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<div class="section" id="efficient-storage-of-data-in-memory">
<h1>Efficient storage of data in memory<a class="headerlink" href="#efficient-storage-of-data-in-memory" title="Permalink to this headline">¶</a></h1>
<p>When dealing with big data, minimizing the amount of memory used is
critical to avoid having to use disk-based access, which can be 100,000
times slower than random access. This notebook deals with ways to
minimizee data storage for several common use cases:</p>
<ul class="simple">
<li>Large arrays of homogenous data (often numbers)</li>
<li>Large string collections</li>
<li>Counting distinct values</li>
<li>Yes/No responses to queries</li>
</ul>
<p>Methods covered range from the mundane (use <code class="docutils literal"><span class="pre">numpy</span></code> arrays rather than
lists), to classic but less well-known data structures (e.g. prefix
trees or tries) to algorithmically ingenious probabilistic data
structures (e.g. bloom filter and hyperloglog).</p>
<div class="section" id="selective-retrieval-from-disk-based-storage">
<h2>Selective retrieval from disk-based storage<a class="headerlink" href="#selective-retrieval-from-disk-based-storage" title="Permalink to this headline">¶</a></h2>
<p>We have alrady seen that there are many ways to retrieve only the parts
of the data we need now into memory at this particular moment. Options
include</p>
<ul class="simple">
<li>generators (e.g. to read a file a line at a time)</li>
<li><code class="docutils literal"><span class="pre">numpy.memmap</span></code></li>
<li>HDF5 via <code class="docutils literal"><span class="pre">h5py</span></code></li>
<li>Key-value stores (e.g. <code class="docutils literal"><span class="pre">redis</span></code>)</li>
<li>SQL and NoSQL databases (e.g. <code class="docutils literal"><span class="pre">sqlite3</span></code>)</li>
</ul>
</div>
<div class="section" id="storing-numbers">
<h2>Storing numbers<a class="headerlink" href="#storing-numbers" title="Permalink to this headline">¶</a></h2>
<p>Less memory is used when storing numbers in numpy arrays rather than
lists.</p>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">sys</span><span class="o">.</span><span class="n">getsizeof</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="mi">1</span><span class="n">e8</span><span class="p">))))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">900000112</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="mi">1</span><span class="n">e8</span><span class="p">))</span><span class="o">.</span><span class="n">nbytes</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">800000000</span>
</pre></div>
</div>
<div class="section" id="using-only-the-precision-needed-can-also-save-memory">
<h3>Using only the precision needed can also save memory<a class="headerlink" href="#using-only-the-precision-needed-can-also-save-memory" title="Permalink to this headline">¶</a></h3>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="mi">1</span><span class="n">e8</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="s1">'float32'</span><span class="p">)</span><span class="o">.</span><span class="n">nbytes</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">400000000</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="mi">1</span><span class="n">e8</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="s1">'float64'</span><span class="p">)</span><span class="o">.</span><span class="n">nbytes</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">800000000</span>
</pre></div>
</div>
</div>
</div>
<div class="section" id="storing-strings">
<h2>Storing strings<a class="headerlink" href="#storing-strings" title="Permalink to this headline">¶</a></h2>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">flatmap</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">items</span><span class="p">):</span>
<span class="k">return</span> <span class="n">it</span><span class="o">.</span><span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">items</span><span class="p">))</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">xss</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">xss</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">)</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'data/Ulysses.txt'</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
