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  25. <div class="section" lang="en">
    26. 

  26. <div class="titlepage"><div><div><h3 class="title">
    27. 

  27. <a name="boost_icl.implementation.complexity"></a><a class="link" href="complexity.html" title="Complexity">Complexity</a>
    28. 

  28. </h3></div></div></div>
    29. 

  29. <a name="boost_icl.implementation.complexity.complexity_of_element_containers"></a><h5>
    30. 

  30. <a name="id1124696"></a>
    31. 

  31.         <a class="link" href="complexity.html#boost_icl.implementation.complexity.complexity_of_element_containers">Complexity
    32. 

  32.         of element containers</a>
    33. 

  33.       </h5>
    34. 

  34. <p>
    35. 

  35.         Since <span class="emphasis"><em>element containers</em></span> <a href="http://www.cplusplus.com/reference/stl/set/" target="_top"><code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">set</span></code>
    36. 

  36.         </a> and <code class="computeroutput"><a class="link" href="../../boost/icl/map.html" title="Class template map">icl::map</a></code> are only
    37. 

  37.         extensions of stl::set and stl::map, their complexity characteristics are
    38. 

  38.         accordingly. So their major operations insertion (addition), deletion and
    39. 

  39.         search are all using logarithmic time.
    40. 

  40.       </p>
    41. 

  41. <a name="boost_icl.implementation.complexity.complexity_of_interval_containers"></a><h5>
    42. 

  42. <a name="id1124742"></a>
    43. 

  43.         <a class="link" href="complexity.html#boost_icl.implementation.complexity.complexity_of_interval_containers">Complexity
    44. 

  44.         of interval containers</a>
    45. 

  45.       </h5>
    46. 

  46. <p>
    47. 

  47.         The operations on <span class="emphasis"><em>interval containers</em></span> behave differently
    48. 

  48.         due to the fact that intervals unlike elements can overlap any number of
    49. 

  49.         other intervals in a container. As long as intervals are relatively small
    50. 

  50.         or just singleton, interval containers behave like containers of elements.
    51. 

  51.         For large intervals however time consumption of operations on interval containers
    52. 

  52.         may be worse, because most or all intervals of a container may have to be
    53. 

  53.         visited. As an example, time complexity of <a class="link" href="../function_reference/addition.html" title="Addition"><span class="emphasis"><em><span class="bold"><strong>Addition</strong></span></em></span></a> on interval containers
    54. 

  54.         is briefly discussed.
    55. 

  55.       </p>
    56. 

  56. <p>
    57. 

  57.         More information on <span class="emphasis"><em><span class="bold"><strong>complexity characteristics</strong></span></em></span>
    58. 

  58.         of <span class="bold"><strong>icl's</strong></span> functions is contained in section
    59. 

  59.         <a class="link" href="../function_reference.html" title="Function Reference">Function Reference</a>
    60. 

  60.       </p>
    61. 

  61. <a name="boost_icl.implementation.complexity.time_complexity_of_addition"></a><h6>
    62. 

  62. <a name="id1127141"></a>
    63. 

  63.         <a class="link" href="complexity.html#boost_icl.implementation.complexity.time_complexity_of_addition">Time
    64. 

  64.         Complexity of Addition</a>
    65. 

  65.       </h6>
    66. 

  66. <p>
    67. 

  67.         The next table gives the time complexities for the overloaded <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+=</span></code>
    68. 

  68.         on interval containers. The instance types of <code class="computeroutput"><span class="identifier">T</span></code>
    69. 

  69.         are given as column headers. Instances of type parameter <code class="computeroutput"><span class="identifier">P</span></code>
    70. 

  70.         are denoted in the second column. The third column contains the specific
    71. 

  71.         kind of complexity statement. If column three is empty <span class="emphasis"><em><span class="bold"><strong>worst case</strong></span></em></span> complexity is given in the related
    72. 

  72.         row.
    73. 

  73.       </p>
    74. 

  74. <div class="table">
    75. 

