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insertionSort.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
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<meta name="author" content="Oliver Holland">
<title>RLO: Data Structures & Algorithms: Insertion Sort</title>
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<li><a href="index.html">Introduction</a></li>
<li><a href="overview.html">Data Structures & Algorithms Overview</a></li>
<li><a href="sortingAlgorithms.html">Sorting Algorithms</a></li>
<li><a href="activity.html">Activity</a></li>
<li><a href="bubbleSort.html">Bubble Sort</a></li>
<li class="active"><a href="insertionSort.html">Insertion Sort</a></li>
<li><a href="quickSort.html">Quick Sort</a></li>
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<article>
<a id="content_start"></a>
<h1>Insertion Sort</h1>
<div class="row">
<div class="col-md-6" id="text-component">
<p>The first algorithm we consider uses the easy-split hard-merge approach with smaller lists of
size 1 and <i>n</i>-1. After sorting the list of size <i>n</i>-1, the merging involves inserting
the single element from the list of size 1 into the sorted list in the correct position.</p>
<p>We present the code in a recursive style as a generic method with an argument of type<code>List<T></code>;
– the type<code><T></code>must implement the<code>Comparable</code>interface so we
need to use a bounded type parameter.</p>
<p>If the list to be sorted is large it would be more efficient to create a list of appropriate
capacity for the result before calling the method, and supply this as an extra argument.</p>
<p>The average number of iterations for the while loop for a list of size <i>n</i>. will be about
<i>n</i>/2 and
in the worst case the number will be <i>n</i>. Since the time for the loop body is
<a href="#" data-toggle="modal" data-target="#gloss_o1">O(1)</a> we can conclude
that the time for the loop is O(<i>n</i>). The addition of the element to the list will also
take O(<i>n</i>)
time in the average and worst cases.</p>
<p>We hence obtain the recurrence relation T(<i>n</i>) ≤ T1+T(n-1) where T1 is O(<i>n</i>) so we can
conclude
that the average and worst case running times for insertion sort are O(<i>n</i><sup>2</sup>).
</p>
</div>
<div class="col-md-6" id="rightText-component">
<p>With the code presented here the time in the best case will also be O(<i>n</i><sup>2</sup>), but
with a linked
list version (efficiently coded using list cells instead of the <code>LinkedList</code> class)
if all
insertions were near the beginning of the list we would get O(<i>n</i>) performance.</p>
</div>
<!-- START code column -->
<div class="col-md-6" id="code-component">
<pre><p>public static <T extends Comparable<T>> List<T> iSort(List<T> l) {
if (l.size()<2)
return new ArrayList<code> List<T></code>(l);
else {
<code>List<T></code> s = iSort(l.subList(1, l.size()));
int n = 0;
while (n < s.size() && l.get(0).compareTo(s.get(n)) > 0) n++;
s.add(n, l.get(0)); <i>// add at location n</i>
return s;
}
}
</p></pre>
<p>Note that although we decided to return an object of type<code>ArrayList<T></code>, the
argument type had
to be<code><T></code>since the argument for the recursive call is the result of a call to
the subList
method and the return type of this method is<code><T></code>.</p>
<p>We could have chosen to write a method that returns a linked list; however since the time
complexity
of the <code>get</code> method for linked lists is O(<i>n</i>) rather than O(1), simply
replacing
<code>ArrayList<T></code>by<code>LinkedList<T></code>would result in the time
for the while loop being O(<i>n</i><sup>2</sup>) instead of O(<i>n</i>). Additionally the <code>subList</code>
method
is more efficient when applied to an <code>ArrayList</code> object.</p>
</div>
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<ul>
<li><b>Subject Matter:</b> Data Structure's & Algorithms</li>
<li><b>Sub-topic:</b> Sorting Algorithms</li>
<li><b>Content Author:</b> Dr Michael Sanderson</li>
<li><b>Developer:</b> Oliver Holland</li>
<li><b>Email:</b> <a href="mailto:[email protected]">[email protected]</a></li>
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<p><b>Page last updated:</b> 19th March 2017</p>
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<h3 class="modal-title" id="gloss_o1_label">O(1)</h3>
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<p>The auxiliary space of insertion sort is O(1), It means the amount of extra memory Insertion Sort
consumes doesn't depend on the input.</p>
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