Maximum sum increasing subsequence from a prefix and a given element after prefix is must

Given an array of n positive integers, write a program to find the maximum sum of increasing subsequence from prefix till ith index and also including a given kth element which is after i, i.e., k > i. 

Examples :

Input: arr[] = {1, 101, 2, 3, 100, 4, 5} i-th index = 4 (Element at 4th index is 100) K-th index = 6 (Element at 6th index is 5.) 
Output: 11 
Explanation:
So we need to calculate the maximum sum of subsequence (1 101 2 3 100 5) such that 5 is necessarily included in the subsequence, so answer is 11 by subsequence (1 2 3 5).

Input: arr[] = {1, 101, 2, 3, 100, 4, 5} i-th index = 2 (Element at 2nd index is 2) K-th index = 5 (Element at 5th index is 4.) 
Output: 7 
Explanation:
So we need to calculate the maximum sum of subsequence (1 101 2 4) such that 4 is necessarily included in the subsequence, so answer is 7 by subsequence (1 2 4). 

Prerequisite : Maximum Sum Increasing Subsequence

Naive Approach:

1. Construct a new array containing elements till ith index and the kth element.
2. Recursively calculate all the increasing subsequences.
3. Discard all the subsequences not having kth element included.
4. Calculate the maximum sum from the left over subsequences and display it.

Time Complexity: O(2ⁿ)

Better Approach: Use a dynamic approach to maintain a table dp[][]. The value of dp[i][k] stores the maximum sum of increasing subsequence till ith index and containing the kth element. 

C++

// C++ program to find maximum sum increasing
// subsequence till i-th index and including
// k-th index.
#include <bits/stdc++.h>
#define ll long long int
using namespace std;

ll pre_compute(ll a[], ll n, ll index, ll k)
{
    ll dp[n][n] = { 0 };

    // Initializing the first row of the dp[][].
    for (int i = 0; i < n; i++) {
        if (a[i] > a[0])
            dp[0][i] = a[i] + a[0];
        else
            dp[0][i] = a[i];
    }

    // Creating the dp[][] matrix.
    for (int i = 1; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (a[j] > a[i] && j > i) {
                if (dp[i - 1][i] + a[j] > dp[i - 1][j])
                    dp[i][j] = dp[i - 1][i] + a[j];
                else
                    dp[i][j] = dp[i - 1][j];
            }
            else
                dp[i][j] = dp[i - 1][j];
        }
    }

    // To calculate for i=4 and k=6.
    return dp[index][k];
}

int main()
{
    ll a[] = { 1, 101, 2, 3, 100, 4, 5 };
    ll n = sizeof(a) / sizeof(a[0]);
    ll index = 4, k = 6;
    printf("%lld", pre_compute(a, n, index, k));
    return 0;
}

Java

// Java program to find maximum sum increasing
// subsequence till i-th index and including
// k-th index.
import java.io.*;

class GFG {

    static int pre_compute(int a[], int n,
                             int index, int k)
    {
        int dp[][] = new int[n][n];

        // Initializing the first row of
        // the dp[][].
        for (int i = 0; i < n; i++) {
            if (a[i] > a[0])
                dp[0][i] = a[i] + a[0];
            else
                dp[0][i] = a[i];
        }

        // Creating the dp[][] matrix.
        for (int i = 1; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                if (a[j] > a[i] && j > i)
                {
                    if (dp[i - 1][i] + a[j] >
                                 dp[i - 1][j])
                        dp[i][j] = dp[i - 1][i]
                                        + a[j];
                    else
                        dp[i][j] = dp[i - 1][j];
                }
                else
                    dp[i][j] = dp[i - 1][j];
            }
        }

        // To calculate for i=4 and k=6.
        return dp[index][k];
    }

    // Driver code
    public static void main(String[] args)
    {
        int a[] = { 1, 101, 2, 3, 100, 4, 5 };
        int n = a.length;
        int index = 4, k = 6;
        System.out.println(
                  pre_compute(a, n, index, k));
    }
}

// This code is contributed by Smitha.

