Longest alternating subsequence in terms of positive and negative integers

Given an array arr[] of positive and negative numbers only. The task is to find the length of the longest alternating (means negative-positive-negative or positive-negative-positive) subsequence present in the array.

Examples:

Input: arr[] = {-4, 3, -5, 9, 10, 12, 2, -1}
Output: 5
Explanation:
The longest sequence is {-4, 3, -5, 9, -1} which of length 5. There can be many more subsequence of this length.



Input: arr[] = {10, 12, 2, -1}
Output: 2
Explanation:
The longest sequence is {10, -1} which is 2. There can be many more subsequence of this length.

Approach:
This problem can be solved using Dynamic Programming. It is a variation Longest Increasing Subsequence(LIS). Following are the steps:

  1. For including and excluding an element in the given array arr[] for LAS(Longest Alternative Subsequence), a variable pos is used, when pos = true means current element needs to be positive and if pos = false then current element needs to be negative.
  2. If we include the current element, change the value of pos and recurr for the next element because we need the next element of opposite sign than the previous included element.
  3. Now LAS[i][pos] can be recursively written as:
    • Base case: If the index called recursively is greater than the last element, then return 0, as there is no such element left to form LAS and if LAS[i][pos] is calculated then return the value.

      if(i == N) {
      return 0;
      }
      if(LAS[i][pos]) {
      return LAS[i][pos];
      }

    • Recursive call: If the base case is not met, then recursively call when current element is included and excluded, then find the maximum of those to find LAS at that index.

      LAS[i][pos] = Longest Alternating Subsequence at index i by including or excluding that element for the value of pos,
      LAS[i][pos] = max(1 + recursive_function(i+1, pos), recursive_function(i+1, pos));

    • Return statement: At each recursive call(except the base case), return the value of LAS[i][pos].

      return LAS[i][pos];

  4. The LAS for the given array arr[] is the maximum of LAS[0][0] and LAS[0][1].

Below is the implementation of the above approach:

C++

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// C++ program to find the
// length of longest alternate
// subsequence
#include <bits/stdc++.h>
using namespace std;
  
// LAS[i][pos] array to find
// the length of LAS till
// index i by including or
// excluding element arr[i]
// on the basis of value of pos
int LAS[1000][2] = { false };
  
int solve(int arr[], int n, int i, bool pos)
{
    // Base Case
    if (i == n)
        return 0;
  
    if (LAS[i][pos])
        return LAS[i][pos];
  
    int inc = 0, exc = 0;
  
    // If current element is
    // positive and pos is true
    // Include the current element
    // and change pos to false
    if (arr[i] > 0 && pos == true) {
        pos = false;
  
        // Recurr for the next
        // iteration
        inc = 1 + solve(arr, n, i + 1, pos);
    }
  
    // If current element is
    // negative and pos is false
    // Include the current element
    // and change pos to true
    else if (arr[i] < 0 && pos == false) {
        pos = true;
  
        // Recurr for the next
        // iteration
        inc = 1 + solve(arr, n, i + 1, pos);
    }
  
    // If current element is
    // excluded, reccur for
    // next iteration
    exc = solve(arr, n, i + 1, pos);
  
    LAS[i][pos] = max(inc, exc);
  
    return LAS[i][pos];
}
  
// Driver's Code
int main()
{
    int arr[] = { -1, 2, 3, 4, 5,
                  -6, 8, -99 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    // Print LAS
    cout << max(solve(arr, n, 0, 0),
                solve(arr, n, 0, 1));
}

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Java

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// Java program to find the
// length of longest alternate
// subsequence
class GFG {
  
// LAS[i][pos] array to find
// the length of LAS till
// index i by including or
// excluding element arr[i]
// on the basis of value of pos
static int LAS[][] = new int[1000][2];
  
static int solve(int arr[], int n, int i,int pos)
{
      
    // Base Case
    if (i == n)
        return 0;
  
    if (LAS[i][pos]== 1)
        return LAS[i][pos];
  
    int inc = 0, exc = 0;
  
    // If current element is
    // positive and pos is 1
    // Include the current element
    // and change pos to 0
    if (arr[i] > 0 && pos == 1) {
        pos = 0;
  
        // Recurr for the next
        // iteration
        inc = 1 + solve(arr, n, i + 1, pos);
    }
  
