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Problems Description: Note that the maximum contiguous subsequence sum algorithms in the

text do not give any indication of the actual sequence. In this assignment, you will modify these

algorithms to return the maximum contiguous subsequence sum including the starting and

ending indexes of the sum. Also, your program will measure their execution times with various

input sizes.

$($a$)$ Modify Algorithms $2,3$&$4$ so that they return in the value of the maximum subsequence

sum and the starting and ending indices of the maximum subsequence. Your algorithm will

return $\text{'}0\text{'}$ if there is no positive number in the input array.

Your algorithm should take input size $($ N $)$ from the user and generate a random sequence of

N integers ranging from $-9999$ to $9999.$ If $N$ is less than $50,$ your program must print the

randomly generated numbers, and find the maximum subsequence independently $($do not

refer the indexes found by other algorithms. Your program should measure $($and print$)$ the

execution time of these algorithms $($Algorithms $2,3$&$4)$ in milliseconds or nanoseconds.

$$ Sample Execution Scenarios $>$

Please enter the size of the problem $($N$)$: $15$

$673-869-153214-139$

$500967$

Algorithm $2$:

MaxSum: $2532,$ S$\_$index: $10,$ E$\_$index: $14$

Execution Time: $\}82048\backslash$mathrm$\{$ nanoseconds

Algorithm $3$:

MaxSum: $2532,$ S$\_$index: $10,$ E$\_$index: $14$

Execution Time: $23165$ nanoseconds

Algorithm $4$:

MaxSum: $2532,$ S$\_$index: $10,$ E$\_$index: $14$

Execution Time: $\}11345\backslash$mathrm$\{$ nanoseconds

$$ Note: all three algorithms must be executed for the same input numbers $>$

$($b$)$ Measure and compare the executions times of your algorithms at least $10$ times, and find

the average execution times of these algorithms. The execution time of each test case

must be recorded in a table. A template of the table for input size $=8000$ is shown below for

your reference. You have to add a similar table for each input size.

Plot the average execution times of Algorithm $2$ & Algorithm $3$ for input sizes

$400,800,1600,3200,6400,12800,$ and $256000.$

Plot the average execution times of Algorithm $3$ & Algorithm $4$ for input sizes $400,$

$800,1600,3200,6400,12800,$ and $25600.$

$($Note: Your graph must show the execution time on the vertical axis $($Y axis$)$ and

input size on the horizontal axis $($ X axis$)$

$($To generate a reasonable graph, please feel free to use different input sizes as needed$)$