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<h1 id="frequent-items">Frequent Items</h1>
<h3 id="error-threshold-table">Error Threshold Table</h3>
<p>This table of thresholds is computed a follows:</p>
<ul>
<li><i>M</i> = Max Map Size. Always a power of 2.</li>
<li><i>N</i> = Stream Length (Total Weights or counts of items)</li>
<li><i>e</i> = Epsilon = 3.5/<i>M</i> (minimum detectable frequency)</li>
<li>
<p><i>T</i> = Threshold = (<i>N</i> &lt; 0.75 * <i>M</i>)? 0 : <i>e * N</i>.</p>
</li>
<li>The error in <i>getEstimate(item)</i> is guaranteed to be at most <i>T</i>.</li>
<li>The function <i>getFrequentItems(NO_FALSE_NEGATIVES)</i> is guaranteed to return a list of items that
contains all items with frequency greater than <i>T</i> and might contain some additional
items, the <i>False Positives</i>, as well.</li>
<li>The function <i>getFrequentItems(NO_FALSE_POSITIVES)</i> is guaranteed return a list of items
that contains all items with frequency greater than 2*<i>T</i>, and no items with frequency less
than <i>T</i>, but it may exclude some items with frequency greater than
<i>T</i> (the <i>False Negatives</i>).</li>
</ul>
<table>
<thead>
<tr>
<th>Max Map Size-&gt;</th>
<th>8</th>
<th>16</th>
<th>32</th>
<th>64</th>
<th>128</th>
<th>256</th>
<th>512</th>
<th>1,024</th>
<th>2,048</th>
<th>4,096</th>
<th>8,192</th>
<th>16,384</th>
<th>32,768</th>
<th>65,536</th>
<th>131,072</th>
<th>262,144</th>
<th>524,288</th>
<th>1,048,576</th>
</tr>
<tr>
<th>Epsilon-&gt;</th>
<th>0.4375</th>
<th>0.2188</th>
<th>0.1094</th>
<th>0.0547</th>
<th>0.0273</th>
<th>0.0137</th>
<th>6.84E-03</th>
<th>3.42E-03</th>
<th>1.71E-03</th>
<th>8.54E-04</th>
<th>4.27E-04</th>
<th>2.14E-04</th>
<th>1.07E-04</th>
<th>5.34E-05</th>
<th>2.67E-05</th>
<th>1.34E-05</th>
<th>6.68E-06</th>
<th>3.34E-06</th>
</tr>
<tr>
<th>Stream Length</th>
<th>Threshold-&gt;</th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
<th> </th>
</tr>
</thead>
<tbody>
<tr>
<td><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong>_</td>
<td><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong><strong>__</strong></td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>100</td>
<td>44</td>
<td>22</td>
<td>11</td>
<td>5</td>
<td>3</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>178</td>
<td>78</td>
<td>39</td>
<td>19</td>
<td>10</td>
<td>5</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>316</td>
<td>138</td>
<td>69</td>
<td>35</td>
<td>17</td>
<td>9</td>
<td>4</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>562</td>
<td>246</td>
<td>123</td>
<td>62</td>
<td>31</td>
<td>15</td>
<td>8</td>
<td>4</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>1,000</td>
<td>438</td>
<td>219</td>
