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<div class="headertitle"><div class="title">graham_scan_algorithm.cpp</div></div>
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<div class="contents">
<div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno"> 2</span><span class="comment"> * @file</span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno"> 3</span><span class="comment"> * @brief Implementation of the [Convex</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno"> 4</span><span class="comment"> * Hull](https://en.wikipedia.org/wiki/Convex_hull) implementation using [Graham</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno"> 5</span><span class="comment"> * Scan](https://en.wikipedia.org/wiki/Graham_scan)</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno"> 6</span><span class="comment"> * @details</span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment"> * In geometry, the convex hull or convex envelope or convex closure of a shape</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span><span class="comment"> * is the smallest convex set that contains it. The convex hull may be defined</span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="comment"> * either as the intersection of all convex sets containing a given subset of a</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="comment"> * Euclidean space, or equivalently as the set of all convex combinations of</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"> 11</span><span class="comment"> * points in the subset. For a bounded subset of the plane, the convex hull may</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span><span class="comment"> * be visualized as the shape enclosed by a rubber band stretched around the</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span><span class="comment"> * subset.</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno"> 14</span><span class="comment"> *</span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno"> 15</span><span class="comment"> * The worst case time complexity of Jarvis’s Algorithm is O(n^2). Using</span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno"> 16</span><span class="comment"> * Graham’s scan algorithm, we can find Convex Hull in O(nLogn) time.</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span><span class="comment"> *</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="comment"> * ### Implementation</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span><span class="comment"> *</span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="comment"> * Sort points</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="comment"> * We first find the bottom-most point. The idea is to pre-process</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="comment"> * points be sorting them with respect to the bottom-most point. Once the points</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span><span class="comment"> * are sorted, they form a simple closed path.</span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="comment"> * The sorting criteria is to use the orientation to compare angles without</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span><span class="comment"> * actually computing them (See the compare() function below) because</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span><span class="comment"> * computation of actual angles would be inefficient since trigonometric</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span><span class="comment"> * functions are not simple to evaluate.</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"> 28</span><span class="comment"> *</span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno"> 29</span><span class="comment"> * Accept or Reject Points</span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span><span class="comment"> * Once we have the closed path, the next step is to traverse the path and</span></div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span><span class="comment"> * remove concave points on this path using orientation. The first two points in</span></div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span><span class="comment"> * sorted array are always part of Convex Hull. For remaining points, we keep</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span><span class="comment"> * track of recent three points, and find the angle formed by them. Let the</span></div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span><span class="comment"> * three points be prev(p), curr(c) and next(n). If the orientation of these</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span><span class="comment"> * points (considering them in the same order) is not counterclockwise, we</span></div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span><span class="comment"> * discard c, otherwise we keep it.</span></div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span><span class="comment"> *</span></div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span><span class="comment"> * @author [Lajat Manekar](https://github.com/Lazeeez)</span></div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span><span class="comment"> *</span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span><span class="preprocessor">#include &lt;cassert&gt;</span> </div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="preprocessor">#include &lt;iostream&gt;</span> </div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span><span class="preprocessor">#include &lt;vector&gt;</span> </div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> </div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span><span class="preprocessor">#include &quot;./graham_scan_functions.hpp&quot;</span> </div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> </div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span><span class="comment">/*******************************************************************************</span></div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span><span class="comment"> * @brief Self-test implementations</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span><span class="comment"> * @returns void</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span><span class="keyword">static</span> <span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/d2c/caesar__cipher_8cpp.html#ae1a3968e7947464bee7714f6d43b7002">test</a>() {</div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> std::vector&lt;geometry::grahamscan::Point&gt; points = {</div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> {0, 3}, {1, 1}, {2, 2}, {4, 4}, {0, 0}, {1, 2}, {3, 1}, {3, 3}};</div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> std::vector&lt;geometry::grahamscan::Point&gt; expected_result = {</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> {0, 3}, {4, 4}, {3, 1}, {0, 0}};</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> std::vector&lt;geometry::grahamscan::Point&gt; derived_result;</div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> std::vector&lt;geometry::grahamscan::Point&gt; res;</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> </div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> derived_result = geometry::grahamscan::convexHull(points, points.size());</div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> </div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> std::cout &lt;&lt; <span class="stringliteral">&quot;1st test: &quot;</span>;</div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; expected_result.size(); i++) {</div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> assert(derived_result[i].x == expected_result[i].x);</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> assert(derived_result[i].y == expected_result[i].y);</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> }</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> std::cout &lt;&lt; <span class="stringliteral">&quot;passed!&quot;</span> &lt;&lt; std::endl;</div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span>}</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> </div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span><span class="comment">/*******************************************************************************</span></div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span><span class="comment"> * @brief Main function</span></div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span><span class="comment"> * @returns 0 on exit</span></div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span><span class="keywordtype">int</span> <a class="code hl_function" href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a>() {</div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> <a class="code hl_function" href="../../d6/d2c/caesar__cipher_8cpp.html#ae1a3968e7947464bee7714f6d43b7002">test</a>(); <span class="comment">// run self-test implementations</span></div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span>}</div>
<div class="ttc" id="acaesar__cipher_8cpp_html_ae1a3968e7947464bee7714f6d43b7002"><div class="ttname"><a href="../../d6/d2c/caesar__cipher_8cpp.html#ae1a3968e7947464bee7714f6d43b7002">test</a></div><div class="ttdeci">void test()</div><div class="ttdef"><b>Definition</b> <a href="../../d6/d2c/caesar__cipher_8cpp_source.html#l00100">caesar_cipher.cpp:100</a></div></div>
<div class="ttc" id="agenerate__parentheses_8cpp_html_gae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdoc">Main function.</div><div class="ttdef"><b>Definition</b> <a href="../../dd/d1e/generate__parentheses_8cpp_source.html#l00110">generate_parentheses.cpp:110</a></div></div>
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