<span class="n">word_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">flatten</span><span class="p">(</span><span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="p">()</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">f</span><span class="p">))</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">sys</span><span class="o">.</span><span class="n">getsizeof</span><span class="p">(</span><span class="n">word_list</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">2258048</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">target</span> <span class="o">=</span> <span class="s1">'WARRANTIES'</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">timeit</span> <span class="o">-</span><span class="n">r1</span> <span class="o">-</span><span class="n">n1</span> <span class="n">word_list</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">target</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="n">loops</span><span class="p">,</span> <span class="n">best</span> <span class="n">of</span> <span class="mi">1</span><span class="p">:</span> <span class="mf">7.97</span> <span class="n">ms</span> <span class="n">per</span> <span class="n">loop</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">word_list</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">bisect</span>
<span class="o">%</span><span class="n">timeit</span> <span class="o">-</span><span class="n">r1</span> <span class="o">-</span><span class="n">n1</span> <span class="n">bisect</span><span class="o">.</span><span class="n">bisect</span><span class="p">(</span><span class="n">word_list</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="n">loops</span><span class="p">,</span> <span class="n">best</span> <span class="n">of</span> <span class="mi">1</span><span class="p">:</span> <span class="mf">31.8</span> <span class="n">µs</span> <span class="n">per</span> <span class="n">loop</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">word_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">word_list</span><span class="p">)</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">sys</span><span class="o">.</span><span class="n">getsizeof</span><span class="p">(</span><span class="n">word_set</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">2097376</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">timeit</span> <span class="o">-</span><span class="n">r1</span> <span class="o">-</span><span class="n">n1</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">word_set</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="n">loops</span><span class="p">,</span> <span class="n">best</span> <span class="n">of</span> <span class="mi">1</span><span class="p">:</span> <span class="mf">2.25</span> <span class="n">µs</span> <span class="n">per</span> <span class="n">loop</span>
</pre></div>
</div>
<div class="code bash highlight-default"><div class="highlight"><pre><span></span>! pip install hat_trie
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">load_ext</span> <span class="n">memory_profiler</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">hat_trie</span> <span class="k">import</span> <span class="n">Trie</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">memit</span> <span class="n">word_trie</span> <span class="o">=</span> <span class="n">Trie</span><span class="p">(</span><span class="n">word_list</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">peak</span> <span class="n">memory</span><span class="p">:</span> <span class="mf">116.97</span> <span class="n">MiB</span><span class="p">,</span> <span class="n">increment</span><span class="p">:</span> <span class="mf">0.39</span> <span class="n">MiB</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">timeit</span> <span class="o">-</span><span class="n">r1</span> <span class="o">-</span><span class="n">n1</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">word_trie</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="n">loops</span><span class="p">,</span> <span class="n">best</span> <span class="n">of</span> <span class="mi">1</span><span class="p">:</span> <span class="mf">5.86</span> <span class="n">µs</span> <span class="n">per</span> <span class="n">loop</span>
</pre></div>
</div>
</div>
<div class="section" id="data-sketches">
<h2>Data Sketches<a class="headerlink" href="#data-sketches" title="Permalink to this headline">¶</a></h2>
<p>A <code class="docutils literal"><span class="pre">sketch</span></code> is a probabilistic algorithm or data structure that
approximates some statistic of interest, typically using very little
memory and processing time. Often they are applied to streaming data,
and so must be able to incrementally process data. Many data sketches
make use of hash functions to distribute data into buckets uniformly.
