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<div class="headertitle"><div class="title">graham_scan_functions.hpp</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 orientation of these points</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span><span class="comment"> * (considering them in same order) is not counterclockwise, we discard c,</span></div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span><span class="comment"> * 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;algorithm&gt;</span> <span class="comment">/// for std::swap</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="preprocessor">#include &lt;cstdlib&gt;</span> <span class="comment">/// for mathematics and datatype conversion</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span><span class="preprocessor">#include &lt;iostream&gt;</span> <span class="comment">/// for IO operations</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span><span class="preprocessor">#include &lt;stack&gt;</span> <span class="comment">/// for std::stack</span></div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span><span class="preprocessor">#include &lt;vector&gt;</span> <span class="comment">/// for std::vector</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"> * @namespace geometry</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span><span class="comment"> * @brief geometric algorithms</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span><span class="comment"> *******************************************************************************/</span></div>
<div class="foldopen" id="foldopen00051" data-start="{" data-end="}">
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"><a class="line" href="../../d5/d5f/namespacegeometry.html"> 51</a></span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../d5/d5f/namespacegeometry.html">geometry</a> {</div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> </div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span><span class="comment"> * @namespace graham scan</span></div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span><span class="comment"> * @brief convex hull algorithm</span></div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span><span class="keyword">namespace </span>grahamscan {</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><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span><span class="comment"> * @struct Point</span></div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span><span class="comment"> * @brief for X and Y co-ordinates of the co-ordinate.</span></div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span><span class="comment"> *******************************************************************************/</span></div>
<div class="foldopen" id="foldopen00063" data-start="{" data-end="};">
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"><a class="line" href="../../d7/d48/structgeometry_1_1grahamscan_1_1_point.html"> 63</a></span><span class="keyword">struct </span><a class="code hl_struct" href="../../d7/d48/structgeometry_1_1grahamscan_1_1_point.html">Point</a> {</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> <span class="keywordtype">int</span> x, y;</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span>};</div>
</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> </div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span><span class="comment">// A global point needed for sorting points with reference</span></div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span><span class="comment">// to the first point Used in compare function of qsort()</span></div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> </div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span><a class="code hl_struct" href="../../d7/d48/structgeometry_1_1grahamscan_1_1_point.html">Point</a> p0;</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</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="comment"> * @brief A utility function to find next to top in a stack.</span></div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span><span class="comment"> * @param S Stack to be used for the process.</span></div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span><span class="comment"> * @returns @param Point Co-ordinates of the Point &lt;int, int&gt;</span></div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span><a class="code hl_struct" href="../../d7/d48/structgeometry_1_1grahamscan_1_1_point.html">Point</a> nextToTop(<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack.html">std::stack&lt;Point&gt;</a> *S) {</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> <a class="code hl_struct" href="../../d7/d48/structgeometry_1_1grahamscan_1_1_point.html">Point</a> p = S-&gt;<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/top.html">top</a>();</div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> S-&gt;<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/pop.html">pop</a>();</div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> <a class="code hl_struct" href="../../d7/d48/structgeometry_1_1grahamscan_1_1_point.html">Point</a> res = S-&gt;<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/top.html">top</a>();</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span> S-&gt;<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/push.html">push</a>(p);</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span> <span class="keywordflow">return</span> res;</div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span>}</div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> </div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span><span class="comment"> * @brief A utility function to return square of distance between p1 and p2.</span></div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span><span class="comment"> * @param p1 Co-ordinates of Point 1 &lt;int, int&gt;.</span></div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span><span class="comment"> * @param p2 Co-ordinates of Point 2 &lt;int, int&gt;.</span></div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span><span class="comment"> * @returns @param int distance between p1 and p2.