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class="clear"></div> + <div class="clear"></div> + <a href="http://maven.apache.org/" title="Built by Maven" class="poweredBy"> + <img class="builtBy" alt="Built by Maven" src="../../images/logos/maven-feather.png" /> + </a> + </div> + </div> + </div> + + + <div id="bodyColumn" class="span9" > + + <h1>HyperLogLogPlus</h1> +<p><a name="HyperLogLogPlus"></a></p> +<div class="section"> +<h2><a name="Preliminaries_-_Distinct_Value_cardinality_Estimation"></a>Preliminaries - Distinct Value (cardinality) Estimation</h2> +<p>Calculating the number of distinct values (DV) in a specific finite-sized list of values using brute force (e.g. a HashSet) techniques is fairly straight forward provided that linear scaling to your number of values fits within your memory constraints. For many cases, the simple calculation serves just fine. However, in many other instances the simple linear approach simply will not scale as well as we would like, and a more robust algorithm is desirable. Enter probabilistic distinct value estimation algorithms: k-minimum-values (KMV), LogLog, HyperLogLog (HLL), and most recently HyperLogLog++ (HLLP). As data volume grows it becomes impractical to store all values in memory in order to reasonably calculate set cardinality. This is where probabilistic estimation algorithms become extremely useful. Often, we are happy to trade a handful of memory and better performance in exchange for an inexact, yet reasonably close approximation of the actual cardinality.</p></div> +<div class="section"> +<h2><a name="The_Case_For_An_Improvement"></a>The Case For An Improvement</h2> +<p>HyperLogLogPlus is an algorithm introduced by Google that builds on the DV estimation capabilities of HLL and improves on accuracy as well as scalability. HLLP not only works well with <i>extremely</i> large data sets (think <i>trillion</i>), but with smaller cardinality sets as well. HyperLogLog is simply not as accurate on the lower end of the cardinality spectrum. In addition, HLLP provides the ability to easily merge estimation sets. Although a data stream may not initially require high cardinality estimates as afforded by HLLP, it will still work accurately and efficiently on the smaller volumes without the need to rewrite code if data volume begins to surpass memory constraints. The paper describing the algorithm can be found here - <a class="externalLink" href="https://research.google.com/pubs/pub40671.html">https://research.google.com/pubs/pub40671.html</a></p></div> +<div class="section"> +<h2><a name="Performance_Metrics"></a>Performance Metrics</h2> +<div class="section"> +<h3><a name="Overview"></a>Overview</h3> +<p>Below is a table detailing the performance characteristics when running HLLP over multiple data set sizes. There are three primary variables manipulated during the test: cardinality, sparse set precision, and normal set precision. Each of these values will have an impact on error rates, execution time, and total memory consumption.</p></div> +<div class="section"> +<h3><a name="Choosing_p_sp"></a>Choosing p, sp</h3> +<p>There is a tradeoff when choosing values for p and sp. A higher value for sp means a lower error rate for instances when the sparse set is being used. However, a higher value for sp also requires more memory, which means the algorithm will cut over to the dense representation more quickly. The Google paper (listed above) recommends settings of p=25, sp=14 for optimal memory use and algorithm accuracy, and these are the defaults provided by the Apache Metron implementation in Stellar. You’ll notice in the tests below, memory consumption tops out around 12KB for this setting, regardless of cardinality size.