Typically, data sketches have the following desirable properties</p>
<ul class="simple">
<li>sub-linear in space</li>
<li>single scan</li>
<li>can be parallelized</li>
<li>can be combined (merge)</li>
</ul>
<p>Some statistics that sketches have been used to estimate include</p>
<ul class="simple">
<li>indicator variables (event detection)</li>
<li>counts</li>
<li>quantiles</li>
<li>moments</li>
<li>entropy</li>
</ul>
<p>Packages for data sketches in Python are relatively immmature, and if
you are interested, you could make a large contribution by creating a
comprehensive open source library of data sketches in Python.</p>
<p>The Morris counter is used as a simple illustration of a probabilistic
data structure, with the standard trade-off of using less memory in
return for less accuracy. The algorithm is extremely simple - keep a
counter <span class="math">\(c\)</span> that represents the <strong>exponent</strong> - that is, when the
Morris counter is <span class="math">\(c\)</span>, the estimated count is <span class="math">\(2^c\)</span>. The
probabilistic part comes from the way that the counter is incremented by
comparing a uniform random variate to <span class="math">\(1/2^c\)</span>.</p>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">random</span> <span class="k">import</span> <span class="n">random</span>
<span class="k">class</span> <span class="nc">MorrisCounter</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">=</span> <span class="n">c</span>
<span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">2</span> <span class="o">**</span> <span class="bp">self</span><span class="o">.</span><span class="n">c</span>
<span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">item</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">+=</span> <span class="n">random</span><span class="p">()</span> <span class="o"><</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="p">)</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">mc</span> <span class="o">=</span> <span class="n">MorrisCounter</span><span class="p">()</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">'True</span><span class="se">\t\t</span><span class="s1">Morris</span><span class="se">\t\t</span><span class="s1">Rel Error'</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">word</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">word_list</span><span class="p">):</span>
<span class="n">mc</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">word</span><span class="p">)</span>
<span class="k">if</span> <span class="n">i</span><span class="o">%</span><span class="nb">int</span><span class="p">(</span><span class="o">.</span><span class="mi">2</span><span class="n">e5</span><span class="p">)</span><span class="o">==</span><span class="mi">0</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%8d</span><span class="se">\t</span><span class="si">%8d</span><span class="se">\t</span><span class="si">%.2f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">mc</span><span class="p">),</span> <span class="mi">0</span> <span class="k">if</span> <span class="n">i</span><span class="o">==</span><span class="mi">0</span> <span class="k">else</span> <span class="nb">abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">mc</span><span class="p">))</span><span class="o">/</span><span class="n">i</span><span class="p">))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kc">True</span> <span class="n">Morris</span> <span class="n">Rel</span> <span class="n">Error</span>
<span class="mi">0</span> <span class="mi">2</span> <span class="mf">0.00</span>
<span class="mi">20000</span> <span class="mi">8192</span> <span class="mf">0.59</span>
<span class="mi">40000</span> <span class="mi">32768</span> <span class="mf">0.18</span>
<span class="mi">60000</span> <span class="mi">65536</span> <span class="mf">0.09</span>
<span class="mi">80000</span> <span class="mi">65536</span> <span class="mf">0.18</span>
<span class="mi">100000</span> <span class="mi">131072</span> <span class="mf">0.31</span>
<span class="mi">120000</span> <span class="mi">131072</span> <span class="mf">0.09</span>
<span class="mi">140000</span> <span class="mi">131072</span> <span class="mf">0.06</span>
<span class="mi">160000</span> <span class="mi">131072</span> <span class="mf">0.18</span>
<span class="mi">180000</span> <span class="mi">131072</span> <span class="mf">0.27</span>
<span class="mi">200000</span> <span class="mi">131072</span> <span class="mf">0.34</span>
<span class="mi">220000</span> <span class="mi">262144</span> <span class="mf">0.19</span>
<span class="mi">240000</span> <span class="mi">262144</span> <span class="mf">0.09</span>
<span class="mi">260000</span> <span class="mi">524288</span> <span class="mf">1.02</span>
</pre></div>
</div>
<p>A simple way to increase the accuracy is to have multiple Morris