</span></div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span><span class="keywordtype">int</span> distSq(<a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> p1, <a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> p2) {</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> <span class="keywordflow">return</span> (p1.x - p2.x) * (p1.x - p2.x) + (p1.<a class="code hl_variable" href="../../d8/dc8/struct_point.html#a2e1b5fb2b2a83571f5c0bc0f66a73cf7">y</a> - p2.<a class="code hl_variable" href="../../d8/dc8/struct_point.html#a2e1b5fb2b2a83571f5c0bc0f66a73cf7">y</a>) * (p1.<a class="code hl_variable" href="../../d8/dc8/struct_point.html#a2e1b5fb2b2a83571f5c0bc0f66a73cf7">y</a> - p2.<a class="code hl_variable" href="../../d8/dc8/struct_point.html#a2e1b5fb2b2a83571f5c0bc0f66a73cf7">y</a>);</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span>}</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> </div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span><span class="comment"> * @brief To find orientation of ordered triplet (p, q, r).</span></div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span><span class="comment"> * @param p Co-ordinates of Point p &lt;int, int&gt;.</span></div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"> 98</span><span class="comment"> * @param q Co-ordinates of Point q &lt;int, int&gt;.</span></div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span><span class="comment"> * @param r Co-ordinates of Point r &lt;int, int&gt;.</span></div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span><span class="comment"> * @returns @param int 0 --&gt; p, q and r are collinear, 1 --&gt; Clockwise,</span></div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span><span class="comment"> * 2 --&gt; Counterclockwise</span></div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span><span class="keywordtype">int</span> orientation(<a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> p, <a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> q, <a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> r) {</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> <span class="keywordtype">int</span> val = (q.y - p.<a class="code hl_variable" href="../../d8/dc8/struct_point.html#a2e1b5fb2b2a83571f5c0bc0f66a73cf7">y</a>) * (r.x - q.x) - (q.x - p.x) * (r.<a class="code hl_variable" href="../../d8/dc8/struct_point.html#a2e1b5fb2b2a83571f5c0bc0f66a73cf7">y</a> - q.y);</div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> </div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> <span class="keywordflow">if</span> (val == 0) {</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> <span class="keywordflow">return</span> 0; <span class="comment">// collinear</span></div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> }</div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span> <span class="keywordflow">return</span> (val &gt; 0) ? 1 : 2; <span class="comment">// clock or counter-clock wise</span></div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span>}</div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> </div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span><span class="comment"> * @brief A function used by library function qsort() to sort an array of</span></div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span><span class="comment"> * points with respect to the first point</span></div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span><span class="comment"> * @param vp1 Co-ordinates of Point 1 &lt;int, int&gt;.</span></div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span><span class="comment"> * @param vp2 Co-ordinates of Point 2 &lt;int, int&gt;.</span></div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span><span class="comment"> * @returns @param int distance between p1 and p2.</span></div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span><span class="keywordtype">int</span> <a class="code hl_struct" href="../../d1/db3/structcompare.html">compare</a>(<span class="keyword">const</span> <span class="keywordtype">void</span> *vp1, <span class="keyword">const</span> <span class="keywordtype">void</span> *vp2) {</div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span> <span class="keyword">auto</span> *p1 = <span class="keyword">static_cast&lt;</span><span class="keyword">const </span><a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> *<span class="keyword">&gt;</span>(vp1);</div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> <span class="keyword">auto</span> *p2 = <span class="keyword">static_cast&lt;</span><span class="keyword">const </span><a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> *<span class="keyword">&gt;</span>(vp2);</div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> </div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> <span class="comment">// Find orientation</span></div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span> <span class="keywordtype">int</span> o = orientation(p0, *p1, *p2);</div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span> <span class="keywordflow">if</span> (o == 0) {</div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> <span class="keywordflow">return</span> (distSq(p0, *p2) &gt;= distSq(p0, *p1)) ? -1 : 1;</div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> }</div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> </div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> <span class="keywordflow">return</span> (o == 2) ? -1 : 1;</div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span>}</div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> </div>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span><span class="comment"> * @brief Prints convex hull of a set of n points.</span></div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span><span class="comment"> * @param points vector of Point&lt;int, int&gt; with co-ordinates.</span></div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span><span class="comment"> * @param size Size of the vector.</span></div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span><span class="comment"> * @returns @param vector vector of Conver Hull.</span></div>