</p></div> +<div class="section"> +<h3><a name="Key"></a>Key</h3> +<hr /> + +<div class="source"> +<div class="source"> +<pre>card: cardinality<br> +sp: sparse precision value<br> +p: normal (dense) precision value<br> +err: error as a percent of the expected cardinality<br> +time: total time to add values to the hllp estimator and calculate a cardinality estimate<br> +size: size of the hllp set in bytes once all values have been added for the specified cardinality<br> +l=low, m=mid(based on percentile chosen), h=high, std=standard deviation<br> +</pre></div></div></div> +<div class="section"> +<h3><a name="Table_Cardinality_200-1000_step_by_200"></a>Table Cardinality 200-1000, step by 200</h3> +<hr /> +<div class="section"> +<h4><a name="Options_Used"></a>Options Used</h4> + +<div class="source"> +<div class="source"> +<pre>num trials: 5000 +card min: 200 +card max: 1000 +card step: 200 +card start: 200 +sp min: 4 +sp max: 32 +sp step: 4 +error percentile: 50.0 +time percentile: 50.0 +size percentile: 50.0 +format err as %: true +</pre></div></div> + +<table border="0" class="table table-striped"> + <thead> + +<tr class="a"> + +<th>card </th> + +<th>sp </th> + +<th>p </th> + +<th>err l/m/h/std (% of actual) </th> + +<th>time l/m/h/std (ms) </th> + +<th>size l/m/h/std (b) </th> + </tr> + </thead> + <tbody> + +<tr class="b"> + +<td>200 </td> + +<td>4 </td> + +<td>4 </td> + +<td>0.000 / 17.000 / 180.000 / 17.470 </td> + +<td>0.070 / 0.118 / 10.092 / 0.204 </td> + +<td>218 / 220 / 220 / 1 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>8 </td> + +<td>4 </td> + +<td>0.000 / 17.000 / 209.500 / 18.060 </td> + +<td>0.071 / 0.085 / 8.786 / 0.128 </td> + +<td>219 / 221 / 221 / 1 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>8 </td> + +<td>8 </td> + +<td>0.000 / 3.500 / 21.500 / 3.185 </td> + +<td>0.082 / 0.085 / 8.375 / 0.119 </td> + +<td>531 / 587 / 637 / 14 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>12 </td> + +<td>4 </td> + +<td>0.000 / 17.000 / 174.000 / 17.376 </td> + +<td>0.070 / 0.085 / 8.708 / 0.128 </td> + +<td>219 / 221 / 221 / 1 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>12 </td> + +<td>8 </td> + +<td>0.000 / 3.000 / 25.500 / 2.847 </td> + +<td>0.086 / 0.119 / 8.984 / 0.151 </td> + +<td>353 / 372 / 627 / 74 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>12 </td> + +<td>12 </td> + +<td>0.000 / 0.500 / 5.500 / 0.659 </td> + +<td>0.082 / 0.096 / 17.208 / 0.243 </td> + +<td>775 / 816 / 834 / 9 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>16 </td> + +<td>4 </td> + +<td>0.000 / 17.500 / 172.000 / 16.622 </td> + +<td>0.068 / 0.075 / 9.553 / 0.137 </td> + +<td>220 / 222 / 222 / 1 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>16 </td> + +<td>8 </td> + +<td>0.000 / 3.500 / 18.000 / 2.899 </td> + +<td>0.095 / 0.118 / 0.663 / 0.031 </td> + +<td>354 / 373 / 390 / 5 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>16 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 1.500 / 0.276 </td> + +<td>0.085 / 0.108 / 13.977 / 0.199 </td> + +<td>802 / 833 / 852 / 6 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>16 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 2.500 / 0.279 </td> + +<td>0.079 / 0.085 / 0.271 / 0.017 </td> + +<td>990 / 1027 / 1044 / 6 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>20 </td> + +<td>4 </td> + +<td>0.000 / 17.000 / 209.500 / 