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<li><a href="/docs/Background/TheChallenge.html">•The Challenge</a></li>
<li><a href="/docs/Background/SketchOrigins.html">•Sketch Origins</a></li>
<li><a href="/docs/Background/SketchElements.html">•Sketch Elements</a></li>
<li><a href="/docs/Background/Presentations.html">•Presentations</a></li>
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<li><a href="/docs/Architecture/MajorSketchFamilies.html">•The Major Sketch Families</a></li>
<li><a href="/docs/Architecture/LargeScale.html">•Large Scale Computing</a></li>
<li><a href="/docs/Architecture/KeyFeatures.html">•Key Features</a></li>
<li><a href="/docs/Architecture/SketchFeaturesMatrix.html">•Sketch Features Matrix</a></li>
<li><a href="/docs/Architecture/Components.html">•Components</a></li>
<li><a href="/docs/Architecture/SketchesByComponent.html">•Sketches by Component</a></li>
<li><a href="/docs/Architecture/SketchCriteria.html">•Sketch Criteria</a></li>
<p id="memory-component">
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<li><a href="/docs/Memory/MemoryComponent.html">•Memory Componet</a></li>
<li><a href="/docs/Memory/MemoryPerformance.html">•Memory Component Performance</a></li>
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<li><a href="/docs/Architecture/OrderSensitivity.html">•Notes on Order Sensitivity</a></li>
<li><a href="/docs/Architecture/Concurrency.html">•Notes on Concurrency</a></li>
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<p id="sketch-families">
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<li><a href="/docs/DistinctCountFeaturesMatrix.html">•Features Matrix</a></li>
<li><a href="/docs/DistinctCountMeritComparisons.html">•Figures-of-Merit Comparison</a></li>
<p id="cpc-sketches">
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<li><a href="/docs/CPC/CPC.html">•CPC Sketch</a></li>
<li><a href="/docs/CPC/CpcPerformance.html">•CPC Sketch Performance</a></li>
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<li><a href="/docs/CPC/CpcJavaExample.html">•CPC Sketch Java Example</a></li>
<li><a href="/docs/CPC/CpcCppExample.html">•CPC Sketch C++ Example</a></li>
<li><a href="/docs/CPC/CpcPigExample.html">•CPC Sketch Pig UDFs</a></li>
<li><a href="/docs/CPC/CpcHiveExample.html">•CPC Sketch Hive UDFs</a></li>
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<p id="hyperloglog-sketches">
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<li><a href="/docs/HLL/HLL.html">•HLL Sketch</a></li>
<li><a href="/docs/HLL/HllMap.html">•HLL Map Sketch</a></li>
<p id="hll-examples">
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<li><a href="/docs/HLL/HllJavaExample.html">•HLL Sketch Java Example</a></li>
<li><a href="/docs/HLL/HllCppExample.html">•HLL Sketch C++ Example</a></li>
<li><a href="/docs/HLL/HllPigUDFs.html">•HLL Sketch Pig UDFs</a></li>
<li><a href="/docs/HLL/HllHiveUDFs.html">•HLL Sketch Hive UDFs</a></li>
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<a data-toggle="collapse" class="menu collapsed" href="#collapse_hll_studies">HLL Studies</a>
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<li><a href="/docs/HLL/HllPerformance.html">•HLL Sketch Performance</a></li>
<li><a href="/docs/HLL/Hll_vs_CS_Hllpp.html">•HLL vs Clearspring HLL++</a></li>
<li><a href="/docs/HLL/HllSketchVsDruidHyperLogLogCollector.html">•HLL Sketch vs Druid HyperLogLogCollector</a></li>
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<a data-toggle="collapse" class="menu collapsed" href="#collapse_theta_sketches">Theta Sketches</a>
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<li><a href="/docs/Theta/ThetaSketchFramework.html">•Theta Sketch Framework</a></li>
<p id="theta-examples">
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<li><a href="/docs/Theta/ConcurrentThetaSketch.html">•Concurrent Theta Sketch</a></li>
<li><a href="/docs/Theta/ThetaJavaExample.html">•Theta Sketch Java Example</a></li>
<li><a href="/docs/Theta/ThetaSparkExample.html">•Theta Sketch Spark Example</a></li>
<li><a href="/docs/Theta/ThetaPigUDFs.html">•Theta Sketch Pig UDFs</a></li>
<li><a href="/docs/Theta/ThetaHiveUDFs.html">•Theta Sketch Hive UDFs</a></li>
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<a data-toggle="collapse" class="menu collapsed" href="#collapse_kmv_tutorial">KMV Tutorial</a>
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<li><a href="/docs/Theta/InverseEstimate.html">•The Inverse Estimate</a></li>
<li><a href="/docs/Theta/KMVempty.html">•Empty Sketch</a></li>
<li><a href="/docs/Theta/KMVfirstEst.html">•First Estimator</a></li>
<li><a href="/docs/Theta/KMVbetterEst.html">•Better Estimator</a></li>
<li><a href="/docs/Theta/KMVrejection.html">•Rejection Rules</a></li>
<li><a href="/docs/Theta/KMVupdateVkth.html">•Update V(kth) Rule</a></li>
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<p id="set-operations-and-p-sampling">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_set_operations_and_p-sampling">Set Operations and P-sampling</a>