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span><a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;Point&gt;</a> convexHull(<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;Point&gt;</a> points, uint64_t size) {</div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> <span class="comment">// Find the bottom-most point</span></div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> <span class="keywordtype">int</span> ymin = points[0].y, <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/min.html">min</a> = 0;</div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt; size; i++) {</div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span> <span class="keywordtype">int</span> y = points[i].y;</div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span> </div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> <span class="comment">// Pick the bottom-most or chose the left-most point in case of tie</span></div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span> <span class="keywordflow">if</span> ((y &lt; ymin) || (ymin == y &amp;&amp; points[i].x &lt; points[min].x)) {</div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> ymin = points[i].y, <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/min.html">min</a> = i;</div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> }</div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span> }</div>
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span> </div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span> <span class="comment">// Place the bottom-most point at first position</span></div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno"> 151</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a>(points[0], points[min]);</div>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> </div>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"> 153</span> <span class="comment">// Sort n-1 points with respect to the first point. A point p1 comes</span></div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> <span class="comment">// before p2 in sorted output if p2 has larger polar angle</span></div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> <span class="comment">// (in counterclockwise direction) than p1.</span></div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> p0 = points[0];</div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/qsort.html">qsort</a>(&amp;points[1], size - 1, <span class="keyword">sizeof</span>(<a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a>), <a class="code hl_struct" href="../../d1/db3/structcompare.html">compare</a>);</div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> </div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> <span class="comment">// If two or more points make same angle with p0, Remove all but the one</span></div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <span class="comment">// that is farthest from p0 Remember that, in above sorting, our criteria</span></div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> <span class="comment">// was to keep the farthest point at the end when more than one points have</span></div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="comment">// same angle.</span></div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> <span class="keywordtype">int</span> m = 1; <span class="comment">// Initialize size of modified array</span></div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt; size; i++) {</div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> <span class="comment">// Keep removing i while angle of i and i+1 is same with respect to p0</span></div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> <span class="keywordflow">while</span> (i &lt; size - 1 &amp;&amp; orientation(p0, points[i], points[i + 1]) == 0) {</div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> i++;</div>
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> }</div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> </div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> points[m] = points[i];</div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> m++; <span class="comment">// Update size of modified array</span></div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> }</div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> </div>
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> <span class="comment">// If modified array of points has less than 3 points, convex hull is not</span></div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> <span class="comment">// possible</span></div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> <span class="keywordflow">if</span> (m &lt; 3) {</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> <span class="keywordflow">return</span> {};</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> };</div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> </div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span> <span class="comment">// Create an empty stack and push first three points to it.</span></div>
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack.html">std::stack&lt;Point&gt;</a> St;</div>
<div class="line"><a id="l00182" name="l00182"></a><span class="lineno"> 182</span> St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/push.html">push</a>(points[0]);</div>
<div class="line"><a id="l00183" name="l00183"></a><span class="lineno"> 183</span> St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/push.html">push</a>(points[1]);</div>
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno"> 184</span> St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/push.html">push</a>(points[2]);</div>
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno"> 185</span> </div>
<div class="line"><a id="l00186" name="l00186"></a><span class="lineno"> 186</span> <span class="comment">// Process remaining n-3 points</span></div>
<div class="line"><a id="l00187" name="l00187"></a><span class="lineno"> 187</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 3; i &lt; m; i++) {</div>
<div class="line"><a id="l00188" name="l00188"></a><span class="lineno"> 188</span> <span class="comment">// Keep removing top while the angle formed by</span></div>
