Dynamic Programming | High-effort vs. Low-effort Tasks Problem

You are given n days and for each day (di) you could either perform a high effort tasks (hi) or a low effort tasks (li) or no task with the constraint that you can choose high-effort tasks only if you choose no task on the previous day. Write a program to find the maximum amount of tasks you can perform within these n days.

Examples:

No. of days (n) = 5
Day      L.E.   H.E
1        1       3
2        5       6
3        4       8
4        5       7
5        3       6

Maximum amount of tasks = 3 + 5 + 4 + 5 + 3 = 20

Method 1:

Optimal Substructure: To find the maximum amount of tasks done till i’th day, we need to compare 2 choices: 

1. Go for high-effort tasks on that day, then find the maximum amount of tasks done till (i – 2) th day.
2. Go for low effort task on that day and find the maximum amount of tasks done till (i – 1) th day.

Let high [1…n] be the input array for high effort task amount on i’th day and low [1…n] be the input array for low effort task amount on ith day. 
Let max_task (high [], low [], i) be the function that returns maximum amount of task done till ith day, so it will return max(high[i] + max_task(high, low, (i – 2)), low [i] + max_task (high, low, (i – 1)))
Therefore, the problem has optimal substructure property as the problem can be solved using solutions to subproblems.

Overlapping Subproblems: Following is a simple recursive implementation of the High-effort vs. Low-effort task problem.
The implementation simply follows the recursive structure mentioned above. So, High-effort vs. Low-effort Task Problem has both properties of a dynamic programming problem. 

Below is the implementation of the above approach:

C++

// A naive recursive C++ program to find maximum
// tasks.
#include <iostream>
using namespace std;

// Returns the maximum among the 2 numbers
int max(int x, int y)
{
    return (x > y ? x : y);
}

// Returns maximum amount of task that can be
// done till day n
int maxTasks(int high[], int low[], int n)
{
    // If n is less than equal to 0, then no
    // solution exists
    if (n <= 0)
        return 0;

    /* Determines which task to choose on day n,
    then returns the maximum till that day */
    return max(high[n - 1] + maxTasks(high, low, (n - 2)),
            low[n - 1] + maxTasks(high, low, (n - 1)));
}

// Driver code
int main()
{
    int n = 5;
    int high[] = {3, 6, 8, 7, 6};
    int low[] = {1, 5, 4, 5, 3};
    cout << maxTasks(high, low, n);
    return 0;
}

// This code is contributed by Shubhamsingh10

C

// A naive recursive C program to find maximum
// tasks.
#include<stdio.h>

// Returns the maximum among the 2 numbers
int max(int x, int y)
{
    return (x > y ? x : y);
}

// Returns maximum amount of task that can be
// done till day n
int maxTasks(int high[], int low[], int n)
{
    // If n is less than equal to 0, then no
    // solution exists
    if (n <= 0)
        return 0;

    /* Determines which task to choose on day n,
       then returns the maximum till that day */
    return max(high[n-1] + maxTasks(high, low, (n-2)),
              low[n-1] + maxTasks(high, low, (n-1)));
}

// Driver program to test above function
int main()
{
    int n = 5;
    int high[] = {3, 6, 8, 7, 6};
    int low[] = {1, 5, 4, 5, 3};
    printf("%d\n", maxTasks(high, low, n));
    return 0;
}

Java

// A naive recursive Java program
// to find maximum tasks.
import java.io.*;
class GFG{

    // Returns maximum amount of task
    // that can be done till day n
    static int maxTasks(int high[], int low[], int n)
    {

        // If n is less than equal to 0,
        // then no solution exists
        if (n <= 0)
            return 0;

        /* Determines which task to choose on day n,
            then returns the maximum till that day */
        return Math.max(high[n - 1] + maxTasks(high, low, (n - 2)),
                low[n - 1] + maxTasks(high, low, (n - 1)));
    }

    // Driver code
    public static void main(String []args)
    {
        int n = 5;
        int high[] = {3, 6, 8, 7, 6};
        int low[] = {1, 5, 4, 5, 3};
        System.out.println( maxTasks(high, low, n));
    }
}

// This code is contributed by Ita_c.

Python3

# A naive recursive Python3 program to
# find maximum tasks.

# Returns maximum amount of task
# that can be done till day n
def maxTasks(high, low, n) :

    # If n is less than equal to 0,
    # then no solution exists
    if (n <= 0) :
        return 0

    # Determines which task to choose on day n,
    # then returns the maximum till that day
    return max(high[n - 1] + maxTasks(high, low, (n - 2)),
               low[n - 1] + maxTasks(high, low, (n - 1)));

# Driver Code
if __name__ == "__main__" :

    n = 5;
    high = [3, 6, 8, 7, 6]
    low = [1, 5, 4, 5, 3]
    print(maxTasks(high, low, n));

# This code is contributed by Ryuga

C#

// A naive recursive C# program
// to find maximum tasks.
using System;

class GFG
{

    // Returns maximum amount of task
    // that can be done till day n
    static int maxTasks(int[] high,
                    int[] low, int n)
    {

        // If n is less than equal to 0,
        // then no solution exists
        if (n <= 0)
            return 0;

        /* Determines which task to choose on day n,
            then returns the maximum till that day */
        return Math.Max(high[n - 1] +
            maxTasks(high, low, (n - 2)), low[n - 1] +
            maxTasks(high, low, (n - 1)));
    }

    // Driver code
    public static void Main()
    {
        int n = 5;
        int[] high = {3, 6, 8, 7, 6};
        int[] low = {1, 5, 4, 5, 3};
        Console.Write( maxTasks(high, low, n));
    }
}

// This code is contributed by Ita_c.

