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| <div class="section boost_icl_implementation_complexity" lang="en"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_icl.implementation.complexity"></a><a class="link" href="complexity.html" title="Complexity">Complexity</a> |
| </h3></div></div></div> |
| <a name="boost_icl.implementation.complexity.complexity_of_element_containers"></a><h5> |
| <a name="id1132060"></a> |
| <a class="link" href="complexity.html#boost_icl.implementation.complexity.complexity_of_element_containers">Complexity |
| of element containers</a> |
| </h5> |
| <p> |
| 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> |
| </a> and <code class="computeroutput"><a class="link" href="../../boost/icl/map.html" title="Class template map">icl::map</a></code> are only |
| extensions of stl::set and stl::map, their complexity characteristics are |
| accordingly. So their major operations insertion (addition), deletion and |
| search are all using logarithmic time. |
| </p> |
| <a name="boost_icl.implementation.complexity.complexity_of_interval_containers"></a><h5> |
| <a name="id1132106"></a> |
| <a class="link" href="complexity.html#boost_icl.implementation.complexity.complexity_of_interval_containers">Complexity |
| of interval containers</a> |
| </h5> |
| <p> |
| The operations on <span class="emphasis"><em>interval containers</em></span> behave differently |
| due to the fact that intervals unlike elements can overlap any number of |
| other intervals in a container. As long as intervals are relatively small |
| or just singleton, interval containers behave like containers of elements. |
| For large intervals however time consumption of operations on interval containers |
| may be worse, because most or all intervals of a container may have to be |
| 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 |
| is briefly discussed. |
| </p> |
| <p> |
| More information on <span class="emphasis"><em><span class="bold"><strong>complexity characteristics</strong></span></em></span> |
| of <span class="bold"><strong>icl's</strong></span> functions is contained in section |
| <a class="link" href="../function_reference.html" title="Function Reference">Function Reference</a> |
| </p> |
| <a name="boost_icl.implementation.complexity.time_complexity_of_addition"></a><h6> |
| <a name="id1132157"></a> |
| <a class="link" href="complexity.html#boost_icl.implementation.complexity.time_complexity_of_addition">Time |
| Complexity of Addition</a> |
| </h6> |
| <p> |
| The next table gives the time complexities for the overloaded <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+=</span></code> |
| on interval containers. The instance types of <code class="computeroutput"><span class="identifier">T</span></code> |
| are given as column headers. Instances of type parameter <code class="computeroutput"><span class="identifier">P</span></code> |
| are denoted in the second column. The third column contains the specific |
| 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 |
| row. |
| </p> |
| <div class="table"> |
| <a name="id1132205"></a><p class="title"><b>Table 1.15. Time Complexity of Addition:</b></p> |
| <div class="table-contents"><table class="table" summary="Time Complexity of Addition:"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| </p> |
| </th> |
| <th> |
| <p> |
| <code class="computeroutput"><span class="identifier">P</span></code> |
| </p> |
| </th> |
| <th> |
| <p> |
| </p> |
| </th> |
| <th> |
| <p> |
| interval<br> set |
| </p> |
| </th> |
| <th> |
| <p> |
| separate<br> interval<br> set |
| </p> |
| </th> |
| <th> |
| <p> |
| split<br> interval<br> set |
| </p> |
| </th> |
| <th> |
| <p> |
| interval<br> map |
| </p> |
| </th> |
| <th> |
| <p> |
| split<br> interval<br> map |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">T</span><span class="special">&</span> |
| <span class="keyword">operator</span> <span class="special">+=(</span><span class="identifier">T</span><span class="special">&</span> |
| <span class="identifier">object</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">P</span><span class="special">&</span> <span class="identifier">addend</span><span class="special">)</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">T</span><span class="special">::</span><span class="identifier">element_type</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">T</span><span class="special">::</span><span class="identifier">segment_type</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| best case |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| worst case |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(n)</em></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| amortized |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(log n)</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">interval_sets</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(m log(n+m))</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(m log(n+m))</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(m log(n+m))</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">interval_maps</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(m log(n+m))</em></span> |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="emphasis"><em>O(m log(n+m))</em></span> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><p> |
| Adding an <span class="emphasis"><em>element</em></span> or <span class="emphasis"><em>element value pair</em></span> |
| is always done in <span class="emphasis"><em>logarithmic time</em></span>, where <span class="emphasis"><em>n</em></span> |
| is the number of intervals in the interval container. The same row of complexities |
| applies to the insertion of a <span class="emphasis"><em>segment</em></span> (an interval or |
| an interval value pair) in the <span class="emphasis"><em><span class="bold"><strong>best case</strong></span></em></span>, |
| where the inserted segment does overlap with only a <span class="emphasis"><em><span class="bold"><strong>small</strong></span></em></span> |
| number of intervals in the container. |
| </p> |
| <p> |
| In the <span class="emphasis"><em><span class="bold"><strong>worst case</strong></span></em></span>, |
| where the inserted segment overlaps with all intervals in the container, |
| the algorithms iterate over all the overlapped segments. Using inplace manipulations |
| of segments and hinted inserts, it is possible to perform all necessary operations |
| on each iteration step in <span class="emphasis"><em>constant time</em></span>. This results |
| in <span class="emphasis"><em><span class="bold"><strong>linear worst case time</strong></span></em></span> |
| complexity for segment addition for all interval containers. |
| </p> |
| <p> |
| 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> |
| 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> |
| adding an interval that overlaps <span class="emphasis"><em>n</em></span> intervals, we need |
| <span class="emphasis"><em>n</em></span> non overlapping additions of <span class="emphasis"><em>logarithmic |
| time</em></span> before we can launch another <span class="emphasis"><em>O(n)</em></span> worst |
| case addition. So we have only a <span class="emphasis"><em><span class="bold"><strong>logarithmic |
| amortized time</strong></span></em></span> for the addition of an interval or interval |
| value pair. |
| </p> |
| <p> |
| For the addition of <span class="emphasis"><em><span class="bold"><strong>interval containers</strong></span></em></span> |
| 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 |
| <span class="emphasis"><em>n</em></span> and <span class="emphasis"><em>m</em></span> are equal and both containers |
| 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 |
| frequently in real world applications performance can be much better. If |
| the added container <code class="computeroutput"><span class="identifier">operand</span></code> |
| is much smaller than <code class="computeroutput"><span class="identifier">object</span></code> |
| and the intervals in <code class="computeroutput"><span class="identifier">operand</span></code> |
| are relatively small, performance can be <span class="emphasis"><em>logarithmic</em></span>. |
| If <span class="emphasis"><em>m</em></span> is small compared with <span class="emphasis"><em>n</em></span> and |
| intervals in <code class="computeroutput"><span class="identifier">operand</span></code> are |
| large, performance tends to be <span class="emphasis"><em>linear</em></span>. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2007 -2010 Joachim Faulhaber<br>Copyright © 1999 -2006 Cortex Software GmbH<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
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