Python3

# Python3 program to find maximum
# sum increasing subsequence till
# i-th index and including k-th index.

def pre_compute(a, n, index, k):

    dp = [[0 for i in range(n)]
             for i in range(n)]

    # Initializing the first
    # row of the dp[][]
    for i in range(n):
        if a[i] > a[0]:
            dp[0][i] = a[i] + a[0]
        else:
            dp[0][i] = a[i]

    # Creating the dp[][] matrix.
    for i in range(1, n):
        for j in range(n):
            if a[j] > a[i] and j > i:
                if dp[i - 1][i] + a[j] > dp[i - 1][j]:
                    dp[i][j] = dp[i - 1][i] + a[j]
                else:
                    dp[i][j] = dp[i - 1][j]
            else:
                dp[i][j] = dp[i - 1][j]

    # To calculate for i=4 and k=6.
    return dp[index][k]

# Driver code
a = [1, 101, 2, 3, 100, 4, 5 ]
n = len(a)
index = 4
k = 6
print(pre_compute(a, n, index, k))

# This code is contributed
# by sahilshelangia

C#

// C# program to find maximum
// sum increasing subsequence
// till i-th index and including
// k-th index.
using System;

class GFG
{
    static int pre_compute(int []a, int n,
                           int index, int k)
    {
    int [,]dp = new int[n, n];

    // Initializing the first
    // row of the dp[][].
    for (int i = 0; i < n; i++)
    {
        if (a[i] > a[0])
            dp[0, i] = a[i] + a[0];
        else
            dp[0, i] = a[i];
    }

    // Creating the dp[][] matrix.
    for (int i = 1; i < n; i++)
    {
        for (int j = 0; j < n; j++)
        {
            if (a[j] > a[i] && j > i)
            {
                if (dp[i - 1, i] + a[j] >
                            dp[i - 1, j])
                    dp[i, j] = dp[i - 1, i] +
                                        a[j];
                else
                    dp[i, j] = dp[i - 1, j];
            }
            else
                dp[i, j] = dp[i - 1, j];
        }
    }

    // To calculate for i=4 and k=6.
    return dp[index, k];
}

// Driver code
static public void Main ()
{
    int []a = {1, 101, 2,
               3, 100, 4, 5};
    int n = a.Length;
    int index = 4, k = 6;
    Console.WriteLine(pre_compute(a, n,
                                  index, k));
}
}

// This code is contributed by @ajit

PHP

<?php
// PHP program to find maximum sum increasing
// subsequence till i-th index and including
// k-th index.

function pre_compute(&$a, $n, $index, $k)
{
    $dp = array_fill(0, $n,
          array_fill(0, $n, NULL));

    // Initializing the first row of the dp[][].
    for ($i = 0; $i < $n; $i++)
    {
        if ($a[$i] > $a[0])
            $dp[0][$i] = $a[$i] + $a[0];
        else
            $dp[0][$i] = $a[$i];
    }

    // Creating the dp[][] matrix.
    for ($i = 1; $i < $n; $i++)
    {
        for ($j = 0; $j < $n; $j++)
        {
            if ($a[$j] > $a[$i] && $j > $i)
            {
                if (($dp[$i - 1][$i] + $a[$j]) >
                                 $dp[$i - 1][$j])
                    $dp[$i][$j] = $dp[$i - 1][$i] +
                                           $a[$j];
                else
                    $dp[$i][$j] = $dp[$i - 1][$j];
            }
            else
                $dp[$i][$j] = $dp[$i - 1][$j];
        }
    }

    // To calculate for i=4 and k=6.
    return $dp[$index][$k];
}

// Driver Code
$a = array( 1, 101, 2, 3, 100, 4, 5 );
$n = sizeof($a);
$index = 4;
$k = 6;
echo pre_compute($a, $n, $index, $k);

// This code is contributed by ita_c
?>

Javascript

<script>

// Javascript program to find maximum
// sum increasing subsequence till
// i-th index and including k-th index.
function pre_compute(a, n, index, k)
{
    let dp = new Array(n);
    for(let i = 0; i < n; i++)
    {
        dp[i] = new Array(n);
        for(let j = 0; j < n; j++)
        {
            dp[i][j] = 0;
        }
    }

    // Initializing the first row of
    // the dp[][].
    for(let i = 0; i < n; i++)
    {
        if (a[i] > a[0])
            dp[0][i] = a[i] + a[0];
        else
            dp[0][i] = a[i];
    }

    // Creating the dp[][] matrix.
    for(let i = 1; i < n; i++)
    {
        for(let j = 0; j < n; j++)
        {
            if (a[j] > a[i] && j > i)
            {
                if (dp[i - 1][i] + a[j] >
                    dp[i - 1][j])
                    dp[i][j] = dp[i - 1][i] +
                                a[j];
                else
                    dp[i][j] = dp[i - 1][j];
            }
            else
                dp[i][j] = dp[i - 1][j];
        }
    }