17.424 </td> + +<td>0.068 / 0.072 / 0.250 / 0.014 </td> + +<td>221 / 223 / 223 / 1 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>20 </td> + +<td>8 </td> + +<td>0.000 / 3.500 / 18.000 / 2.944 </td> + +<td>0.096 / 0.106 / 13.581 / 0.192 </td> + +<td>357 / 374 / 388 / 5 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>20 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 1.000 / 0.066 </td> + +<td>0.084 / 0.088 / 1.096 / 0.019 </td> + +<td>915 / 945 / 972 / 8 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>20 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.500 / 0.070 </td> + +<td>0.083 / 0.087 / 10.272 / 0.145 </td> + +<td>912 / 953 / 982 / 8 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>20 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 1.000 / 0.069 </td> + +<td>0.079 / 0.081 / 0.210 / 0.017 </td> + +<td>1028 / 1056 / 1058 / 2 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>24 </td> + +<td>4 </td> + +<td>0.000 / 17.500 / 136.000 / 16.589 </td> + +<td>0.068 / 0.071 / 0.952 / 0.025 </td> + +<td>221 / 223 / 223 / 1 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>24 </td> + +<td>8 </td> + +<td>0.000 / 3.500 / 18.000 / 2.955 </td> + +<td>0.095 / 0.106 / 10.387 / 0.148 </td> + +<td>353 / 373 / 390 / 5 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>24 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.500 / 0.016 </td> + +<td>0.082 / 0.083 / 0.235 / 0.014 </td> + +<td>1035 / 1048 / 1056 / 3 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>24 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.500 / 0.012 </td> + +<td>0.082 / 0.084 / 10.904 / 0.154 </td> + +<td>1038 / 1051 / 1059 / 3 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>24 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 0.500 / 0.010 </td> + +<td>0.082 / 0.085 / 0.410 / 0.019 </td> + +<td>1049 / 1064 / 1082 / 5 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>24 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.500 / 0.023 </td> + +<td>0.079 / 0.084 / 11.926 / 0.168 </td> + +<td>1247 / 1264 / 1275 / 4 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>28 </td> + +<td>4 </td> + +<td>0.000 / 17.500 / 229.000 / 17.294 </td> + +<td>0.068 / 0.072 / 0.369 / 0.015 </td> + +<td>222 / 224 / 224 / 1 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>28 </td> + +<td>8 </td> + +<td>83.000 / 83.000 / 84.000 / 0.118 </td> + +<td>0.093 / 0.098 / 10.918 / 0.157 </td> + +<td>272 / 276 / 278 / 1 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>28 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.500 / 0.016 </td> + +<td>0.082 / 0.083 / 10.157 / 0.143 </td> + +<td>1032 / 1067 / 1075 / 3 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>28 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.500 / 0.007 </td> + +<td>0.082 / 0.083 / 0.617 / 0.020 </td> + +<td>1058 / 1069 / 1077 / 3 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>28 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 0.500 / 0.010 </td> + +<td>0.082 / 0.085 / 11.619 / 0.165 </td> + +<td>1060 / 1071 / 1080 / 3 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>28 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.500 / 0.007 </td> + +<td>0.083 / 0.085 / 0.516 / 0.022 </td> + +<td>1065 / 1083 / 1100 / 4 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>28 </td> + +<td>28 </td> + +<td>0.000 / 0.000 / 0.500 / 0.019 </td> + +<td>0.080 / 0.087 / 11.278 / 0.160 </td> + +<td>1250 / 1267 / 1278 / 4 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>32 </td> + +<td>4 </td> + +<td>0.000 / 18.000 / 175.500 / 17.157 </td> + +<td>0.068 / 0.078 / 0.223 / 0.019 </td> + +<td>222 / 224 / 224 / 1 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>32 </td> + +<td>8 </td> + +<td>99.000 / 99.000 / 99.000 / 0.000 </td> + +<td>0.093 / 0.107 / 11.695 / 0.167 </td> + +<td>258 / 258 / 258 / 0 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>32 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.500 / 0.014 </td> + +<td>0.081 / 0.084 / 8.906 / 0.126 </td> + +<td>1060 / 1071 / 1075 / 2 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>32 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.500 / 0.010 </td> + +<td>0.082 / 0.085 / 1.458 / 0.027 </td> + +<td>1061 / 1073 / 1077 / 2 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>32 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 0.500 / 0.012 </td> + +<td>0.082 / 0.084 / 10.499 / 0.148 </td> + +<td>1065 / 1074 / 1079 / 2 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>32 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.500 / 0.012 </td> + +<td>0.082 / 0.085 / 0.420 / 0.020 </td> + +<td>1066 / 1075 / 1082 / 2 </td> + </tr> + +<tr class="b"> + +<td>200 </td> + +<td>32 </td> + +<td>28 </td> + +<td>0.000 / 0.000 / 0.500 / 0.016 </td> + +<td>0.082 / 0.086 / 8.753 / 0.124 </td> + +<td>1076 / 1088 / 1102 / 4 </td> + </tr> + +<tr class="a"> + +<td>200 </td> + +<td>32 </td> + +<td>32 </td> + +<td>0.000 / 0.000 / 0.500 / 0.010 </td> + +<td>0.080 / 0.081 / 13.248 / 0.188 </td> + +<td>1249 / 1268 / 1279 / 4 </td> + </tr> + +<tr class="b"> + +<td>400 </td> + +<td>4 </td> + +<td>4 </td> + +<td>0.000 / 17.500 / 175.250 / 17.530 </td> + +<td>0.136 / 0.142 / 0.591 / 0.035 </td> + +<td>218 / 220 / 220 / 0 </td> + </tr> + +<tr class="a"> + +<td>400 </td> + +<td>8 </td> + +<td>4 </td> + +<td>0.000 / 17.000 / 148.500 / 17.174 </td> + +<td>0.136 / 0.156 / 12.882 / 0.241 </td> + +<td>219 / 221 / 221 / 0 </td> + </tr> + +<tr class="b"> + +<td>400 </td> + +<td>8 </td> + +<td>8 </td> + 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0.443 / 11.219 / 0.279 </td> + +<td>2971 / 3242 / 3364 / 39 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>16 </td> + +<td>4 </td> + +<td>0.000 / 17.600 / 184.300 / 17.148 </td> + +<td>0.336 / 0.368 / 21.656 / 0.350 </td> + +<td>221 / 222 / 222 / 0 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>16 </td> + +<td>8 </td> + +<td>0.000 / 3.800 / 21.900 / 3.350 </td> + +<td>0.369 / 0.385 / 15.840 / 0.277 </td> + +<td>388 / 399 / 412 / 3 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>16 </td> + +<td>12 </td> + +<td>0.000 / 0.200 / 1.200 / 0.174 </td> + +<td>0.422 / 0.438 / 16.035 / 0.335 </td> + +<td>3428 / 3530 / 3628 / 16 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>16 </td> + +<td>16 </td> + +<td>0.000 / 0.200 / 1.200 / 0.173 </td> + +<td>0.405 / 0.420 / 13.370 / 0.305 </td> + +<td>4314 / 4484 / 4573 / 20 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>20 </td> + +<td>4 </td> + +<td>0.000 / 17.600 / 223.000 / 16.968 </td> + +<td>0.336 / 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/ 12.804 / 0.275 </td> + +<td>222 / 223 / 223 / 0 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>24 </td> + +<td>8 </td> + +<td>0.000 / 3.800 / 24.100 / 3.403 </td> + +<td>0.371 / 0.406 / 15.710 / 0.298 </td> + +<td>390 / 400 / 412 / 3 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>24 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.200 / 0.018 </td> + +<td>0.417 / 0.449 / 12.253 / 0.294 </td> + +<td>4494 / 4614 / 4636 / 7 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>24 