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno"> 189</span> <span class="comment">// points next-to-top, top, and points[i] makes</span></div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno"> 190</span> <span class="comment">// a non-left turn</span></div>
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno"> 191</span> <span class="keywordflow">while</span> (St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/size.html">size</a>() &gt; 1 &amp;&amp;</div>
<div class="line"><a id="l00192" name="l00192"></a><span class="lineno"> 192</span> orientation(nextToTop(&amp;St), St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/top.html">top</a>(), points[i]) != 2) {</div>
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno"> 193</span> St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/pop.html">pop</a>();</div>
<div class="line"><a id="l00194" name="l00194"></a><span class="lineno"> 194</span> }</div>
<div class="line"><a id="l00195" name="l00195"></a><span class="lineno"> 195</span> St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/push.html">push</a>(points[i]);</div>
<div class="line"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span> }</div>
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno"> 197</span> </div>
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;Point&gt;</a> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>;</div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span> <span class="comment">// Now stack has the output points, push them into the resultant vector</span></div>
<div class="line"><a id="l00200" name="l00200"></a><span class="lineno"> 200</span> <span class="keywordflow">while</span> (!St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/empty.html">empty</a>()) {</div>
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno"> 201</span> <a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> p = St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/top.html">top</a>();</div>
<div class="line"><a id="l00202" name="l00202"></a><span class="lineno"> 202</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.push_back(p);</div>
<div class="line"><a id="l00203" name="l00203"></a><span class="lineno"> 203</span> St.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/stack/pop.html">pop</a>();</div>
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span> }</div>
<div class="line"><a id="l00205" name="l00205"></a><span class="lineno"> 205</span> </div>
<div class="line"><a id="l00206" name="l00206"></a><span class="lineno"> 206</span> <span class="keywordflow">return</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>; <span class="comment">// return resultant vector with Convex Hull co-ordinates.</span></div>
<div class="line"><a id="l00207" name="l00207"></a><span class="lineno"> 207</span>}</div>
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span>} <span class="comment">// namespace grahamscan</span></div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span>} <span class="comment">// namespace geometry</span></div>
</div>
<div class="ttc" id="aempty_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/stack/empty.html">std::stack::empty</a></div><div class="ttdeci">T empty(T... args)</div></div>
<div class="ttc" id="afibonacci__sum_8cpp_html_aadb40ac4c74a7efc0680b83eeee138aa"><div class="ttname"><a href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">math::fibonacci_sum::result</a></div><div class="ttdeci">uint64_t result(uint64_t n)</div><div class="ttdef"><b>Definition</b> fibonacci_sum.cpp:76</div></div>
<div class="ttc" id="amin_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/min.html">std::min</a></div><div class="ttdeci">T min(T... args)</div></div>
<div class="ttc" id="anamespacegeometry_html"><div class="ttname"><a href="../../d5/d5f/namespacegeometry.html">geometry</a></div><div class="ttdoc">for std::vector</div><div class="ttdef"><b>Definition</b> graham_scan_functions.hpp:51</div></div>
<div class="ttc" id="apop_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/stack/pop.html">std::stack::pop</a></div><div class="ttdeci">T pop(T... args)</div></div>
<div class="ttc" id="apush_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/stack/push.html">std::stack::push</a></div><div class="ttdeci">T push(T... args)</div></div>
<div class="ttc" id="aqsort_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/qsort.html">std::qsort</a></div><div class="ttdeci">T qsort(T... args)</div></div>
<div class="ttc" id="asize_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/stack/size.html">std::stack::size</a></div><div class="ttdeci">T size(T... args)</div></div>
<div class="ttc" id="astack_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/stack.html">std::stack</a></div></div>
<div class="ttc" id="astruct_point_html"><div class="ttname"><a href="../../d8/dc8/struct_point.html">Point</a></div><div class="ttdef"><b>Definition</b> line_segment_intersection.cpp:12</div></div>
<div class="ttc" id="astruct_point_html_a2e1b5fb2b2a83571f5c0bc0f66a73cf7"><div class="ttname"><a href="../../d8/dc8/struct_point.html#a2e1b5fb2b2a83571f5c0bc0f66a73cf7">Point::y</a></div><div class="ttdeci">int y</div><div class="ttdoc">Point respect to x coordinate.</div><div class="ttdef"><b>Definition</b> line_segment_intersection.cpp:14</div></div>
<div class="ttc" id="astructcompare_html"><div class="ttname"><a href="../../d1/db3/structcompare.html">compare</a></div><div class="ttdef"><b>Definition</b> huffman.cpp:28</div></div>
<div class="ttc" id="astructgeometry_1_1grahamscan_1_1_point_html"><div class="ttname"><a href="../../d7/d48/structgeometry_1_1grahamscan_1_1_point.html">geometry::grahamscan::Point</a></div><div class="ttdef"><b>Definition</b> graham_scan_functions.hpp:63</div></div>
<div class="ttc" id="aswap_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a></div><div class="ttdeci">T swap(T... args)</div></div>
<div class="ttc" id="atop_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/stack/top.html">std::stack::top</a></div><div class="ttdeci">T top(T... args)</div></div>
<div class="ttc" id="avector_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a></div></div>
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