    // To calculate for i=4 and k=6.
    return dp[index][k];
}

// Driver code
let a = [ 1, 101, 2, 3, 100, 4, 5 ];
let n = a.length;
let index = 4, k = 6;

document.write(pre_compute(a, n, index, k));

// This code is contributed by mukesh07

</script>

Output

11

Time Complexity: O(n²) 
Auxiliary Space: O(n²)

Efficient approach: This problem is basically finding of maximum sum of increasing sub-sequence up to the given index i that all the elements of the sub-sequence is less than the kth (index) element or arr[k]. Hence, find the Maximum Sum Increasing Subsequence.

For example: arr[] = {1, 101, 2, 3, 100, 4, 5}, index = 4; k = 6; 

Now, we need to just find the max sum sub-sequence from the array till index 4 given that all the elements of that sub-sequence is less that arr[k] which is 5. Now, iterating through the array. 

For i = 0; as 1 < 5; max increasing sub-sequence {1}, max = 1. 
For i = 1; as 101 > 5; skip this entry. Max increasing sub-sequence {1}, max = 1. 
For i = 2; as 2 < 5; max increasing sub-sequence {1, 2}, max = 3. 
For i = 3; as 3 < 5; max increasing sub-sequence {1, 2, 3}, max = 6. 
For i = 4; as 100 > 5; skip this entry. Max increasing sub-sequence {1, 2, 3}, max = 6.
as index = 4; hence stop here and answer will be max + a[k] = 6 + 5 = 11.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
#include <limits.h>
using namespace std;

// Function to find the
// maximum of two numbers
int max(int a, int b)
{
    if (a > b) {
        return a;
    }
    return b;
}

// Function to find the sum
int pre_compute(int a[], int n, int index, int k)
{
    // Base case
    if (index >= k) {
        return -1;
    }
    // Initialize the dp table
    int dp[index] = { 0 };

    int i;

    // Initialize the dp array with
    // corresponding array index value
    for (i = 0; i <= index; i++) {
        dp[i] = a[i];
    }

    int maxi = INT_MIN;

    for (i = 0; i <= index; i++) {
        // Only include values
        // which are less than a[k]
        if (a[i] >= a[k]) {
            continue;
        }

        for (int j = 0; j < i; j++) {
            // Check if a[i] is
            // greater than a[j]
            if (a[i] > a[j]) {
                dp[i] = dp[j] + a[i];
            }

            // Update maxi
            maxi = max(maxi, dp[i]);
        }
    }

    // Incase all the elements in
    // the array upto ith index
    // are greater or equal to a[k]
    if (maxi == INT_MIN) {
        return a[k];
    }
    return maxi + a[k];
    // Contributed by Mainak Dutta
}

// Driver code
int main()
{
    int a[] = { 1, 101, 2, 3, 100, 4, 5 };
    int n = sizeof(a) / sizeof(a[0]);
    int index = 4, k = 6;

    // Function call
    printf("%d", pre_compute(a, n, index, k));
    return 0;
}

Java

// Java program for the above approach
import java.io.*;

class GFG{

// Function to find the
// maximum of two numbers
static int max(int a, int b)
{
    if (a > b)
    {
        return a;
    }
    return b;
}

// Function to find the sum
static int pre_compute(int a[], int n,
                       int index, int k)
{

    // Base case
    if (index >= k)
    {
        return -1;
    }

    // Initialize the dp table
    int[] dp = new int[index + 1];

    int i;

    // Initialize the dp array with
    // corresponding array index value
    for(i = 0; i <= index; i++)
    {
        dp[i] = a[i];
    }

    int maxi = Integer.MIN_VALUE;

    for(i = 0; i <= index; i++)
    {

        // Only include values
        // which are less than a[k]
        if (a[i] >= a[k])
        {
            continue;
        }

        for(int j = 0; j < i; j++)
        {

            // Check if a[i] is
            // greater than a[j]
            if (a[i] > a[j])
            {
                dp[i] = dp[j] + a[i];
            }

            // Update maxi
            maxi = max(maxi, dp[i]);
        }
    }

    // Incase all the elements in
    // the array upto ith index
    // are greater or equal to a[k]
    if (maxi == Integer.MIN_VALUE)
    {
        return a[k];
    }
    return maxi + a[k];
}

