Longest alternating subsequence which has maximum sum of elements


Given a list of length N with positive and negative integers. The task is to choose the longest alternating subsequence of the given sequence (i.e. the sign of each next element is the opposite from the sign of the current element). Among all such subsequences, we have to choose one which has the maximum sum of elements and display that sum.

Examples:

Input: list = [-2 10 3 -8 -4 -1 5 -2 -3 1]
Output: 11
Explaination:
The largest subsequence with the greatest sum is [-2 10 -1 5 -2 1] with length 6.

Input: list=[12 4 -5 7 -9]
Output: 5
Explaination:
The largest subsequence with greatest sum is [12 -5 7 -9] with length 4.

Approach: The solution can be reached by following approach:-



  • For getting alternating subsequence with maximum length and largest sum we will be traversing the whole list (length of list)-1 times for comparing signs of consecutive elements.
  • During traversal if we are getting more than 1 consecutive elements of same sign(exp. 1 2 4), then we will append the maximum element out of them to another list named large. so from 1 2 and 4 we will append 4 to another list.
  • If we have consecutive element of opposite sign we will simply add those elements to that list named large.
  • Finally the list named large will have longest alternating subsequence with largest elements.
  • Now, we will have to calculate the sum of all elements from that list named large.

Lets take an example, we have a list [1, 2, 3, -2, -5, 1, -7, -1].

  1. In traversing this list length-1 times, we are getting 1, 2, 3 with the same sign so we will append greatest of these (i.e 3) to another list named large here.
    Hence large=[3]
  2. Now -2 and -5 have the same sign so we will append -2 to another List.
    large=[3, -2]
  3. Now, the sign of 1 and -7 are opposite, so we will append 1 to large.
    large=[3, -2, 1]
  4. For -7, -1 signs are same, Hence append -1 to large.
    large=[3, -2, 1, -1]
  5. Calculate the sum = 3 – 2 + 1 – 1 = 1

Below is Python implementation of the above approach:-

C++

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// C++ implementation to find the 
// longest alternating subsequence 
// which has the maximum sum
#include<bits/stdc++.h>
using namespace std;
  
int calculateMaxSum(int n, int li[])
{
      
    // Creating a temporary list ar to 
    // every time store same sign element
    // to calculate maximum element from 
    // that list ar
    vector<int> ar;
      
    // Appending 1st element of list li 
    // to the ar
    ar.push_back(li[0]);
      
    // Creating list to store maximum 
    // values 
    vector<int> large;
  
    for(int j = 0; j < n - 1; j++)
    {
         
       // If both number are positive 
       // then append (j + 1)th element 
       // to temporary list ar
       if(li[j] > 0 and li[j + 1] > 0)
       {
           ar.push_back(li[j + 1]);
       }
       else if(li[j] > 0 and li[j + 1] < 0)
       {
             
           // If opposite elements found 
           // then append maximum element 
           // to large list
           large.push_back(*max_element(ar.begin(), 
                                        ar.end()));
                                          
           // Empty ar list to re-append
           // next elements 
           ar.clear();
           ar.push_back(li[j + 1]);
       }
       else if(li[j] < 0 and li[j + 1] > 0)
       {
             
           // If opposite elements found 
           // then append maximum element
           // to large list
           large.push_back(*max_element(ar.begin(),
                                        ar.end()));
             
           // Empty ar list to re-append 
           // next elements 
           ar.clear();
           ar.push_back(li[j + 1]);
       }
       else
       {
           // If both number are negative
           // then append (j + 1)th element
           // to temporary list ar 
           ar.push_back(li[j + 1]);
       }
    }
      
    // The final Maximum element in ar list
    // also needs to be appended to large list 
    large.push_back(*max_element(ar.begin(),
                                 ar.end()));
      
    // Returning the sum of all elements 
    // from largest elements list with
    // largest alternating subsequence size
    int sum = 0;
    for(int i = 0; i < large.size(); i++)
       sum += large[i];
    return sum;
}
      
// Driver code
int main()
{
    int list[] = { -2, 8, 3, 8, -4, -15,
                    5, -2, -3, 1 };
    int N = sizeof(list) / sizeof(list[0]);
  
    cout << (calculateMaxSum(N, list));
}
  
// This code is contributed by Bhupendra_Singh

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Python3

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# Python3 implementation to find the 
# longest alternating subsequence 
# which has the maximum sum
  
def calculateMaxSum(n, li):
    # Creating a temporary list ar to every
    # time store same sign element to 
    # calculate maximum element from 
    # that list ar
    ar =[]
      
    # Appending 1st element of list li 
    # to the ar
    ar.append(li[0])
      
    # Creating list to store maximum 
    # values 
    large =[]
      
    for j in range(0, n-1):
          
        # If both number are positive 
        # then append (j + 1)th element 
        # to temporary list ar
        if(li[j]>0 and li[j + 1]>0):
            ar.append(li[j + 1])
        elif(li[j]>0 and li[j + 1]<0):
              
            # If opposite elements found 
            # then append maximum element 
            # to large list
            large.append(max(ar))
              
            # Empty ar list to re-append
            # next elements  
            ar =[]
            ar.append(li[j + 1])
        elif(li[j]<0 and li[j + 1]>0):
              
            # If opposite elements found 
            # then append maximum element
            # to large list
            large.append(max(ar))
              
            # Empty ar list to re-append 
            # next elements 
            ar =[]
            ar.append(li[j + 1])
        else:
            # If both number are negative
            # then append (j + 1)th element
            # to temporary list ar 
            ar.append(li[j + 1])
              
    # The final Maximum element in ar list
    # also needs to be appended to large list 
    large.append(max(ar))
      
    # returning the sum of all elements 
    # from largest elements list with
    # largest alternating subsequence size
    return sum(large)
  
  
# Driver code
list =[-2, 8, 3, 8, -4, -15, 5, -2, -3, 1]
N = len(list)
  
print(calculateMaxSum(N, list))

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Output:

6

Time Complexity:O(N)

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