PHP

<?php
// A naive recursive PHP program to find maximum
// tasks.

// Returns maximum amount of task that can be
// done till day n
function maxTasks($high, $low, $n)
{
    // If n is less than equal to 0, then no
    // solution exists
    if ($n <= 0)
        return 0;

    /* Determines which task to choose on day n,
    then returns the maximum till that day */
    return max($high[$n - 1] + maxTasks($high, $low, ($n - 2)),
                $low[$n - 1] + maxTasks($high, $low, ($n - 1)));
}

// Driver Code
$n = 5;
$high = array(3, 6, 8, 7, 6);
$low = array(1, 5, 4, 5, 3);
print(maxTasks($high, $low, $n));

// This code is contributed by mits
?>

Javascript

<script>

// A naive recursive Java program
// to find maximum tasks.

    // Returns maximum amount of task
    // that can be done till day n
    function maxTasks(high, low, n)
    {

        // If n is less than equal to 0,
        // then no solution exists
        if (n <= 0)
            return 0;

        /* Determines which task to choose on day n,
            then returns the maximum till that day */
        return Math.max(high[n - 1] + maxTasks(high, low, (n - 2)),
                low[n - 1] + maxTasks(high, low, (n - 1)));
    }


    // Driver code

        let n = 5;
        let high = [3, 6, 8, 7, 6];
        let low = [1, 5, 4, 5, 3];
        document.write( maxTasks(high, low, n));;

</script>

Output

20

It should be noted that the above function computes the same subproblems again and again. 
Therefore, this problem has Overlapping Subproblems Property. So the High-effort vs. Low-effort Task Problem has both the properties of a dynamic programming problem.

Method 2: Dynamic Programming Solution

C++

// A DP based C++ program to find maximum tasks.
#include <iostream>
using namespace std;

// Returns the maximum among the 2 numbers
int max(int x, int y)
{
    return (x > y ? x : y);
}

// Returns maximum amount of task that can be
// done till day n
int maxTasks(int high[], int low[], int n)
{
    // An array task_dp that stores the maximum
    // task done
    int task_dp[n+1];

    // If n = 0, no solution exists
    task_dp[0] = 0;

    // If n = 1, high effort task on that day will
    // be the solution
    task_dp[1] = high[0];

    // Fill the entire array determining which
    // task to choose on day i
    for (int i = 2; i <= n; i++)
        task_dp[i] = max(high[i-1] + task_dp[i-2],
                         low[i-1] + task_dp[i-1]);
    return task_dp[n];
}

// Driver program to test above function
int main()
{
    int n = 5;
    int high[] = {3, 6, 8, 7, 6};
    int low[] = {1, 5, 4, 5, 3};
    cout << maxTasks(high, low, n);
    return 0;
}

// This code is contributed by shivanisinghss2110

C

// A DP based C++ program to find maximum tasks.
#include<stdio.h>

// Returns the maximum among the 2 numbers
int max(int x, int y)
{
    return (x > y ? x : y);
}

// Returns maximum amount of task that can be
// done till day n
int maxTasks(int high[], int low[], int n)
{
    // An array task_dp that stores the maximum
    // task done
    int task_dp[n+1];

    // If n = 0, no solution exists
    task_dp[0] = 0;

    // If n = 1, high effort task on that day will
    // be the solution
    task_dp[1] = high[0];

    // Fill the entire array determining which
    // task to choose on day i
    for (int i = 2; i <= n; i++)
        task_dp[i] = max(high[i-1] + task_dp[i-2],
                         low[i-1] + task_dp[i-1]);
    return task_dp[n];
}

// Driver program to test above function
int main()
{
    int n = 5;
    int high[] = {3, 6, 8, 7, 6};
    int low[] = {1, 5, 4, 5, 3};
    printf("%d", maxTasks(high, low, n));
    return 0;
}

Java

// A DP based Java program to find maximum tasks.
import java.io.*;
class GFG {

    // Returns the maximum among the 2 numbers
    static int max(int x, int y) { return (x > y ? x : y); }

    // Returns maximum amount of task that can be
    // done till day n
    static int maxTasks(int[] high, int[] low, int n)
    {
        // An array task_dp that stores the maximum
        // task done
        int[] task_dp = new int[n + 1];

        // If n = 0, no solution exists
        task_dp[0] = 0;

        // If n = 1, high effort task on that day will
        // be the solution
        task_dp[1] = high[0];

        // Fill the entire array determining which
        // task to choose on day i
        for (int i = 2; i <= n; i++)
            task_dp[i]
                = Math.max(high[i - 1] + task_dp[i - 2],
                           low[i - 1] + task_dp[i - 1]);
        return task_dp[n];
    }

    // Driver code
    public static void main(String[] args)
    {
        int n = 5;
        int[] high = { 3, 6, 8, 7, 6 };
        int[] low = { 1, 5, 4, 5, 3 };
        System.out.println(maxTasks(high, low, n));
    }
}

// This code is contributed by Code_Mech.