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<li><a href="/docs/Theta/ThetaSketchSetOps.html">•Set Operations</a></li>
<li><a href="/docs/Theta/ThetaSetOpsCornerCases.html">•Model & Test Set Operations</a></li>
<li><a href="/docs/Theta/ThetaPSampling.html"><i>p</i>-Sampling</a></li>
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<p id="accuracy">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_accuracy">Accuracy</a>
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<li><a href="/docs/Theta/ThetaAccuracy.html">•Basic Accuracy</a></li>
<li><a href="/docs/Theta/ThetaAccuracyPlots.html">•Accuracy Plots</a></li>
<li><a href="/docs/Theta/ThetaErrorTable.html">•Relative Error Table</a></li>
<li><a href="/docs/Theta/ThetaSketchSetOpsAccuracy.html">•SetOp Accuracy</a></li>
<li><a href="/docs/Theta/AccuracyOfDifferentKUnions.html">•Unions With Different k</a></li>
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<p id="size">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_size">Size</a>
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<li><a href="/docs/Theta/ThetaSize.html">•Theta Sketch Size</a></li>
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<p id="speed">
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<li><a href="/docs/Theta/ThetaUpdateSpeed.html">•Update Speed</a></li>
<li><a href="/docs/Theta/ThetaMergeSpeed.html">•Merge Speed</a></li>
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<p id="theta-sketch-theory">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_theta_sketch_theory">Theta Sketch Theory</a>
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<li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/ThetaSketchFramework.pdf">•Theta Sketch Framework (PDF)</a></li>
<li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/ThetaSketchEquations.pdf">•Theta Sketch Equations (PDF)</a></li>
<li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/DataSketches.pdf">•DataSketches (PDF)</a></li>
<li><a href="/docs/Theta/ThetaConfidenceIntervals.html">•Confidence Intervals Notes</a></li>
<li><a href="/docs/Theta/ThetaMergingAlgorithm.html">•Merging Algorithm Notes</a></li>
<li><a href="/docs/Theta/ThetaReferences.html">•Theta References</a></li>
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<p id="tuple-sketches">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_tuple_sketches">Tuple Sketches</a>
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<li><a href="/docs/Tuple/TupleOverview.html">•Tuple Overview</a></li>
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<a data-toggle="collapse" class="menu collapsed" href="#collapse_tuple_examples">Tuple Examples</a>
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<li><a href="/docs/Tuple/TupleJavaExample.html">•Tuple Java Example</a></li>
<li><a href="/docs/Tuple/TupleEngagementExample.html">•Tuple Engagement Example</a></li>
<li><a href="/docs/Tuple/TuplePigUDFs.html">•Tuple Pig UDFs</a></li>
<li><a href="/docs/Tuple/TupleHiveUDFs.html">•Tuple Hive UDFs</a></li>
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<p id="most-frequent">
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<li><a href="/docs/Frequency/FrequencySketchesOverview.html">•Frequency Sketches Overview</a></li>
<p id="frequent-item-sketches">
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<li><a href="/docs/Frequency/FrequentItemsOverview.html">•Frequent Items Overview</a></li>
<li><a href="/docs/Frequency/FrequentItemsErrorTable.html">•Frequent Items Error Table</a></li>
<li><a href="/docs/Frequency/FrequentItemsReferences.html">•Frequent Items References</a></li>
<li><a href="/docs/Frequency/FrequentItemsPerformance.html">•Frequent Items Performance</a></li>
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<a data-toggle="collapse" class="menu collapsed" href="#collapse_most_frequent_examples">Most Frequent Examples</a>
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<li><a href="/docs/Frequency/FrequentItemsJavaExample.html">•Frequent Items Java Example</a></li>
<li><a href="/docs/Frequency/FrequentItemsCppExample.html">•Frequent Items C++ Example</a></li>
<li><a href="/docs/Frequency/FrequentItemsPigUDFs.html">•Frequent Items Pig UDFs</a></li>
<li><a href="/docs/Frequency/FrequentItemsHiveUDFs.html">•Frequent Items Hive UDFs</a></li>
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<p id="frequent-distinct-sketches">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_frequent_distinct_sketches">Frequent Distinct Sketches</a>
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<li><a href="/docs/Frequency/FrequentDistinctTuplesSketch.html">•Frequent Distinct Tuples Sketch</a></li>
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<p id="quantiles-and-histograms">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_and_histograms">Quantiles And Histograms</a>