counters and take the average. These two ideas of using a probabilistic
calculation and multiple samples to improve precision are the basis for
the more useful probabilisitc data structures described below.</p>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">mcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">MorrisCounter</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">'True</span><span class="se">\t\t</span><span class="s1">Morris</span><span class="se">\t\t</span><span class="s1">Rel Error'</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">word</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">word_list</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
<span class="n">mcs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">word</span><span class="p">)</span>
<span class="n">estimate</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">mcs</span><span class="p">])</span>
<span class="k">if</span> <span class="n">i</span><span class="o">%</span><span class="nb">int</span><span class="p">(</span><span class="o">.</span><span class="mi">2</span><span class="n">e5</span><span class="p">)</span><span class="o">==</span><span class="mi">0</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%8d</span><span class="se">\t</span><span class="si">%8d</span><span class="se">\t</span><span class="si">%.2f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">estimate</span><span class="p">,</span> <span class="mi">0</span> <span class="k">if</span> <span class="n">i</span><span class="o">==</span><span class="mi">0</span> <span class="k">else</span> <span class="nb">abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">estimate</span><span class="p">)</span><span class="o">/</span><span class="n">i</span><span class="p">))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kc">True</span> <span class="n">Morris</span> <span class="n">Rel</span> <span class="n">Error</span>
<span class="mi">0</span> <span class="mi">2</span> <span class="mf">0.00</span>
<span class="mi">20000</span> <span class="mi">20889</span> <span class="mf">0.04</span>
<span class="mi">40000</span> <span class="mi">26214</span> <span class="mf">0.34</span>
<span class="mi">60000</span> <span class="mi">36044</span> <span class="mf">0.40</span>
<span class="mi">80000</span> <span class="mi">49152</span> <span class="mf">0.39</span>
<span class="mi">100000</span> <span class="mi">72089</span> <span class="mf">0.28</span>
<span class="mi">120000</span> <span class="mi">75366</span> <span class="mf">0.37</span>
<span class="mi">140000</span> <span class="mi">95027</span> <span class="mf">0.32</span>
<span class="mi">160000</span> <span class="mi">111411</span> <span class="mf">0.30</span>
<span class="mi">180000</span> <span class="mi">131072</span> <span class="mf">0.27</span>
<span class="mi">200000</span> <span class="mi">137625</span> <span class="mf">0.31</span>
<span class="mi">220000</span> <span class="mi">137625</span> <span class="mf">0.37</span>
<span class="mi">240000</span> <span class="mi">144179</span> <span class="mf">0.40</span>
<span class="mi">260000</span> <span class="mi">170393</span> <span class="mf">0.34</span>
</pre></div>
</div>
<p>The Morris counter is less useful because the degree of memory saved as
compared to counting the number of elements exactly is not much unless
the numbers are staggeringly huge. In contrast, counting the number of
<strong>distinct</strong> elements exactly requires storage of all distinct elements
(e.g. in a set) and hence grows with the cardinality <span class="math">\(n\)</span>.
Probabilistic data structures known as Distinct Value Sketches can do
this with a tiny and fixed memory size.</p>
<p>Examples where counting distinct values is useful:</p>
<ul class="simple">
<li>number of unique users in a Twitter stream</li>
<li>number of distinct records to be fetched by a databse query</li>
<li>number of unique IP addresses accessing a website</li>
<li>number of distinct queries submitted to a search engine</li>
<li>number of distinct DNA motifs in genomics data sets (e.g. microbiome)</li>
</ul>
<p>A hash function takes data of arbitrary size and converts it into a
number in a fixed range. Ideally, given an arbitrary set of data items,
the hash function generates numbers that follow a uniform distribution
within the fixed range. Hash functions are immensely useful throughout
computer science (for example - they power Python sets and
dictionaries), and especially for the generation of probabilistic data
structures.</p>
<div class="section" id="a-simple-hash-function-mapping">
<h3>A simple hash function mapping<a class="headerlink" href="#a-simple-hash-function-mapping" title="Permalink to this headline">¶</a></h3>
<p>Note the <strong>collisions</strong>. If not handled, there is a loss of information.
