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    <p><code>O(nlogn)</code> is "faster" then <code>O(n^2)</code>, but let's recall what the big O notation mean.</p> <p>It means that if algorithm A has complexity of <code>O(nlogn)</code>, for some constants <code>N_1</code>, and <code>c1</code>, for each <code>n&gt;N</code> - the algorithm is "faster" then <code>c1*n*log(n)</code>. If algorithm B has <code>O(n^2)</code>, there are some constants <code>N_2</code>,<code>c2</code> such that the algorithm is "faster" then <code>c2 * n^2 * log(n)</code> for <code>n &gt; N_2</code>.</p> <p>However - what happens before this <code>N</code>? and what is this constant <code>C</code>? We do not know. <strong>Algorithm <code>B</code> could still be "faster" then algorithm <code>A</code> for small inputs</strong>, but for large inputs - indeed, <code>A</code> will be faster (the asymptotic bound is better).</p> <p>For example, let's take the case where algorithm <code>A</code> runs in <code>T_1(n) = 10* nlog(n)</code> ops, while algorithm <code>B</code> runs in <code>T_2(n) = n^2</code>. for <code>n=3</code> - we get that <code>T_1(3) = 10*3*2 = 60</code> (we tool the ceil for <code>log(n)</code>), and <code>T_2(3) = 9</code> - thus algorithm <code>B</code>, though <code>O(n^2)</code> is faster then <code>A</code> for this input.</p> <p><strong>Regarding quick sort and insertion sort:</strong> <br>Quick sort is usually extremely fast, and decays into quadric time in very rare cases (the probability of this happening is slim if we chose a random element as a pivot).</p> <p>However, the constant behind the big O notation in quicksort is greater then insertion sort. Thus - <strong>a possible optimization is: use quicksort until you get to a certain threshold</strong> (say 30 elements), and then sort this subarray with insertion sort rather then quicksort. <a href="https://stackoverflow.com/questions/12622015/why-should-insertion-sort-be-used-after-threshold-crossover-in-merge-sort">This post</a> discusses this optimization.</p> <p>Bubble sort is (empirically) terrible for random arrays, but could be good if the array is almost sorted and the "out of place" elements are in its beginning.</p>
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