// Driver code
public static void main (String[] args)
{
    int a[] = { 1, 101, 2, 3, 100, 4, 5 };
    int n = a.length;
    int index = 4, k = 6;

    System.out.println(pre_compute(a, n, index, k));
}
}

// This code is contributed by rag2127

Python3

# Python3 program for the above approach

# Function to find the sum
def pre_compute(a, n, index, k):

    # Base case
    if (index >= k):
        return -1

    # Initialize the dp table
    dp = [0 for i in range(index)]

    # Initialize the dp array with
    # corresponding array index value
    for i in range(index):
        dp[i] = a[i]

    maxi = -float('inf')

    for i in range(index):

        # Only include values
        # which are less than a[k]
        if (a[i] >= a[k]):
            continue

        for j in range(i):

            # Check if a[i] is
            # greater than a[j]
            if (a[i] > a[j]):
                dp[i] = dp[j] + a[i]

            # Update maxi
            maxi = max(maxi, dp[i])

    # Incase all the elements in
    # the array upto ith index
    # are greater or equal to a[k]
    if (maxi == -float('inf')):
        return a[k]

    return maxi + a[k]

# Driver code
a = [ 1, 101, 2, 3, 100, 4, 5 ]
n = len(a)
index = 4
k = 6

# Function call
print(pre_compute(a, n, index, k))

# This code is contributed by rohitsingh07052

C#

// C# program for the above approach
using System;

class GFG{

// Function to find the
// maximum of two numbers
static int max(int a, int b)
{
    if (a > b)
    {
        return a;
    }
    return b;
}

// Function to find the sum
static int pre_compute(int[] a, int n,
                       int index, int k)
{

    // Base case
    if (index >= k)
    {
        return -1;
    }

    // Initialize the dp table
    int[] dp = new int[index + 1];

    int i;

    // Initialize the dp array with
    // corresponding array index value
    for(i = 0; i <= index; i++)
    {
        dp[i] = a[i];
    }

    int maxi = Int32.MinValue;

    for(i = 0; i <= index; i++)
    {

        // Only include values
        // which are less than a[k]
        if (a[i] >= a[k])
        {
            continue;
        }

        for(int j = 0; j < i; j++)
        {

            // Check if a[i] is
            // greater than a[j]
            if (a[i] > a[j])
            {
                dp[i] = dp[j] + a[i];
            }

            // Update maxi
            maxi = Math.Max(maxi, dp[i]);
        }
    }

    // Incase all the elements in
    // the array upto ith index
    // are greater or equal to a[k]
    if (maxi == Int32.MinValue)
    {
        return a[k];
    }
    return maxi + a[k];
}

// Driver code
static public void Main()
{
    int[] a = { 1, 101, 2, 3, 100, 4, 5 };
    int n = a.Length;
    int index = 4, k = 6;

    Console.WriteLine(pre_compute(a, n, index, k));
}
}

// This code is contributed by avanitrachhadiya2155

Javascript

<script>

// Javascript program for the above approach

// Function to find the
// maximum of two numbers
function max(a, b)
{
    if (a > b)
    {
        return a;
    }
    return b;
}

// Function to find the sum
function pre_compute(a, n, index, k)
{

    // Base case
    if (index >= k)
    {
        return -1;
    }

    // Initialize the dp table
    let dp = new Array(index + 1);
    dp.fill(0);

    let i;

    // Initialize the dp array with
    // corresponding array index value
    for(i = 0; i <= index; i++)
    {
        dp[i] = a[i];
    }

    let maxi = Number.MIN_VALUE;

    for(i = 0; i <= index; i++)
    {

        // Only include values
        // which are less than a[k]
        if (a[i] >= a[k])
        {
            continue;
        }

        for(let j = 0; j < i; j++)
        {

            // Check if a[i] is
            // greater than a[j]
            if (a[i] > a[j])
            {
                dp[i] = dp[j] + a[i];
            }

            // Update maxi
            maxi = Math.max(maxi, dp[i]);
        }
    }

    // Incase all the elements in
    // the array upto ith index
    // are greater or equal to a[k]
    if (maxi == Number.MIN_VALUE)
    {
        return a[k];
    }
    return maxi + a[k];
}

// Driver code
let a = [ 1, 101, 2, 3, 100, 4, 5 ];
let n = a.length;
let index = 4, k = 6;

document.write(pre_compute(a, n, index, k));

// This code is contributed by divyeshrabadiya07

</script>

Output

11

Time Complexity: O( index² ) 
Auxiliary Space: O( index )