  75. <a name="id1127189"></a><p class="title"><b>Table&#160;1.15.&#160;Time Complexity of Addition:</b></p>
    76. 

  76. <div class="table-contents"><table class="table" summary="Time Complexity of Addition:">
    77. 

  77. <colgroup>
    78. 

  78. <col>
    79. 

  79. <col>
    80. 

  80. <col>
    81. 

  81. <col>
    82. 

  82. <col>
    83. 

  83. <col>
    84. 

  84. <col>
    85. 

  85. <col>
    86. 

  86. </colgroup>
    87. 

  87. <thead><tr>
    88. 

  88. <th>
    89. 

  89.             <p>
    90. 

  90.             </p>
    91. 

  91.             </th>
    92. 

  92. <th>
    93. 

  93.             <p>
    94. 

  94.               <code class="computeroutput"><span class="identifier">P</span></code>
    95. 

  95.             </p>
    96. 

  96.             </th>
    97. 

  97. <th>
    98. 

  98.             <p>
    99. 

  99.             </p>
    100. 

  100.             </th>
    101. 

  101. <th>
    102. 

  102.             <p>
    103. 

  103.               interval<br> set
    104. 

  104.             </p>
    105. 

  105.             </th>
    106. 

  106. <th>
    107. 

  107.             <p>
    108. 

  108.               separate<br> interval<br> set
    109. 

  109.             </p>
    110. 

  110.             </th>
    111. 

  111. <th>
    112. 

  112.             <p>
    113. 

  113.               split<br> interval<br> set
    114. 

  114.             </p>
    115. 

  115.             </th>
    116. 

  116. <th>
    117. 

  117.             <p>
    118. 

  118.               interval<br> map
    119. 

  119.             </p>
    120. 

  120.             </th>
    121. 

  121. <th>
    122. 

  122.             <p>
    123. 

  123.               split<br> interval<br> map
    124. 

  124.             </p>
    125. 

  125.             </th>
    126. 

  126. </tr></thead>
    127. 

  127. <tbody>
    128. 

  128. <tr>
    129. 

  129. <td>
    130. 

  130.             <p>
    131. 

  131.               <code class="computeroutput"><span class="identifier">T</span><span class="special">&amp;</span>
    132. 

  132.               <span class="keyword">operator</span> <span class="special">+=(</span><span class="identifier">T</span><span class="special">&amp;</span>
    133. 

  133.               <span class="identifier">object</span><span class="special">,</span>
    134. 

  134.               <span class="keyword">const</span> <span class="identifier">P</span><span class="special">&amp;</span> <span class="identifier">addend</span><span class="special">)</span></code>
    135. 

  135.             </p>
    136. 

  136.             </td>
    137. 

  137. <td>
    138. 

  138.             <p>
    139. 

  139.               <code class="computeroutput"><span class="identifier">T</span><span class="special">::</span><span class="identifier">element_type</span></code>
    140. 

  140.             </p>
    141. 

  141.             </td>
    142. 

  142. <td>
    143. 

  143.             <p>
    144. 

  144.             </p>
    145. 

  145.             </td>
    146. 

  146. <td>
    147. 

  147.             <p>
    148. 

  148.               <span class="emphasis"><em>O(log n)</em></span>
    149. 

  149.             </p>
    150. 

  150.             </td>
    151. 

  151. <td>
    152. 

  152.             <p>
    153. 

  153.               <span class="emphasis"><em>O(log n)</em></span>
    154. 

  154.             </p>
    155. 

  155.             </td>
    156. 

  156. <td>
    157. 

  157.             <p>
    158. 

  158.               <span class="emphasis"><em>O(log n)</em></span>
    159. 

  159.             </p>
    160. 

  160.             </td>
    161. 

  161. <td>
    162. 

  162.             <p>
    163. 

  163.               <span class="emphasis"><em>O(log n)</em></span>
    164. 

  164.             </p>
    165. 

  165.             </td>
    166. 

  166. <td>
    167. 

  167.             <p>
    168. 

  168.               <span class="emphasis"><em>O(log n)</em></span>
    169. 