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<li><a href="/docs/Quantiles/SketchingQuantilesAndRanksTutorial.html">•Quantiles and Ranks Tutorial</a></li>
<li><a href="/docs/Quantiles/QuantilesOverview.html">•Quantiles Overview</a></li>
<li><a href="/docs/KLL/KLLSketch.html">•KLL Floats sketch</a></li>
<li><a href="/docs/KLL/KLLAccuracyAndSize.html">•KLL Sketch Accuracy and Size</a></li>
<li><a href="/docs/REQ/ReqSketch.html">•REQ Floats sketch</a></li>
<li><a href="/docs/Quantiles/OrigQuantilesSketch.html">•Original QuantilesSketch</a></li>
<p id="quantiles-examples">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_examples">Quantiles Examples</a>
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<li><a href="/docs/Quantiles/QuantilesJavaExample.html">•Quantiles Sketch Java Example</a></li>
<li><a href="/docs/KLL/KLLCppExample.html">•KLL Quantiles Sketch C++ Example</a></li>
<li><a href="/docs/Quantiles/QuantilesPigUDFs.html">•Quantiles Sketch Pig UDFs</a></li>
<li><a href="/docs/Quantiles/QuantilesHiveUDFs.html">•Quantiles Sketch Hive UDFs</a></li>
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<p id="quantiles-studies">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_studies">Quantiles Studies</a>
</p>
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<li><a href="/docs/QuantilesStudies/KllSketchVsTDigest.html">•KLL sketch vs t-digest</a></li>
<li><a href="/docs/QuantilesStudies/DruidApproxHistogramStudy.html">•Druid Approximate Histogram</a></li>
<li><a href="/docs/QuantilesStudies/MomentsSketchStudy.html">•Moments Sketch Study</a></li>
<li><a href="/docs/QuantilesStudies/QuantilesStreamAStudy.html">•Quantiles StreamA Study</a></li>
<li><a href="/docs/QuantilesStudies/ExactQuantiles.html">•Exact Quantiles for Studies</a></li>
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<p id="quantiles-sketch-theory">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_sketch_theory">Quantiles Sketch Theory</a>
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<li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/Quantiles_KLL.pdf">•Optimal Quantile Approximation in Streams</a></li>
<li><a href="/docs/Quantiles/QuantilesReferences.html">•Quantiles References</a></li>
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<p id="sampling">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_sampling">Sampling</a>
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<li><a href="/docs/Sampling/ReservoirSampling.html">•Reservoir Sampling</a></li>
<li><a href="/docs/Sampling/ReservoirSamplingPerformance.html">•Reservoir Sampling Performance</a></li>
<li><a href="/docs/Sampling/VarOptSampling.html">•VarOpt Sampling</a></li>
<p id="sampling-examples">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_sampling_examples">Sampling Examples</a>
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<div class="collapse" id="collapse_sampling_examples">
<li><a href="/docs/Sampling/ReservoirSamplingJava.html">•Reservoir Sampling Java Example</a></li>
<li><a href="/docs/Sampling/ReservoirSamplingPigUDFs.html">•Reservoir Sampling Pig UDFs</a></li>
<li><a href="/docs/Sampling/VarOptSamplingJava.html">•VarOpt Sampling Java Example</a></li>
<li><a href="/docs/Sampling/VarOptPigUDFs.html">•VarOpt Sampling Pig UDFs</a></li>
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<p id="system-integrations">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_system_integrations">System Integrations</a>
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<li><a href="/docs/SystemIntegrations/ApacheDruidIntegration.html">•Using Sketches in ApacheDruid</a></li>
<li><a href="/docs/SystemIntegrations/ApacheHiveIntegration.html">•Using Sketches in Apache Hive</a></li>
<li><a href="/docs/SystemIntegrations/ApachePigIntegration.html">•Using Sketches in Apache Pig</a></li>
<li><a href="/docs/SystemIntegrations/PostgreSQLIntegration.html">•Using Sketches in PostgreSQL</a></li>
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<p id="community">
<a data-toggle="collapse" class="menu collapsed" href="#collapse_community">Community</a>
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<li><a href="/docs/Community/index.html">•Community</a></li>
<li><a href="/docs/Community/Downloads.html">•Downloads</a></li>
<li><a href="/docs/Community/NewCommitterProcess.html">•Committer Process</a></li>
<li><a href="/docs/Community/ReleaseProcessForCppComponents.html">•Release Process For CPP Components</a></li>
<li><a href="/docs/Community/ReleaseProcessForJavaComponents.html">•Release Process For Java Components</a></li>
<li><a href="/docs/Community/Transitioning.html">•Transitioning from prior GitHub Site</a></li>
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<h2 id="kll-sketch">KLL Sketch</h2>
<p>Implementation of a very compact quantiles sketch with lazy compaction scheme and nearly optimal accuracy per bit.