Commonly, practical hash functions return a 32 or 64 bit integer. Also
note that there are an arbitrary number of hash functions that can
return numbers within a given range.</p>
<p>Note also that because the hash function is deterministic, the same item
will always map to the same bin.</p>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">string_hash</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">ord</span><span class="p">(</span><span class="n">char</span><span class="p">)</span> <span class="k">for</span> <span class="n">char</span> <span class="ow">in</span> <span class="n">word</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">sentence</span> <span class="o">=</span> <span class="s2">"The quick brown fox jumps over the lazy dog."</span>
<span class="k">for</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">sentence</span><span class="o">.</span><span class="n">split</span><span class="p">():</span>
<span class="nb">print</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="n">string_hash</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">The</span> <span class="mi">9</span>
<span class="n">quick</span> <span class="mi">1</span>
<span class="n">brown</span> <span class="mi">2</span>
<span class="n">fox</span> <span class="mi">3</span>
<span class="n">jumps</span> <span class="mi">9</span>
<span class="n">over</span> <span class="mi">4</span>
<span class="n">the</span> <span class="mi">1</span>
<span class="n">lazy</span> <span class="mi">8</span>
<span class="n">dog</span><span class="o">.</span> <span class="mi">0</span>
</pre></div>
</div>
</div>
<div class="section" id="built-in-python-hash-function">
<h3>Built-in Python hash function<a class="headerlink" href="#built-in-python-hash-function" title="Permalink to this headline">¶</a></h3>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">help</span><span class="p">(</span><span class="nb">hash</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">Help</span> <span class="n">on</span> <span class="n">built</span><span class="o">-</span><span class="ow">in</span> <span class="n">function</span> <span class="nb">hash</span> <span class="ow">in</span> <span class="n">module</span> <span class="n">builtins</span><span class="p">:</span>
<span class="nb">hash</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="o">/</span><span class="p">)</span>
<span class="n">Return</span> <span class="n">the</span> <span class="nb">hash</span> <span class="n">value</span> <span class="k">for</span> <span class="n">the</span> <span class="n">given</span> <span class="nb">object</span><span class="o">.</span>
<span class="n">Two</span> <span class="n">objects</span> <span class="n">that</span> <span class="n">compare</span> <span class="n">equal</span> <span class="n">must</span> <span class="n">also</span> <span class="n">have</span> <span class="n">the</span> <span class="n">same</span> <span class="nb">hash</span> <span class="n">value</span><span class="p">,</span> <span class="n">but</span> <span class="n">the</span>
<span class="n">reverse</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">necessarily</span> <span class="n">true</span><span class="o">.</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">sentence</span><span class="o">.</span><span class="n">split</span><span class="p">():</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">{:<10s}</span><span class="s1"> </span><span class="si">{:24}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="nb">hash</span><span class="p">(</span><span class="n">word</span><span class="p">)))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">The</span> <span class="o">-</span><span class="mi">184990475008303844</span>
<span class="n">quick</span> <span class="mi">3616884800889772302</span>
<span class="n">brown</span> <span class="mi">6377133929055916905</span>
<span class="n">fox</span> <span class="o">-</span><span class="mi">611579958660588990</span>
<span class="n">jumps</span> <span class="o">-</span><span class="mi">5271806623556898369</span>
<span class="n">over</span> <span class="mi">7546130948312823661</span>
<span class="n">the</span> <span class="mi">6678103606492090842</span>
<span class="n">lazy</span> <span class="o">-</span><span class="mi">1515512778017190090</span>
<span class="n">dog</span><span class="o">.</span> <span class="mi">3069897472948403276</span>
</pre></div>
</div>
</div>
<div class="section" id="using-a-hash-function-from-the-murmurhash3-library">
<h3>Using a hash function from the MurmurHash3 library<a class="headerlink" href="#using-a-hash-function-from-the-murmurhash3-library" title="Permalink to this headline">¶</a></h3>
<p>Note that the hash function accepts a seed, allowing the creation of
multiple hash functions. We also display the hash result as a 32-bit