  169.             </p>
    170. 

  170.             </td>
    171. 

  171. </tr>
    172. 

  172. <tr>
    173. 

  173. <td>
    174. 

  174.             <p>
    175. 

  175.             </p>
    176. 

  176.             </td>
    177. 

  177. <td>
    178. 

  178.             <p>
    179. 

  179.               <code class="computeroutput"><span class="identifier">T</span><span class="special">::</span><span class="identifier">segment_type</span></code>
    180. 

  180.             </p>
    181. 

  181.             </td>
    182. 

  182. <td>
    183. 

  183.             <p>
    184. 

  184.               best case
    185. 

  185.             </p>
    186. 

  186.             </td>
    187. 

  187. <td>
    188. 

  188.             <p>
    189. 

  189.               <span class="emphasis"><em>O(log n)</em></span>
    190. 

  190.             </p>
    191. 

  191.             </td>
    192. 

  192. <td>
    193. 

  193.             <p>
    194. 

  194.               <span class="emphasis"><em>O(log n)</em></span>
    195. 

  195.             </p>
    196. 

  196.             </td>
    197. 

  197. <td>
    198. 

  198.             <p>
    199. 

  199.               <span class="emphasis"><em>O(log n)</em></span>
    200. 

  200.             </p>
    201. 

  201.             </td>
    202. 

  202. <td>
    203. 

  203.             <p>
    204. 

  204.               <span class="emphasis"><em>O(log n)</em></span>
    205. 

  205.             </p>
    206. 

  206.             </td>
    207. 

  207. <td>
    208. 

  208.             <p>
    209. 

  209.               <span class="emphasis"><em>O(log n)</em></span>
    210. 

  210.             </p>
    211. 

  211.             </td>
    212. 

  212. </tr>
    213. 

  213. <tr>
    214. 

  214. <td>
    215. 

  215.             <p>
    216. 

  216.             </p>
    217. 

  217.             </td>
    218. 

  218. <td>
    219. 

  219.             <p>
    220. 

  220.             </p>
    221. 

  221.             </td>
    222. 

  222. <td>
    223. 

  223.             <p>
    224. 

  224.               worst case
    225. 

  225.             </p>
    226. 

  226.             </td>
    227. 

  227. <td>
    228. 

  228.             <p>
    229. 

  229.               <span class="emphasis"><em>O(n)</em></span>
    230. 

  230.             </p>
    231. 

  231.             </td>
    232. 

  232. <td>
    233. 

  233.             <p>
    234. 

  234.               <span class="emphasis"><em>O(n)</em></span>
    235. 

  235.             </p>
    236. 

  236.             </td>
    237. 

  237. <td>
    238. 

  238.             <p>
    239. 

  239.               <span class="emphasis"><em>O(n)</em></span>
    240. 

  240.             </p>
    241. 

  241.             </td>
    242. 

  242. <td>
    243. 

  243.             <p>
    244. 

  244.               <span class="emphasis"><em>O(n)</em></span>
    245. 

  245.             </p>
    246. 

  246.             </td>
    247. 

  247. <td>
    248. 

  248.             <p>
    249. 

  249.               <span class="emphasis"><em>O(n)</em></span>
    250. 

  250.             </p>
    251. 

  251.             </td>
    252. 

  252. </tr>
    253. 

  253. <tr>
    254. 

  254. <td>
    255. 

  255.             <p>
    256. 

  256.             </p>
    257. 

  257.             </td>
    258. 

  258. <td>
    259. 

  259.             <p>
    260. 

  260.             </p>
    261. 

  261.             </td>
    262. 

  262. <td>
    263. 

  263.             <p>
    264. 

  264.               amortized
    265. 

  265.             </p>
    266. 

  266.             </td>
    267. 

  267. <td>
    268. 

  268.             <p>
    269. 

  269.               <span class="emphasis"><em>O(log n)</em></span>
    270. 

  270.             </p>
    271. 

  271.             </td>
    272. 

  272. <td>
    273. 

  273.             <p>
    274. 

