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<div class="headertitle"><div class="title">partition_problem.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 [Partition</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno"> 4</span><span class="comment"> * Problem](https://en.wikipedia.org/wiki/Partition_problem )</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno"> 5</span><span class="comment"> * @details</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno"> 6</span><span class="comment"> * The partition problem, or number partitioning, is the task of deciding</span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment"> * whether a given multiset S of positive integers can be partitioned into two</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span><span class="comment"> * subsets S1 and S2 such that the sum of the numbers in S1 equals the sum of</span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="comment"> * the numbers in S2. Although the partition problem is NP-complete, there is a</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="comment"> * pseudo-polynomial time dynamic programming solution, and there are heuristics</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"> 11</span><span class="comment"> * that solve the problem in many instances, either optimally or approximately.</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span><span class="comment"> * For this reason, it has been called &quot;the easiest hard problem&quot;.</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span><span class="comment"> *</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno"> 14</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="l00015" name="l00015"></a><span class="lineno"> 15</span><span class="comment"> * Graham’s scan algorithm, we can find Convex Hull in O(nLogn) time.</span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno"> 16</span><span class="comment"> *</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span><span class="comment"> * ### Implementation</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="comment"> *</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span><span class="comment"> * Step 1</span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="comment"> * Calculate sum of the array. If sum is odd, there can not be two subsets with</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="comment"> * equal sum, so return false.</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="comment"> *</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span><span class="comment"> * Step 2</span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="comment"> * If sum of array elements is even, calculate sum/2 and find a subset of array</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span><span class="comment"> * with sum equal to sum/2.</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span><span class="comment"> *</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span><span class="comment"> * @author [Lajat Manekar](https://github.com/Lazeeez)</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"> *******************************************************************************/</span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span><span class="preprocessor">#include &lt;cassert&gt;</span> </div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span><span class="preprocessor">#include &lt;cstdint&gt;</span> </div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span><span class="preprocessor">#include &lt;iostream&gt;</span> </div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span><span class="preprocessor">#include &lt;numeric&gt;</span> </div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span><span class="preprocessor">#include &lt;vector&gt;</span> </div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span><span class="comment"> * @namespace dp</span></div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span><span class="comment"> * @brief Dynamic programming algorithms</span></div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span><span class="comment"> *******************************************************************************/</span></div>
<div class="foldopen" id="foldopen00039" data-start="{" data-end="}">
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"><a class="line" href="../../df/d88/namespacedp.html"> 39</a></span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../df/d88/namespacedp.html">dp</a> {</div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span> </div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span><span class="comment">/******************************************************************************</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="comment"> * @namespace partitionProblem</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span><span class="comment"> * @brief Partition problem algorithm</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span><span class="keyword">namespace </span>partitionProblem {</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 Returns true if arr can be partitioned in two subsets of equal sum,</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span><span class="comment"> * otherwise false</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span><span class="comment"> * @param arr vector containing elements</span></div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span><span class="comment"> * @param size Size of the vector.</span></div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span><span class="comment"> * @returns @param bool whether the vector can be partitioned or not.</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="keywordtype">bool</span> findPartiion(<span class="keyword">const</span> std::vector&lt;uint64_t&gt; &amp;arr, uint64_t size) {</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> uint64_t sum = std::accumulate(arr.begin(), arr.end(),</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> 0); <span class="comment">// Calculate sum of all elements</span></div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> </div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> <span class="keywordflow">if</span> (sum % 2 != 0) {</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> <span class="keywordflow">return</span> <span class="keyword">false</span>; <span class="comment">// if sum is odd, it cannot be divided into two equal sum</span></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::vector&lt;bool&gt; part;</div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> <span class="comment">// bool part[sum / 2 + 1];</span></div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> </div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> <span class="comment">// Initialize the part array as 0</span></div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> <span class="keywordflow">for</span> (uint64_t it = 0; it &lt;= sum / 2; ++it) {</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> part.push_back(<span class="keyword">false</span>);</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">// Fill the partition table in bottom up manner</span></div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> <span class="keywordflow">for</span> (uint64_t it = 0; it &lt; size; ++it) {</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> <span class="comment">// The element to be included in the sum cannot be greater than the sum</span></div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> <span class="keywordflow">for</span> (uint64_t it2 = sum / 2; it2 &gt;= arr[it];</div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span> --it2) { <span class="comment">// Check if sum - arr[i]</span></div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> <span class="comment">// ould be formed from a subset using elements before index i</span></div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> <span class="keywordflow">if</span> (part[it2 - arr[it]] == 1 || it2 == arr[it]) {</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span> part[it2] = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> }</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> }</div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> }</div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> <span class="keywordflow">return</span> part[sum / 2];</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span>}</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span>} <span class="comment">// namespace partitionProblem</span></div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span>} <span class="comment">// namespace dp</span></div>
</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 Self-test implementations</span></div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span><span class="comment"> * @returns void</span></div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span><span class="comment"> *******************************************************************************/</span></div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span><span class="keyword">static</span> <span class="keywordtype">void</span> <a class="code hl_function" href="../../dd/d1e/generate__parentheses_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>() {</div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> std::vector&lt;uint64_t&gt; arr = {{1, 3, 3, 2, 3, 2}};</div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span> uint64_t n = arr.size();</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> <span class="keywordtype">bool</span> expected_result = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span> <span class="keywordtype">bool</span> derived_result = dp::partitionProblem::findPartiion(arr, n);</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> std::cout &lt;&lt; <span class="stringliteral">&quot;1st test: &quot;</span>;</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span> assert(expected_result == derived_result);</div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Passed!&quot;</span> &lt;&lt; std::endl;</div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span>}</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"> 98</span> </div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span><span class="comment">/*******************************************************************************</span></div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span><span class="comment"> * @brief Main function</span></div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span><span class="comment"> * @returns 0 on exit</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> <a class="code hl_function" href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a>() {</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> <a class="code hl_function" href="../../dd/d1e/generate__parentheses_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// run self-test implementations</span></div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span>}</div>
<div class="ttc" id="agenerate__parentheses_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../dd/d1e/generate__parentheses_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Self-test implementations.</div><div class="ttdef"><b>Definition</b> <a href="../../dd/d1e/generate__parentheses_8cpp_source.html#l00082">generate_parentheses.cpp:82</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>
<div class="ttc" id="anamespacedp_html"><div class="ttname"><a href="../../df/d88/namespacedp.html">dp</a></div><div class="ttdoc">for std::vector</div><div class="ttdef"><b>Definition</b> <a href="#l00039">partition_problem.cpp:39</a></div></div>
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