<td>109</td>
<td>55</td>
<td>27</td>
<td>14</td>
<td>7</td>
<td>3</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>1,778</td>
<td>778</td>
<td>389</td>
<td>194</td>
<td>97</td>
<td>49</td>
<td>24</td>
<td>12</td>
<td>6</td>
<td>3</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>3,162</td>
<td>1,383</td>
<td>692</td>
<td>346</td>
<td>173</td>
<td>86</td>
<td>43</td>
<td>22</td>
<td>11</td>
<td>5</td>
<td>3</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>5,623</td>
<td>2,460</td>
<td>1,230</td>
<td>615</td>
<td>308</td>
<td>154</td>
<td>77</td>
<td>38</td>
<td>19</td>
<td>10</td>
<td>5</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>10,000</td>
<td>4,375</td>
<td>2,188</td>
<td>1,094</td>
<td>547</td>
<td>273</td>
<td>137</td>
<td>68</td>
<td>34</td>
<td>17</td>
<td>9</td>
<td>4</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>17,783</td>
<td>7,780</td>
<td>3,890</td>
<td>1,945</td>
<td>972</td>
<td>486</td>
<td>243</td>
<td>122</td>
<td>61</td>
<td>30</td>
<td>15</td>
<td>8</td>
<td>4</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>31,623</td>
<td>13,835</td>
<td>6,917</td>
<td>3,459</td>
<td>1,729</td>
<td>865</td>
<td>432</td>
<td>216</td>
<td>108</td>
<td>54</td>
<td>27</td>
<td>14</td>
<td>7</td>
<td>3</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>56,234</td>
<td>24,602</td>
<td>12,301</td>
<td>6,151</td>
<td>3,075</td>
<td>1,538</td>
<td>769</td>
<td>384</td>
<td>192</td>
<td>96</td>
<td>48</td>
<td>24</td>
<td>12</td>
<td>6</td>
<td>3</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>100,000</td>
<td>43,750</td>
<td>21,875</td>
<td>10,938</td>
<td>5,469</td>
<td>2,734</td>
<td>1,367</td>
<td>684</td>
<td>342</td>
<td>171</td>
<td>85</td>
<td>43</td>
<td>21</td>
<td>11</td>
<td>5</td>
<td>3</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>177,828</td>
<td>77,800</td>
<td>38,900</td>
<td>19,450</td>
<td>9,725</td>
<td>4,862</td>
<td>2,431</td>
<td>1,216</td>
<td>608</td>
<td>304</td>
<td>152</td>
<td>76</td>
<td>38</td>
<td>19</td>
<td>9</td>
<td>5</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>316,228</td>
<td>138,350</td>
<td>69,175</td>
<td>34,587</td>
<td>17,294</td>
<td>8,647</td>
<td>4,323</td>
<td>2,162</td>
<td>1,081</td>
<td>540</td>
<td>270</td>
<td>135</td>
<td>68</td>
<td>34</td>
<td>17</td>
<td>8</td>
<td>4</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>562,341</td>
<td>246,024</td>
<td>123,012</td>
<td>61,506</td>
<td>30,753</td>
<td>15,377</td>
<td>7,688</td>
<td>3,844</td>
<td>1,922</td>
<td>961</td>
<td>481</td>
<td>240</td>
<td>120</td>
<td>60</td>
<td>30</td>
<td>15</td>
<td>8</td>
<td>4</td>
<td>0</td>
</tr>
<tr>
<td>1.0E+06</td>
<td>437,500</td>
<td>218,750</td>
