182.4) scale(3.94)' color='rgb(255, 0, 0)'/>
<use xlink:href='#gpPt6' transform='translate(322.4,176.3) scale(3.94)' color='rgb(255, 0, 0)'/>
<use xlink:href='#gpPt6' transform='translate(374.2,122.0) scale(3.94)' color='rgb(255, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_8" ><title>gnuplot_plot_8</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<g transform="translate(275.7,189.3)" stroke="none" fill="rgb(0,0,255)" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" font-weight="bold" >⬠</tspan></text>
</g>
</g>
</g>
<g id="gnuplot_plot_9" ><title>gnuplot_plot_9</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt12' transform='translate(316.6,81.7) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_10" ><title>gnuplot_plot_10</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt8' transform='translate(255.0,182.6) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_11" ><title>gnuplot_plot_11</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt6' transform='translate(403.5,25.2) scale(3.94)' color='rgb( 0, 255, 0)'/>
</g>
</g>
<g id="gnuplot_plot_12" ><title>gnuplot_plot_12</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt9' transform='translate(256.9,184.2) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_13" ><title>gnuplot_plot_13</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt0' transform='translate(273.9,188.1) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_14" ><title>gnuplot_plot_14</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<use xlink:href='#gpPt1' transform='translate(257.7,171.9) scale(5.25)' color='rgb( 0, 0, 0)'/>
</g>
</g>
<g id="gnuplot_plot_15" ><title>gnuplot_plot_15</title>
<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
<g transform="translate(206.5,196.5)" stroke="none" fill="black" font-family="Helvetica" font-size="14.00" text-anchor="middle">
<text><tspan font-family="Helvetica" >▩</tspan></text>
</g>
</g>
</g>
<g id="gnuplot_plot_16" ><title>gnuplot_plot_16</title>
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<g id="gnuplot_plot_17" ><title>gnuplot_plot_17</title>
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<g id="gnuplot_plot_10" ><title>gnuplot_plot_10</title>
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<g id="gnuplot_plot_11" ><title>gnuplot_plot_11</title>
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<g id="gnuplot_plot_12" ><title>gnuplot_plot_12</title>
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<g id="gnuplot_plot_13" ><title>gnuplot_plot_13</title>
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<g id="gnuplot_plot_14" ><title>gnuplot_plot_14</title>
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<g id="gnuplot_plot_15" ><title>gnuplot_plot_15</title>
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<g id="gnuplot_plot_16" ><title>gnuplot_plot_16</title>
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<g id="gnuplot_plot_17" ><title>gnuplot_plot_17</title>
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