One of the simplest sorting algorithms is the insertion sort. Insertion sort consists of N−1 passes. For pass p=1 through N−1, insertion sort ensures that the elements in positions 0 through p are in sorted order. Insertion sort makes use of the fact that elements in positions 0 through p−1 are already known to be in sorted order.

**Discuss in detail about Insertion
Sort**

One of
the simplest sorting algorithms is the **insertion
sort**. Insertion sort consists of *N*−1
**passes. **For pass** ***p*=1 through** ***N*−1,
insertion sort ensures that the elements in positions 0 through** ***p
*are in sorted order. Insertion sort makes use of the fact that elements in
positions 0 through* p*−1* *are already known to be in sorted
order. Figure below shows a sample array after each pass of insertion sort.
Figure belowshows the general strategy. In pass *p*, we move the element in position *p* left until its correct place is found among the first *p*+1 elements. The code implements this
strategy. Lines 11 to 14 implement that data movement without the explicit use
of swaps.

The
element in position *p* is moved to
tmp, and all larger elements (prior to position *p*) are moved one spot to the right. Then tmp is moved to the
correct spot. This is the same technique that was used in the implementation of
binary heaps.

**Insertion Sort After each Pass**

*1 */**

*2 ** Simple
insertion sort.

*3 **/

*4 *template
<typename Comparable>

*5 *void
insertionSort( vector<Comparable> & a )

*6 *{

*7 *for( int
p = 1; p < a.size( ); ++p )

*8 *{

*9 *Comparable
tmp = std::move( a[ p ] );

*10*

*11 *int j;

*12 *for( j =
p; j > 0 && tmp < a[ j - 1 ]; --j )* 13 *a[ j ] = std::move( a[ j - 1 ] );

*14 *a[ j ] =
std::move( tmp );

*15*}

*16*}

**Insertion sort routine**

**Implementation of Insertion Sort**

In the
STL, instead of having the sort routines take an array of comparable items as a
single parameter, the sort routines receive a pair of iterators that represent
the start and endmarker of a range. A two-parameter sort routine uses just that
pair of iterators and presumes that the items can be ordered, while a
three-parameter sort routine has a function object as a third parameter.
Converting the algorithm in Figure 7.2 to use the STL introduces several
issues. The obvious issues are

1. We must
write a two-parameter sort and a three-parameter sort. Presumably, the two
parameter sort invokes the three-parameter sort, with less<Object>{ } as
the third parameter.

2. Array
access must be converted to iterator access.

3. Line
11 of the original code requires that we create tmp, which in the new code will
have type Object.

The first
issue is the trickiest because the template type parameters (i.e., the generic
types) for the two-parameter sort are both Iterator; however, Object is not one
of the generic type parameters. Prior to C++11, one had to write extra routines
to solve this problem. As shown in Figure 7.3, C++11 introduces decltype which
cleanly expresses the intent. Algorithm shows the main sorting code that
replaces array indexing with use of the iterator, and that replaces calls to operator<
with calls to the lessThan function object. Observe that once we actually code
the insertionSort algorithm, every statement in the original code is replaced
with a corresponding statement in the new code that makes

*1 */*

*2 ** The
two-parameter version calls the three-parameter version,

*3 ** using
C++11 decltype

*4 **/

*5 *template
<typename Iterator>

*6 *void
insertionSort( const Iterator & begin, const Iterator & end )

*7 *{

*8 *insertionSort(
begin, end, less<decltype(*begin)>{ } );

*9 *}

Two-parameter
sort invokes three-parameter sort via C++11 decltype

*1 *template
<typename Iterator, typename Comparator>

*2 *void
insertionSort( const Iterator & begin, const Iterator & end,* 3 *Comparator lessThan )

*4 *{

*5 *if( begin
== end )

*6 *return;

*7*

*8 *Iterator
j;

*9*

*10 *for(
Iterator p = begin+1; p != end; ++p )

*11 *{

*12 *auto tmp
= std::move( *p );

*13 *for( j =
p; j != begin && lessThan( tmp, *( j-1 ) ); --j )

*14 **j =
std::move( *(j-1) );

*15 **j =
std::move( tmp );

*16 *}

*17 *}

**Three-parameter sort using iterators**

straightforward
use of iterators and the function object. The original code is arguably much
simpler to read, which is why we use our simpler interface rather than the STL
interface when coding our sorting algorithms.

**Analysis of Insertion Sort**

Because
of the nested loops, each of which can take *N*
iterations, insertion sort is *O*(*N*2). Furthermore, this bound is tight,
because input in reverse order can achieve this bound. A precise calculation
shows that the number of tests in the inner loop in is at most *p* + 1 for each value of *p*.

Summing
over all *p* gives a total of On the
other hand, if the input is presorted, the running time is *O*(*N*), because the test in
the inner for loop always fails immediately. Indeed, if the input is almost
sorted (this term will be more rigorously defined in the next section), insertion
sort will run quickly. Because of this wide variation, it is worth analyzing
the average-case behavior of this algorithm. It turns out that the average case
is *_*(*N*2) for insertion sort, as well as for a variety of other sorting
algorithms.

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