<td>109,375</td>
<td>54,688</td>
<td>27,344</td>
<td>13,672</td>
<td>6,836</td>
<td>3,418</td>
<td>1,709</td>
<td>854</td>
<td>427</td>
<td>214</td>
<td>107</td>
<td>53</td>
<td>27</td>
<td>13</td>
<td>7</td>
<td>3</td>
</tr>
<tr>
<td>1.8E+06</td>
<td>777,997</td>
<td>388,999</td>
<td>194,499</td>
<td>97,250</td>
<td>48,625</td>
<td>24,312</td>
<td>12,156</td>
<td>6,078</td>
<td>3,039</td>
<td>1,520</td>
<td>760</td>
<td>380</td>
<td>190</td>
<td>95</td>
<td>47</td>
<td>24</td>
<td>12</td>
<td>6</td>
</tr>
<tr>
<td>3.2E+06</td>
<td>1.38E+06</td>
<td>691,748</td>
<td>345,874</td>
<td>172,937</td>
<td>86,469</td>
<td>43,234</td>
<td>21,617</td>
<td>10,809</td>
<td>5,404</td>
<td>2,702</td>
<td>1,351</td>
<td>676</td>
<td>338</td>
<td>169</td>
<td>84</td>
<td>42</td>
<td>21</td>
<td>11</td>
</tr>
<tr>
<td>5.6E+06</td>
<td>2.46E+06</td>
<td>1.23E+06</td>
<td>615,061</td>
<td>307,530</td>
<td>153,765</td>
<td>76,883</td>
<td>38,441</td>
<td>19,221</td>
<td>9,610</td>
<td>4,805</td>
<td>2,403</td>
<td>1,201</td>
<td>601</td>
<td>300</td>
<td>150</td>
<td>75</td>
<td>38</td>
<td>19</td>
</tr>
<tr>
<td>1.0E+07</td>
<td>4.38E+06</td>
<td>2.19E+06</td>
<td>1.09E+06</td>
<td>546,875</td>
<td>273,438</td>
<td>136,719</td>
<td>68,359</td>
<td>34,180</td>
<td>17,090</td>
<td>8,545</td>
<td>4,272</td>
<td>2,136</td>
<td>1,068</td>
<td>534</td>
<td>267</td>
<td>134</td>
<td>67</td>
<td>33</td>
</tr>
<tr>
<td>1.8E+07</td>
<td>7.78E+06</td>
<td>3.89E+06</td>
<td>1.94E+06</td>
<td>972,497</td>
<td>486,248</td>
<td>243,124</td>
<td>121,562</td>
<td>60,781</td>
<td>30,391</td>
<td>15,195</td>
<td>7,598</td>
<td>3,799</td>
<td>1,899</td>
<td>950</td>
<td>475</td>
<td>237</td>
<td>119</td>
<td>59</td>
</tr>
<tr>
<td>3.2E+07</td>
<td>1.38E+07</td>
<td>6.92E+06</td>
<td>3.46E+06</td>
<td>1.73E+06</td>
<td>864,685</td>
<td>432,343</td>
<td>216,171</td>
<td>108,086</td>
<td>54,043</td>
<td>27,021</td>
<td>13,511</td>
<td>6,755</td>
<td>3,378</td>
<td>1,689</td>
<td>844</td>
<td>422</td>
<td>211</td>
<td>106</td>
</tr>
<tr>
<td>5.6E+07</td>
<td>2.46E+07</td>
<td>1.23E+07</td>
<td>6.15E+06</td>
<td>3.08E+06</td>
<td>1.54E+06</td>
<td>768,826</td>
<td>384,413</td>
<td>192,207</td>
<td>96,103</td>
<td>48,052</td>
<td>24,026</td>
<td>12,013</td>
<td>6,006</td>
<td>3,003</td>
<td>1,502</td>
<td>751</td>
<td>375</td>
<td>188</td>
</tr>
<tr>
<td>1.0E+08</td>
<td>4.38E+07</td>
<td>2.19E+07</td>
<td>1.09E+07</td>
<td>5.47E+06</td>
<td>2.73E+06</td>
<td>1.37E+06</td>
<td>683,594</td>
<td>341,797</td>
<td>170,898</td>
<td>85,449</td>
<td>42,725</td>
<td>21,362</td>
<td>10,681</td>
<td>5,341</td>
<td>2,670</td>
<td>1,335</td>
<td>668</td>
<td>334</td>
</tr>
<tr>
<td>1.8E+08</td>
<td>7.78E+07</td>
<td>3.89E+07</td>