Python3

# A DP based Python3 program to find maximum tasks.
# Returns the maximum among the 2 numbers
def max1(x, y):

    return x if(x > y) else y;

# Returns maximum amount of task
# that can be done till day n
def maxTasks(high, low, n):

    # An array task_dp that stores
    # the maximum task done
    task_dp = [0] * (n + 1);

    # If n = 0, no solution exists
    task_dp[0] = 0;

    # If n = 1, high effort task
    # on that day will be the solution
    task_dp[1] = high[0];

    # Fill the entire array determining
    # which task to choose on day i
    for i in range(2, n + 1):
        task_dp[i] = max(high[i - 1] + task_dp[i - 2],
                          low[i - 1] + task_dp[i - 1]);
    return task_dp[n];

# Driver code
n = 5;
high = [3, 6, 8, 7, 6];
low = [1, 5, 4, 5, 3];
print(maxTasks(high, low, n));

# This code is contributed by mits

C#

// A DP based C# program to find maximum tasks.
using System;

class GFG
{

// Returns the maximum among the 2 numbers
static int max(int x, int y)
{
    return (x > y ? x : y);
}

// Returns maximum amount of task that can be
// done till day n
static int maxTasks(int []high, int []low, int n)
{
    // An array task_dp that stores the maximum
    // task done
    int[] task_dp = new int[n + 1];

    // If n = 0, no solution exists
    task_dp[0] = 0;

    // If n = 1, high effort task on that day will
    // be the solution
    task_dp[1] = high[0];

    // Fill the entire array determining which
    // task to choose on day i
    for (int i = 2; i <= n; i++)
        task_dp[i] = max(high[i - 1] + task_dp[i - 2],
                        low[i - 1] + task_dp[i - 1]);
    return task_dp[n];
}

// Driver program to test above function
static void Main()
{
    int n = 5;
    int []high = {3, 6, 8, 7, 6};
    int []low = {1, 5, 4, 5, 3};
    Console.WriteLine(maxTasks(high, low, n));
}
}

// This code is contributed by mits

PHP

<?php
// A DP based PHP program to find maximum tasks.
// Returns the maximum among the 2 numbers
function max1($x, $y)
{
    return ($x > $y ? $x : $y);
}

// Returns maximum amount of task that can be
// done till day n
function maxTasks($high, $low, $n)
{
    // An array task_dp that stores the maximum
    // task done
    $task_dp = array($n + 1);

    // If n = 0, no solution exists
    $task_dp[0] = 0;

    // If n = 1, high effort task on that day will
    // be the solution
    $task_dp[1] = $high[0];

    // Fill the entire array determining which
    // task to choose on day i
    for ($i = 2; $i <= $n; $i++)
        $task_dp[$i] = max($high[$i - 1] + $task_dp[$i - 2],
                        $low[$i - 1] + $task_dp[$i - 1]);
    return $task_dp[$n];
}

// Driver code
{
    $n = 5;
    $high = array(3, 6, 8, 7, 6);
    $low = array(1, 5, 4, 5, 3);
    echo(maxTasks($high, $low, $n));
}

// This code is contributed by Code_Mech.

Javascript

<script>

// A DP based javascript program to find maximum tasks.


// Returns the maximum among the 2 numbers
function max(x , y)
{
    return (x > y ? x : y);
}

// Returns maximum amount of task that can be
// done till day n
function maxTasks(high , low , n)
{
    // An array task_dp that stores the maximum
    // task done
    var task_dp = Array.from({length: n+1}, (_, i) => 0);

    // If n = 0, no solution exists
    task_dp[0] = 0;

    // If n = 1, high effort task on that day will
    // be the solution
    task_dp[1] = high[0];

    // Fill the entire array determining which
    // task to choose on day i
    for (i = 2; i <= n; i++)
        task_dp[i] = Math.max(high[i - 1] + task_dp[i - 2],
                        low[i - 1] + task_dp[i - 1]);
    return task_dp[n];
}

// Driver code
var n = 5;
var high = [3, 6, 8, 7, 6];
var low = [1, 5, 4, 5, 3];
document.write(maxTasks(high, low, n));

// This code is contributed by Amit Katiyar
</script>

Output

20

Time Complexity : O(n)
Auxiliary Space: O(n)

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