$1$ Problem Description

Instructions. You are provided the skeleton code named Sort.java. The source file is available on Canvas in a folder named HW$1.$ Please modify the skeleton code to solve the following tasks.

Task $1(80$ pts$).$ Implement the Insertion Sort algorithm as discussed in Lecture $1.($Hint: use the function checked sorted to check if your output is indeed sorted.$)$

Task $2(20$ pts$).$ Generate a report to discuss the time performance of the algorithm. Compare it with their theoretical time complexity as discussed in the lecture. Plots and figures are encouraged to help draw the conclusion. See Figure $1$ for an example of the plot.

Figure $1$: An example or tne time periormance piot

Code:

package sorting;

import java.util.$*$;

public class Sort $\{$

public static int$[]$ insertion$\_$sort $($int$[]$ array$)\{$

$/*$

$*$ fill in your program

$*/$

return array;

$\}$

$/*$

$*$ n: the size of the output array

$*$ k: the maximum value in the array

$*/$

public static int$[]$ generate$\_$random$\_$array $($int n$,$ int k$)\{$

List list;

int$[]$ array;

Random rnd;

rnd $=$ new Random$($System$.$currentTimeMillis$())$;

list $=$ new ArrayList $()$;

for $($int i $=1$; i n; i$++)$

list.add$($new Integer$($rnd$.$nextInt$($k$+1)))$;

Collections.shuffle$($list$,$ rnd$)$;

array $=$ new int$[$n$]$;

for $($int i $=0$; i $$ n; i$++)$

array$[$i$]=$ list.get$($i$).$intValue$()$;

return array;

$\}$

$/*$

$*$ n: the size of the output array

$*/$

public static int$[]$ generate$\_$random$\_$array $($int n$)\{$

List list;

int$[]$ array;

list $=$ new ArrayList $()$;

for $($int i $=1$; i n; i$++)$

list.add$($new Integer$($i$))$;

Collections.shuffle$($list$,$ new Random$($System$.$currentTimeMillis$()))$;

array $=$ new int$[$n$]$;

for $($int i $=0$; i $$ n; i$++)$

array$[$i$]=$ list.get$($i$).$intValue$()$;

return array;

$\}$

$/*$

$*$ Input: an integer array

$*$ Output: true if the array is acsendingly sorted, otherwise return false

$*/$

public static boolean check$\_$sorted $($int$[]$ array$)\{$

for $($int i $=1$; i $$ array.length; i$++)\{$

if $($array$[$i$-1]>$ array$[$i$])$

return false;

$\}$

return true;

$\}$

public static void print$\_$array $($int$[]$ array$)\{$

for $($int i $=0$; i $$ array.length; i$++)$

System.out.print$($array$[$i$]+",")$;

System.out.println$()$;

$\}$

public static void main$($String$[]$ args$)\{$

$//$ TODO Auto$-$generated method stub

System.out.println$("$Insertion sort starts $------------------")$;

for $($int n $=100000$; n ; n$=$n$+100000)\{$

int$[]$ array $=$ Sort.generate$\_$random$\_$array$($n$)$;

long t$1=$ System.currentTimeMillis$()$;

array $=$ Sort.insertion$\_$sort$($array$)$;

long t$2=$ System.currentTimeMillis$()$;

long t $=$ t$2-$ t$1$;

boolean flag $=$ Sort.check$\_$sorted$($array$)$;

System.out.println$($n $+","+$ t $+","+$ flag$)$;

$\}$

System.out.println$("$Insertion sort ends $------------------")$;

$\}$

$\}$