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.100 / 0.018 </td> + +<td>0.417 / 0.434 / 11.046 / 0.254 </td> + +<td>4594 / 4620 / 4641 / 7 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>24 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 0.200 / 0.019 </td> + +<td>0.422 / 0.458 / 13.433 / 0.270 </td> + +<td>4645 / 4682 / 4722 / 11 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>24 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.200 / 0.016 </td> + +<td>0.405 / 0.435 / 11.470 / 0.189 </td> + +<td>5635 / 5672 / 5703 / 9 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>28 </td> + +<td>4 </td> + +<td>0.000 / 17.200 / 148.500 / 17.046 </td> + +<td>0.338 / 0.363 / 20.446 / 0.380 </td> + +<td>224 / 224 / 224 / 0 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>28 </td> + +<td>8 </td> + +<td>0.000 / 15.100 / 30.700 / 4.794 </td> + +<td>0.371 / 0.389 / 18.272 / 0.359 </td> + +<td>390 / 400 / 411 / 3 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>28 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.100 / 0.013 </td> + +<td>0.417 / 0.432 / 13.575 / 0.264 </td> + +<td>4677 / 4700 / 4719 / 6 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>28 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.200 / 0.013 </td> + +<td>0.417 / 0.435 / 18.060 / 0.381 </td> + +<td>4545 / 4702 / 4723 / 6 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>28 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 0.200 / 0.013 </td> + +<td>0.417 / 0.448 / 17.596 / 0.344 </td> + +<td>4679 / 4707 / 4728 / 6 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>28 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.200 / 0.012 </td> + +<td>0.421 / 0.454 / 13.807 / 0.297 </td> + +<td>4733 / 4764 / 4801 / 10 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>28 </td> + +<td>28 </td> + +<td>0.000 / 0.000 / 0.200 / 0.013 </td> + +<td>0.406 / 0.456 / 20.533 / 0.353 </td> + +<td>5646 / 5675 / 5704 / 9 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>32 </td> + +<td>4 </td> + +<td>0.000 / 18.150 / 177.600 / 19.537 </td> + +<td>0.338 / 0.352 / 12.293 / 0.252 </td> + +<td>223 / 224 / 224 / 0 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>32 </td> + +<td>8 </td> + +<td>1.500 / 22.100 / 35.100 / 4.399 </td> + +<td>0.371 / 0.407 / 17.782 / 0.386 </td> + +<td>388 / 398 / 412 / 3 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>32 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.200 / 0.011 </td> + +<td>0.417 / 0.435 / 11.830 / 0.247 </td> + +<td>4698 / 4717 / 4730 / 5 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>32 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.200 / 0.013 </td> + +<td>0.417 / 0.443 / 18.214 / 0.276 </td> + +<td>4700 / 4719 / 4733 / 5 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>32 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 0.200 / 0.012 </td> + +<td>0.417 / 0.437 / 11.208 / 0.232 </td> + +<td>4703 / 4720 / 4737 / 5 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>32 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.200 / 0.013 </td> + +<td>0.417 / 0.448 / 18.288 / 0.405 </td> + +<td>4706 / 4724 / 4740 / 5 </td> + </tr> + +<tr class="b"> + +<td>1000 </td> + +<td>32 </td> + +<td>28 </td> + +<td>0.000 / 0.000 / 0.200 / 0.011 </td> + +<td>0.420 / 0.437 / 13.389 / 0.245 </td> + +<td>4613 / 4782 / 4814 / 9 </td> + </tr> + +<tr class="a"> + +<td>1000 </td> + +<td>32 </td> + +<td>32 </td> + +<td>0.000 / 0.000 / 0.100 / 0.012 </td> + +<td>0.406 / 0.424 / 12.513 / 0.232 </td> + +<td>5641 / 5675 / 