  274.               <span class="emphasis"><em>O(log n)</em></span>
    275. 

  275.             </p>
    276. 

  276.             </td>
    277. 

  277. <td>
    278. 

  278.             <p>
    279. 

  279.             </p>
    280. 

  280.             </td>
    281. 

  281. <td>
    282. 

  282.             <p>
    283. 

  283.             </p>
    284. 

  284.             </td>
    285. 

  285. <td>
    286. 

  286.             <p>
    287. 

  287.             </p>
    288. 

  288.             </td>
    289. 

  289. </tr>
    290. 

  290. <tr>
    291. 

  291. <td>
    292. 

  292.             <p>
    293. 

  293.             </p>
    294. 

  294.             </td>
    295. 

  295. <td>
    296. 

  296.             <p>
    297. 

  297.               <code class="computeroutput"><span class="identifier">interval_sets</span></code>
    298. 

  298.             </p>
    299. 

  299.             </td>
    300. 

  300. <td>
    301. 

  301.             <p>
    302. 

  302.             </p>
    303. 

  303.             </td>
    304. 

  304. <td>
    305. 

  305.             <p>
    306. 

  306.               <span class="emphasis"><em>O(m log(n+m))</em></span>
    307. 

  307.             </p>
    308. 

  308.             </td>
    309. 

  309. <td>
    310. 

  310.             <p>
    311. 

  311.               <span class="emphasis"><em>O(m log(n+m))</em></span>
    312. 

  312.             </p>
    313. 

  313.             </td>
    314. 

  314. <td>
    315. 

  315.             <p>
    316. 

  316.               <span class="emphasis"><em>O(m log(n+m))</em></span>
    317. 

  317.             </p>
    318. 

  318.             </td>
    319. 

  319. <td>
    320. 

  320.             <p>
    321. 

  321.             </p>
    322. 

  322.             </td>
    323. 

  323. <td>
    324. 

  324.             <p>
    325. 

  325.             </p>
    326. 

  326.             </td>
    327. 

  327. </tr>
    328. 

  328. <tr>
    329. 

  329. <td>
    330. 

  330.             <p>
    331. 

  331.             </p>
    332. 

  332.             </td>
    333. 

  333. <td>
    334. 

  334.             <p>
    335. 

  335.               <code class="computeroutput"><span class="identifier">interval_maps</span></code>
    336. 

  336.             </p>
    337. 

  337.             </td>
    338. 

  338. <td>
    339. 

  339.             <p>
    340. 

  340.             </p>
    341. 

  341.             </td>
    342. 

  342. <td>
    343. 

  343.             <p>
    344. 

  344.             </p>
    345. 

  345.             </td>
    346. 

  346. <td>
    347. 

  347.             <p>
    348. 

  348.             </p>
    349. 

  349.             </td>
    350. 

  350. <td>
    351. 

  351.             <p>
    352. 

  352.             </p>
    353. 

  353.             </td>
    354. 

  354. <td>
    355. 

  355.             <p>
    356. 

  356.               <span class="emphasis"><em>O(m log(n+m))</em></span>
    357. 

  357.             </p>
    358. 

  358.             </td>
    359. 

  359. <td>
    360. 

  360.             <p>
    361. 

  361.               <span class="emphasis"><em>O(m log(n+m))</em></span>
    362. 

  362.             </p>
    363. 

  363.             </td>
    364. 

  364. </tr>
    365. 

  365. </tbody>
    366. 

  366. </table></div>
    367. 

  367. </div>
    368. 

  368. <br class="table-break"><p>
    369. 

  369.         Adding an <span class="emphasis"><em>element</em></span> or <span class="emphasis"><em>element value pair</em></span>
    370. 

  370.         is always done in <span class="emphasis"><em>logarithmic time</em></span>, where <span class="emphasis"><em>n</em></span>
    371. 

  371.         is the number of intervals in the interval container. The same row of complexities
    372. 

  372.         applies to the insertion of a <span class="emphasis"><em>segment</em></span> (an interval or
    373. 