binary string.</p>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">mmh3</span>
<span class="k">for</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">sentence</span><span class="o">.</span><span class="n">split</span><span class="p">():</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">{:<10}</span><span class="s1"> </span><span class="si">{:+032b}</span><span class="s1"> </span><span class="si">{:+032b}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">word</span><span class="o">.</span><span class="n">ljust</span><span class="p">(</span><span class="mi">10</span><span class="p">),</span> <span class="n">mmh3</span><span class="o">.</span><span class="n">hash</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">1234</span><span class="p">),</span>
<span class="n">mmh3</span><span class="o">.</span><span class="n">hash</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">4321</span><span class="p">)))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">The</span> <span class="o">+</span><span class="mi">0001000011111110001001110101100</span> <span class="o">+</span><span class="mi">1110110100100101010111100011010</span>
<span class="n">quick</span> <span class="o">-</span><span class="mi">0101111111011110110101100101000</span> <span class="o">+</span><span class="mi">1000100001101010110000101101100</span>
<span class="n">brown</span> <span class="o">+</span><span class="mi">1000101010000110110010001110101</span> <span class="o">-</span><span class="mi">1101101110000000010001100010100</span>
<span class="n">fox</span> <span class="o">-</span><span class="mi">1000000010010010000111001111011</span> <span class="o">+</span><span class="mi">0111011111000011001001001110111</span>
<span class="n">jumps</span> <span class="o">+</span><span class="mi">0000010111000011010000100101010</span> <span class="o">+</span><span class="mi">0010010001111110100010010110011</span>
<span class="n">over</span> <span class="o">-</span><span class="mi">0110101101111001001101011111011</span> <span class="o">-</span><span class="mi">1101110111110010000101101000100</span>
<span class="n">the</span> <span class="o">-</span><span class="mi">1000000101110000000110011111001</span> <span class="o">+</span><span class="mi">0001000111100111011000011100101</span>
<span class="n">lazy</span> <span class="o">-</span><span class="mi">1101011000111111110011111001100</span> <span class="o">+</span><span class="mi">0010101110101100001000101110000</span>
<span class="n">dog</span><span class="o">.</span> <span class="o">+</span><span class="mi">0100110101101111101011110111111</span> <span class="o">-</span><span class="mi">0101111000110000001011110001011</span>
</pre></div>
</div>
</div>
<div class="section" id="loglog-family">
<h3>LogLog family<a class="headerlink" href="#loglog-family" title="Permalink to this headline">¶</a></h3>
<p>The binary digits in a (say) 32-bit hash are effectively random, and
equivalent to a sequence of fair coin tosses. Hence the probability that
we see a run of 5 zeros in the smallest hash so far suggests that we
have added <span class="math">\(2^5\)</span> unique items so far. This is the intuition behind
the loglog family of Distinct Value Sketches. Note that the biggest
count we can track with 32 bits is <span class="math">\(2^{32} = 4294967296\)</span>.</p>
<p>The accuracy of the sketch can be improved by averaging results with
multiple coin flippers. In practice, this is done by using the first
<span class="math">\(k\)</span> bit registers to identify <span class="math">\(2^k\)</span> different coin flippers.
Hence, the max count is now <span class="math">\(2 ** (32 - k)\)</span>. The hyperloglog
algorithm uses the harmonic mean of the <span class="math">\(2^k\)</span> flippers which
reduces the effect of outliers and hence the variance of the estimate.</p>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">15</span><span class="p">):</span>
<span class="n">k</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="n">i</span>
<span class="n">hashes</span> <span class="o">=</span> <span class="p">[</span><span class="s1">''</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">32</span><span class="p">)))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">)]</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%6d</span><span class="se">\t</span><span class="si">%s</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">hashes</span><span class="p">)))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span> <span class="mi">2</span> <span class="mi">00010001101110111100011011110100</span>
<span class="mi">4</span> <span class="mi">00001111000110101110111110111011</span>
<span class="mi">8</span> <span class="mi">00011100101100110010101100011110</span>
<span class="mi">16</span> <span class="mi">01000001100111101011001100101001</span>