5704 / 9 </td> + </tr> + </tbody> +</table></div></div></div> +<div class="section"> +<h2><a name="Table_Cardinality_1500-5000_step_by_500"></a>Table Cardinality 1500-5000, step by 500</h2> +<div class="section"> +<div class="section"> +<h4><a name="Options_Used"></a>Options Used</h4> + +<div class="source"> +<div class="source"> +<pre>num trials: 5000 +card min: 1500 +card max: 5000 +card step: 500 +card start: 1500 +sp min: 4 +sp max: 32 +sp step: 4 +error percentile: 50.0 +time percentile: 50.0 +size percentile: 50.0 +format err as %: true +</pre></div></div> + +<table border="0" class="table table-striped"> + <thead> + +<tr class="a"> + +<th>card </th> + +<th>sp </th> + +<th>p </th> + +<th>err l/m/h/std (% of actual) </th> + +<th>time l/m/h/std (ms) </th> + +<th>size l/m/h/std (b) </th> + </tr> + </thead> + <tbody> + +<tr class="b"> + +<td>1500 </td> + +<td>4 </td> + +<td>4 </td> + +<td>0.000 / 17.600 / 254.400 / 17.627 </td> + +<td>0.501 / 0.612 / 19.629 / 0.593 </td> + +<td>220 / 220 / 220 / 0 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>8 </td> + +<td>4 </td> + +<td>0.000 / 17.600 / 158.133 / 16.980 </td> + +<td>0.506 / 0.594 / 16.843 / 0.493 </td> + +<td>221 / 221 / 221 / 0 </td> + </tr> + +<tr class="b"> + +<td>1500 </td> + +<td>8 </td> + +<td>8 </td> + +<td>0.000 / 4.000 / 23.933 / 3.590 </td> + +<td>0.533 / 0.567 / 11.059 / 0.259 </td> + +<td>392 / 404 / 416 / 3 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>12 </td> + +<td>4 </td> + +<td>0.000 / 18.067 / 208.733 / 17.934 </td> + +<td>0.505 / 0.531 / 20.544 / 0.368 </td> + +<td>221 / 221 / 221 / 0 </td> + </tr> + +<tr class="b"> + +<td>1500 </td> + +<td>12 </td> + +<td>8 </td> + +<td>0.000 / 4.067 / 22.133 / 3.579 </td> + +<td>0.529 / 0.576 / 22.421 / 0.453 </td> + +<td>391 / 404 / 416 / 3 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>12 </td> + +<td>12 </td> + +<td>0.000 / 0.800 / 4.133 / 0.707 </td> + +<td>0.607 / 0.638 / 15.124 / 0.329 </td> + 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/ 7248 / 9 </td> + </tr> + +<tr class="b"> + +<td>1500 </td> + +<td>28 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.133 / 0.012 </td> + +<td>0.619 / 0.662 / 11.359 / 0.279 </td> + +<td>7258 / 7305 / 7354 / 13 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>28 </td> + +<td>28 </td> + +<td>0.000 / 0.000 / 0.133 / 0.012 </td> + +<td>0.609 / 0.648 / 12.260 / 0.300 </td> + +<td>8643 / 8687 / 8732 / 12 </td> + </tr> + +<tr class="b"> + +<td>1500 </td> + +<td>32 </td> + +<td>4 </td> + +<td>0.000 / 17.267 / 175.933 / 18.415 </td> + +<td>0.508 / 0.529 / 11.408 / 0.321 </td> + +<td>224 / 224 / 224 / 0 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>32 </td> + +<td>8 </td> + +<td>0.000 / 14.667 / 29.200 / 4.897 </td> + +<td>0.529 / 0.563 / 11.784 / 0.278 </td> + +<td>393 / 404 / 416 / 3 </td> + </tr> + +<tr class="b"> + +<td>1500 </td> + +<td>32 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.133 / 0.012 </td> + +<td>0.611 / 0.668 / 13.335 / 0.342 </td> + +<td>7226 / 7252 / 7273 / 6 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>32 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.133 / 0.012 </td> + +<td>0.611 / 0.655 / 19.309 / 0.454 </td> + +<td>7230 / 7254 / 7276 / 6 </td> + </tr> + +<tr class="b"> + +<td>1500 </td> + +<td>32 </td> + +<td>20 </td> + +<td>0.000 / 0.000 / 0.133 / 0.012 </td> + +<td>0.612 / 0.643 / 12.975 / 0.314 </td> + +<td>7231 / 7256 / 7277 / 6 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>32 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.067 / 0.013 </td> + +<td>0.612 / 0.666 / 12.496 / 0.357 </td> + +<td>7236 / 7262 / 7286 / 7 </td> + </tr> + +<tr class="b"> + +<td>1500 </td> + +<td>32 </td> + +<td>28 </td> + +<td>0.000 / 0.000 / 0.133 / 0.013 </td> + +<td>0.625 / 0.657 / 21.381 / 0.401 </td> + +<td>7305 / 7347 / 7383 / 11 </td> + </tr> + +<tr class="a"> + +<td>1500 </td> + +<td>32 </td> + +<td>32 </td> + +<td>0.000 / 0.000 / 0.133 / 0.012 </td> + +<td>0.609 / 0.652 / 19.148 / 0.387 </td> + +<td>8643 / 8687 / 8724 / 12 </td> + </tr> + +<tr class="b"> + +<td>2000 </td> + +<td>4 </td> + +<td>4 </td> + +<td>0.050 / 17.950 / 200.300 / 17.701 </td> + +<td>0.678 / 0.712 / 13.274 / 0.370 </td> + +<td>220 / 220 / 220 / 0 </td> + </tr> + +<tr class="a"> + +<td>2000 </td> + +<td>8 </td> + +<td>4 </td> + +<td>0.050 / 17.200 / 227.450 / 17.112 </td> + +<td>0.678 / 0.706 / 12.428 / 0.357 </td> + +<td>221 / 221 / 221 / 0 </td> + </tr> + +<tr class="b"> + +<td>2000 </td> + +<td>8 </td> + +<td>8 </td> + +<td>0.000 / 4.150 / 28.100 / 3.687 </td> + +<td>0.705 / 0.746 / 11.845 / 0.297 </td> + +<td>397 / 408 / 418 / 3 </td> + </tr> + +<tr class="a"> + +<td>2000 </td> + +<td>12 </td> + +<td>4 </td> + +<td>0.050 / 17.075 / 166.300 / 17.289 </td> + +<td>0.679 / 0.746 / 19.810 / 0.436 </td> + +<td>221 / 221 / 221 / 0 </td> + </tr> + +<tr class="b"> + +<td>2000 </td> + +<td>12 </td> + +<td>8 </td> + +<td>0.000 / 4.050 / 26.000 / 3.697 </td> + +<td>0.700 / 0.772 / 19.166 / 0.385 </td> + +<td>396 / 408 / 419 / 3 </td> + </tr> + +<tr class="a"> + +<td>2000 </td> + +<td>12 </td> + +<td>12 </td> + +<td>0.050 / 0.850 / 4.150 / 0.733 </td> + +<td>0.818 / 0.904 / 20.237 / 0.460 </td> + +<td>5109 / 5354 / 5633 / 103 </td> + </tr> + +<tr class="b"> + +<td>2000 </td> + +<td>16 </td> + +<td>4 </td> + +<td>0.050 / 17.650 / 196.200 / 17.539 </td> + +<td>0.678 / 0.747 / 17.211 / 0.435 </td> + +<td>222 / 222 / 222 / 0 </td> + </tr> + +<tr class="a"> + +<td>2000 </td> + +<td>16 </td> + +<td>8 </td> + +<td>0.000 / 4.100 / 23.650 / 3.720 </td> + +<td>0.699 / 0.770 / 17.887 / 0.378 </td> + +<td>397 / 409 / 420 / 3 </td> + </tr> + +<tr class="b"> + +<td>2000 </td> + +<td>16 </td> + +<td>12 </td> + +<td>0.000 / 0.200 / 1.100 / 0.163 </td> + +<td>0.835 / 0.915 / 24.825 / 0.515 </td> + +<td>6456 / 6547 / 6647 / 23 </td> + </tr> + +<tr class="a"> + +<td>2000 </td> + +<td>16 </td> + +<td>16 </td> + +<td>0.000 / 0.200 / 1.100 / 0.167 </td> + +<td>0.820 / 0.906 / 15.149 / 0.427 </td> + +<td>8177 / 8278 / 8384 / 27 </td> + 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class="b"> + +<td>2000 </td> + +<td>24 </td> + +<td>24 </td> + +<td>0.000 / 0.000 / 0.150 / 0.017 </td> + +<td>0.823 / 0.874 / 13.270 / 0.398 </td> + +<td>10491 / 10531 / 10569 / 11 </td> + </tr> + +<tr class="a"> + +<td>2000 </td> + +<td>28 </td> + +<td>4 </td> + +<td>0.050 / 17.600 / 206.800 / 17.750 </td> + +<td>0.675 / 0.726 / 18.063 / 0.424 </td> + +<td>224 / 224 / 224 / 0 </td> + </tr> + +<tr class="b"> + +<td>2000 </td> + +<td>28 </td> + +<td>8 </td> + +<td>0.000 / 6.050 / 23.850 / 4.359 </td> + +<td>0.698 / 0.752 / 24.616 / 0.531 </td> + +<td>400 / 411 / 421 / 3 </td> + </tr> + +<tr class="a"> + +<td>2000 </td> + +<td>28 </td> + +<td>12 </td> + +<td>0.000 / 0.000 / 0.100 / 0.013 </td> + +<td>0.826 / 0.879 / 16.497 / 0.458 </td> + +<td>8600 / 8623 / 8641 / 6 </td> + </tr> + +<tr class="b"> + +<td>2000 </td> + +<td>28 </td> + +<td>16 </td> + +<td>0.000 / 0.000 / 0.100 / 0.012 </td> + +<td>0.826 / 0.898 / 16.948 / 0.417 </td> + +<td>8602 / 8626 / 8645 / 6 </td> + </tr> + +<tr 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