  373.         an interval value pair) in the <span class="emphasis"><em><span class="bold"><strong>best case</strong></span></em></span>,
    374. 

  374.         where the inserted segment does overlap with only a <span class="emphasis"><em><span class="bold"><strong>small</strong></span></em></span>
    375. 

  375.         number of intervals in the container.
    376. 

  376.       </p>
    377. 

  377. <p>
    378. 

  378.         In the <span class="emphasis"><em><span class="bold"><strong>worst case</strong></span></em></span>,
    379. 

  379.         where the inserted segment overlaps with all intervals in the container,
    380. 

  380.         the algorithms iterate over all the overlapped segments. Using inplace manipulations
    381. 

  381.         of segments and hinted inserts, it is possible to perform all necessary operations
    382. 

  382.         on each iteration step in <span class="emphasis"><em>constant time</em></span>. This results
    383. 

  383.         in <span class="emphasis"><em><span class="bold"><strong>linear worst case time</strong></span></em></span>
    384. 

  384.         complexity for segment addition for all interval containers.
    385. 

  385.       </p>
    386. 

  386. <p>
    387. 

  387.         After performing a worst case addition for an <code class="computeroutput"><a class="link" href="../../boost/icl/interval_set.html" title="Class template interval_set">interval_set</a></code>
    388. 

  388.         or a <code class="computeroutput"><a class="link" href="../../boost/icl/separate_interval_set.html" title="Class template separate_interval_set">separate_interval_sets</a></code>
    389. 

  389.         adding an interval that overlaps <span class="emphasis"><em>n</em></span> intervals, we need
    390. 

  390.         <span class="emphasis"><em>n</em></span> non overlapping additions of <span class="emphasis"><em>logarithmic
    391. 

  391.         time</em></span> before we can launch another <span class="emphasis"><em>O(n)</em></span> worst
    392. 

  392.         case addition. So we have only a <span class="emphasis"><em><span class="bold"><strong>logarithmic
    393. 

  393.         amortized time</strong></span></em></span> for the addition of an interval or interval
    394. 

  394.         value pair.
    395. 

  395.       </p>
    396. 

  396. <p>
    397. 

  397.         For the addition of <span class="emphasis"><em><span class="bold"><strong>interval containers</strong></span></em></span>
    398. 

  398.         complexity is <span class="emphasis"><em>O(m log(n+m))</em></span>. So for the <span class="emphasis"><em><span class="bold"><strong>worst case</strong></span></em></span>, where the container sizes
    399. 

  399.         <span class="emphasis"><em>n</em></span> and <span class="emphasis"><em>m</em></span> are equal and both containers
    400. 

  400.         cover the same ranges, the complexity of container addition is <span class="emphasis"><em><span class="bold"><strong>loglinear</strong></span></em></span>. For other cases, that occur
    401. 

  401.         frequently in real world applications performance can be much better. If
    402. 

  402.         the added container <code class="computeroutput"><span class="identifier">operand</span></code>
    403. 

  403.         is much smaller than <code class="computeroutput"><span class="identifier">object</span></code>
    404. 

  404.         and the intervals in <code class="computeroutput"><span class="identifier">operand</span></code>
    405. 

  405.         are relatively small, performance can be <span class="emphasis"><em>logarithmic</em></span>.
    406. 

  406.         If <span class="emphasis"><em>m</em></span> is small compared with <span class="emphasis"><em>n</em></span> and
    407. 

  407.         intervals in <code class="computeroutput"><span class="identifier">operand</span></code> are
    408. 

  408.         large, performance tends to be <span class="emphasis"><em>linear</em></span>.
    409. 

  409.       </p>
    410. 

  410. </div>
    411. 

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    412. 

  412. <td align="left"></td>
    413. 

  413. <td align="right"><div class="copyright-footer">Copyright &#169; 2007 -2010 Joachim Faulhaber<br>Copyright &#169; 1999 -2006 Cortex Software GmbH<p>
    414. 

  414.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    415. 

  415.         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
    416. 

  416.       </p>
    417. 

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