<span class="mi">32</span> <span class="mi">00001011000101110101000110101010</span>
<span class="mi">64</span> <span class="mi">00000000110100000010001101011101</span>
<span class="mi">128</span> <span class="mi">00000000100101100011100101100111</span>
<span class="mi">256</span> <span class="mi">00000000001101010111001001101010</span>
<span class="mi">512</span> <span class="mi">00000000001001011011001100001100</span>
<span class="mi">1024</span> <span class="mi">00000000001010000100011000011011</span>
<span class="mi">2048</span> <span class="mi">00000000010100111100001101001100</span>
<span class="mi">4096</span> <span class="mi">00000000000000110001101010011100</span>
<span class="mi">8192</span> <span class="mi">00000000000000001100110110000101</span>
<span class="mi">16384</span> <span class="mi">00000000000000001101011110101001</span>
</pre></div>
</div>
<div class="code bash highlight-default"><div class="highlight"><pre><span></span><span class="n">pip</span> <span class="n">install</span> <span class="n">hyperloglog</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">hyperloglog</span> <span class="k">import</span> <span class="n">HyperLogLog</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">hll</span> <span class="o">=</span> <span class="n">HyperLogLog</span><span class="p">(</span><span class="mf">0.01</span><span class="p">)</span> <span class="c1"># accept 1% counting error</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s1">'True</span><span class="se">\t\t</span><span class="s1">HLL</span><span class="se">\t\t</span><span class="s1">Rel Error'</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">word</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">word_list</span><span class="p">):</span>
<span class="n">s</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">word</span><span class="p">)</span>
<span class="n">hll</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">word</span><span class="p">)</span>
<span class="k">if</span> <span class="n">i</span><span class="o">%</span><span class="nb">int</span><span class="p">(</span><span class="o">.</span><span class="mi">2</span><span class="n">e5</span><span class="p">)</span><span class="o">==</span><span class="mi">0</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%8d</span><span class="se">\t</span><span class="si">%8d</span><span class="se">\t\t</span><span class="si">%.2f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">h1</span><span class="p">),</span> <span class="mi">0</span> <span class="k">if</span> <span class="n">i</span><span class="o">==</span><span class="mi">0</span> <span class="k">else</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">h1</span><span class="p">))</span><span class="o">/</span><span class="n">i</span><span class="p">))</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="kc">True</span> <span class="n">HLL</span> <span class="n">Rel</span> <span class="n">Error</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="o">---------------------------------------------------------------------------</span>
<span class="ne">NameError</span> <span class="n">Traceback</span> <span class="p">(</span><span class="n">most</span> <span class="n">recent</span> <span class="n">call</span> <span class="n">last</span><span class="p">)</span>
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="mi">22150</span><span class="n">e0b1002</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span><span class="p">()</span>
<span class="mi">1</span> <span class="nb">print</span><span class="p">(</span><span class="s1">'True</span><span class="se">\t\t</span><span class="s1">HLL</span><span class="se">\t\t</span><span class="s1">Rel Error'</span><span class="p">)</span>
<span class="mi">2</span> <span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
<span class="o">----></span> <span class="mi">3</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">word</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">word_list</span><span class="p">):</span>
<span class="mi">4</span> <span class="n">s</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">word</span><span class="p">)</span>
<span class="mi">5</span> <span class="n">hll</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">word</span><span class="p">)</span>
<span class="ne">NameError</span><span class="p">:</span> <span class="n">name</span> <span class="s1">'word_list'</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">defined</span>
</pre></div>
</div>
<div class="section" id="bloom-filters">
<h4>Bloom filters<a class="headerlink" href="#bloom-filters" title="Permalink to this headline">¶</a></h4>
<p>Bloom filters are designed to answer queries about whether a specific
item is in a collection. If the answer is NO, then it is definitive.