<td>1.94E+07</td>
<td>9.72E+06</td>
<td>4.86E+06</td>
<td>2.43E+06</td>
<td>1.22E+06</td>
<td>607,810</td>
<td>303,905</td>
<td>151,953</td>
<td>75,976</td>
<td>37,988</td>
<td>18,994</td>
<td>9,497</td>
<td>4,749</td>
<td>2,374</td>
<td>1,187</td>
<td>594</td>
</tr>
<tr>
<td>3.2E+08</td>
<td>1.38E+08</td>
<td>6.92E+07</td>
<td>3.46E+07</td>
<td>1.73E+07</td>
<td>8.65E+06</td>
<td>4.32E+06</td>
<td>2.16E+06</td>
<td>1.08E+06</td>
<td>540,428</td>
<td>270,214</td>
<td>135,107</td>
<td>67,554</td>
<td>33,777</td>
<td>16,888</td>
<td>8,444</td>
<td>4,222</td>
<td>2,111</td>
<td>1,056</td>
</tr>
<tr>
<td>5.6E+08</td>
<td>2.46E+08</td>
<td>1.23E+08</td>
<td>6.15E+07</td>
<td>3.08E+07</td>
<td>1.54E+07</td>
<td>7.69E+06</td>
<td>3.84E+06</td>
<td>1.92E+06</td>
<td>961,033</td>
<td>480,516</td>
<td>240,258</td>
<td>120,129</td>
<td>60,065</td>
<td>30,032</td>
<td>15,016</td>
<td>7,508</td>
<td>3,754</td>
<td>1,877</td>
</tr>
<tr>
<td>1.0E+09</td>
<td>4.38E+08</td>
<td>2.19E+08</td>
<td>1.09E+08</td>
<td>5.47E+07</td>
<td>2.73E+07</td>
<td>1.37E+07</td>
<td>6.84E+06</td>
<td>3.42E+06</td>
<td>1.71E+06</td>
<td>854,492</td>
<td>427,246</td>
<td>213,623</td>
<td>106,812</td>
<td>53,406</td>
<td>26,703</td>
<td>13,351</td>
<td>6,676</td>
<td>3,338</td>
</tr>
<tr>
<td>1.8E+09</td>
<td>7.78E+08</td>
<td>3.89E+08</td>
<td>1.94E+08</td>
<td>9.72E+07</td>
<td>4.86E+07</td>
<td>2.43E+07</td>
<td>1.22E+07</td>
<td>6.08E+06</td>
<td>3.04E+06</td>
<td>1.52E+06</td>
<td>759,763</td>
<td>379,881</td>
<td>189,941</td>
<td>94,970</td>
<td>47,485</td>
<td>23,743</td>
<td>11,871</td>
<td>5,936</td>
</tr>
<tr>
<td>3.2E+09</td>
<td>1.38E+09</td>
<td>6.92E+08</td>
<td>3.46E+08</td>
<td>1.73E+08</td>
<td>8.65E+07</td>
<td>4.32E+07</td>
<td>2.16E+07</td>
<td>1.08E+07</td>
<td>5.40E+06</td>
<td>2.70E+06</td>
<td>1.35E+06</td>
<td>675,535</td>
<td>337,768</td>
<td>168,884</td>
<td>84,442</td>
<td>42,221</td>
<td>21,110</td>
<td>10,555</td>
</tr>
<tr>
<td>5.6E+09</td>
<td>2.46E+09</td>
<td>1.23E+09</td>
<td>6.15E+08</td>
<td>3.08E+08</td>
<td>1.54E+08</td>
<td>7.69E+07</td>
<td>3.84E+07</td>
<td>1.92E+07</td>
<td>9.61E+06</td>
<td>4.81E+06</td>
<td>2.40E+06</td>
<td>1.20E+06</td>
<td>600,645</td>
<td>300,323</td>
<td>150,161</td>
<td>75,081</td>
<td>37,540</td>
<td>18,770</td>
</tr>
<tr>
<td>1.0E+10</td>
<td>4.38E+09</td>
<td>2.19E+09</td>
<td>1.09E+09</td>
<td>5.47E+08</td>
<td>2.73E+08</td>
<td>1.37E+08</td>
<td>6.84E+07</td>
<td>3.42E+07</td>
<td>1.71E+07</td>
<td>8.54E+06</td>
<td>4.27E+06</td>
<td>2.14E+06</td>
<td>1.07E+06</td>
<td>534,058</td>
<td>267,029</td>
<td>133,514</td>
<td>66,757</td>
<td>33,379</td>
</tr>
</tbody>
</table>
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