See <a href="https://arxiv.org/abs/1603.05346v2">Optimal Quantile Approximation in Streams, by Zohar Karnin, Kevin Lang, Edo Liberty</a>.
The name KLL is composed of the initial letters of the last names of the authors.</p>
<p>The usage of KllSketch is very similar to the classic quantiles DoublesSketch.</p>
<ul>
<li>The key feature of this sketch is its compactness for a given accuracy.</li>
<li>It is separately implemented for both float and double values and can be configured for use on-heap or off-heap (Direct mode).</li>
<li>The parameter K that affects the accuracy and the size of the sketch is not restricted to powers of 2.</li>
<li>The default of 200 was chosen to yield approximately the same normalized rank error (1.65%) as the classic quantiles DoublesSketch (K=128, error 1.73%).</li>
</ul>
<h3 id="java-example">Java example</h3>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>import org.apache.datasketches.kll.KllFloatsSketch;
KllFloatsSketch sketch = KllFloatsSketch.newHeapInstance();
int n = 1000000;
for (int i = 0; i &lt; n; i++) {
sketch.update(i);
}
float median = sketch.getQuantile(0.5);
double rankOf1000 = sketch.getRank(1000);
</code></pre></div></div>
<h3 id="differences-of-kllsketch-from-the-original-quantiles-doublessketch">Differences of KllSketch from the original quantiles DoublesSketch</h3>
<ul>
<li>KLL has a smaller size for the same accuracy.</li>
<li>KLL is slightly faster to update.</li>
<li>The KLL parameter K doesn’t have to be power of 2.</li>
<li>KLL operates with either float values or double values.</li>
<li>KLL uses a merge method rather than a union object.</li>
</ul>
<p>The starting point for the comparison is setting K in such a way that rank error would be approximately the same. As pointed out above, the default K for both sketches should achieve this. Here is the comparison of the single-sided normalized rank error (getRank() method) for the default K:</p>
<p><img class="doc-img-full" src="/docs/img/kll/kll200-vs-ds128-rank-error.png" alt="RankError" /></p>
<p>The classic quantiles DoublesSketch has two forms with different serialized sizes: UpdateDoublesSketch and CompactDoublesSketch. The KLL sketches makes this distinction differently. When the KllSketch is serialized using <em>toByteArray()</em> it is always in a compact form and immutable. When the KllSketch is on-heap it is always updatable. It can be created off-heap using the static factory method <em>newDirectInstance(…)</em> method, which is also updatable. It is possible to move from off-heap (Direct) to on-heap using the <em>heapify(Memory)</em> method. The <em>merge(…)</em> method will work with off-heap sketches, on-heap sketches and Memory wrapped compact byte arrays.</p>
<p>Here is the comparison of serialized sizes:</p>
<p><img class="doc-img-full" src="/docs/img/kll/kll200-vs-ds128-size.png" alt="SerializedSize" /></p>
<p>Here is the comparison of the number of retained items to see the difference with no influence of the size of the item type:</p>
<p><img class="doc-img-full" src="/docs/img/kll/kll200-vs-ds128-items.png" alt="NumberOfRetainedItems" /></p>
<p>Below is the accuracy per byte measure (the higher the better). Suppose rank error is 1%, so the corresponding accuracy would be 99%. Divide that by size in bytes to arrive at this measure of how expensive each percentage point is in terms of space needed:</p>
<p><img class="doc-img-full" src="/docs/img/kll/kll200-vs-ds128-accuracy-per-byte-log.png" alt="AccuracyPerByte" /></p>
<p>Below is the update() method speed:</p>
<p><img class="doc-img-full" src="/docs/img/kll/kll200-vs-ds128-update.png" alt="UpdateTime" /></p>
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