However, if the answer is yes, it might be a false positive. The
possibility of a false positive makes the Bloom filter a probabilistic
data structure.</p>
<p>A bloom filter consists of a bit vector of length <span class="math">\(k\)</span> initially
set to zero, and <span class="math">\(n\)</span> different hash functions that return a hash
value that will fall into one of the <span class="math">\(k\)</span> bins. In the construction
phase, for every item in the collection, <span class="math">\(n\)</span> hash values are
generated by the <span class="math">\(n\)</span> hash functions, and every position indicated
by a hash value is flipped to one. In the query phase, given an item,
<span class="math">\(n\)</span> hash values are calculated as before - if any of these
<span class="math">\(n\)</span> positions is a zero, then the item is definitely not in the
collection. However, because of the possibility of hash collisions, even
if all the positions are one, this could be a false positive. Clearly,
the rate of false positives depends on the ratio of zero and one bits,
and there are Bloom filter implementations that will dynamically bound
the ratio and hence the false positive rate.</p>
<p>Possible uses of a Bloom filter include:</p>
<ul class="simple">
<li>Does a particular sequence motif appear in a DNA string?</li>
<li>Has this book been recommended to this customer before?</li>
<li>Check if an element exists on disk before performing I/O</li>
<li>Check if URL is a potential malware site using in-browser Bloom
filter to minimize network communication</li>
<li>As an alternative way to generate distinct value counts cheaply (only
increment count if Bloom filter says NO)</li>
</ul>
<div class="code bash highlight-default"><div class="highlight"><pre><span></span><span class="n">pip</span> <span class="n">install</span> <span class="n">git</span><span class="o">+</span><span class="n">https</span><span class="p">:</span><span class="o">//</span><span class="n">github</span><span class="o">.</span><span class="n">com</span><span class="o">/</span><span class="n">jaybaird</span><span class="o">/</span><span class="n">python</span><span class="o">-</span><span class="n">bloomfilter</span><span class="o">.</span><span class="n">git</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">pybloom</span> <span class="k">import</span> <span class="n">ScalableBloomFilter</span>
<span class="c1"># The Scalable Bloom Filter grows as needed to keep the error rate small</span>
<span class="c1"># The default error_rate=0.001</span>
<span class="n">sbf</span> <span class="o">=</span> <span class="n">ScalableBloomFilter</span><span class="p">()</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">word_set</span><span class="p">:</span>
<span class="n">sbf</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">word</span><span class="p">)</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="n">test_words</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'banana'</span><span class="p">,</span> <span class="s1">'artist'</span><span class="p">,</span> <span class="s1">'Dublin'</span><span class="p">,</span> <span class="s1">'masochist'</span><span class="p">,</span> <span class="s1">'Obama'</span><span class="p">]</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">test_words</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">sbf</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">banana</span> <span class="kc">True</span>
<span class="n">artist</span> <span class="kc">True</span>
<span class="n">Dublin</span> <span class="kc">True</span>
<span class="n">masochist</span> <span class="kc">False</span>
<span class="n">Obama</span> <span class="kc">False</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="c1">### Chedck</span>
<span class="k">for</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">test_words</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">word_set</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">banana</span> <span class="kc">True</span>
<span class="n">artist</span> <span class="kc">True</span>
<span class="n">Dublin</span> <span class="kc">True</span>
<span class="n">masochist</span> <span class="kc">False</span>
<span class="n">Obama</span> <span class="kc">False</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">load_ext</span> <span class="n">version_information</span>
</pre></div>
</div>
<div class="code python highlight-default"><div class="highlight"><pre><span></span><span class="o">%</span><span class="n">version_information</span> <span class="n">pybloom</span><span class="p">,</span> <span class="n">hyperloglog</span><span class="p">,</span> <span class="n">hat_trie</span>
</pre></div>
</div>
<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.5.1 64bit [GCC 4.2.1 (Apple Inc. build 5577)]</td></tr><tr><td>IPython</td><td>4.0.3</td></tr><tr><td>OS</td><td>Darwin 15.4.0 x86_64 i386 64bit</td></tr><tr><td>pybloom</td><td>2.0.0</td></tr><tr><td>hyperloglog</td><td>0.0.10</td></tr><tr><td>hat_trie</td><td>0.3</td></tr><tr><td colspan='2'>Thu Apr 14 16:01:59 2016 EDT</td></tr></table></div>
</div>
</div>
</div>
</div>
</div>
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<li class="toctree-l1"><a class="reference internal" href="17A_Intermediate_Sized_Data.html">Biggish Data</a></li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">Efficient storage of data in memory</a><ul>
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<li class="toctree-l2"><a class="reference internal" href="#storing-numbers">Storing numbers</a></li>
<li class="toctree-l2"><a class="reference internal" href="#storing-strings">Storing strings</a></li>
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