    // If current element is
    // negative and pos is o
    // Include the current element
    // and change pos to 1
    else if (arr[i] < 0 && pos == 0) {
        pos = 1;
  
        // Recurr for the next
        // iteration
        inc = 1 + solve(arr, n, i + 1, pos);
    }
  
    // If current element is
    // excluded, reccur for
    // next iteration
    exc = solve(arr, n, i + 1, pos);
  
    LAS[i][pos] = Math.max(inc, exc);
  
    return LAS[i][pos];
}
  
// Driver's Code
public static void main (String[] args) 
{
    int arr[] = { -1, 2, 3, 4, 5, -6, 8, -99 };
    int n = arr.length;
  
    // Print LAS
    System.out.println(Math.max(solve(arr, n, 0, 0),solve(arr, n, 0, 1)));
}
}
  
// This code is contributed by AnkitRai01

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Python3

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# Python3 program to find the 
# length of longest alternate 
# subsequence 
import numpy as np
  
# LAS[i][pos] array to find 
# the length of LAS till 
# index i by including or 
# excluding element arr[i] 
# on the basis of value of pos 
LAS = np.zeros((1000, 2))
  
for i in range(1000) :
    for j in range(2) :
        LAS[i][j] = False
  
def solve(arr, n, i, pos) :
  
    # Base Case 
    if (i == n) :
        return 0
  
    if (LAS[i][pos]) :
        return LAS[i][pos]; 
  
    inc = 0; exc = 0
  
    # If current element is 
    # positive and pos is true 
    # Include the current element 
    # and change pos to false 
    if (arr[i] > 0 and pos == True) :
        pos = False
  
        # Recurr for the next 
        # iteration 
        inc = 1 + solve(arr, n, i + 1, pos); 
  
    # If current element is 
    # negative and pos is false 
    # Include the current element 
    # and change pos to true 
    elif (arr[i] < 0 and pos == False) :
        pos = True
  
        # Recurr for the next 
        # iteration 
        inc = 1 + solve(arr, n, i + 1, pos); 
      
    # If current element is 
    # excluded, reccur for 
    # next iteration 
    exc = solve(arr, n, i + 1, pos); 
  
    LAS[i][pos] = max(inc, exc); 
  
    return LAS[i][pos]; 
  
# Driver's Code 
if __name__ == "__main__"
  
    arr = [ -1, 2, 3, 4, 5, -6, 8, -99 ]; 
    n = len(arr); 
  
    # Print LAS 
    print(max(solve(arr, n, 0, 0), solve(arr, n, 0, 1))); 
      
# This code is contributed by AnkitRai01

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C#

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// C# program to find the
// length of longest alternate
// subsequence
  
using System;
  
public class GFG {
  
// LAS[i][pos] array to find
// the length of LAS till
// index i by including or
// excluding element arr[i]
// on the basis of value of pos
static int [,]LAS = new int[1000,2];
  
static int solve(int []arr, int n, int i,int pos)
{
      
    // Base Case
    if (i == n)
        return 0;
  
    if (LAS[i,pos]== 1)
        return LAS[i,pos];
  
    int inc = 0, exc = 0;
  
    // If current element is
    // positive and pos is 1
    // Include the current element
    // and change pos to 0
    if (arr[i] > 0 && pos == 1) {
        pos = 0;
  
        // Recurr for the next
        // iteration
        inc = 1 + solve(arr, n, i + 1, pos);
    }
  
    // If current element is
    // negative and pos is o
    // Include the current element
    // and change pos to 1
    else if (arr[i] < 0 && pos == 0) {
        pos = 1;
  
        // Recurr for the next
        // iteration
        inc = 1 + solve(arr, n, i + 1, pos);
    }
  
    // If current element is
    // excluded, reccur for
    // next iteration
    exc = solve(arr, n, i + 1, pos);
  
    LAS[i,pos] = Math.Max(inc, exc);
  
    return LAS[i,pos];
}
  
// Driver's Code
public static void Main() 
{
    int []arr = { -1, 2, 3, 4, 5, -6, 8, -99 };
    int n = arr.Length;
  
    // Print LAS
    Console.WriteLine(Math.Max(solve(arr, n, 0, 0),solve(arr, n, 0, 1)));
}
}
  
// This code is contributed by AnkitRai01

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Output:

5

Time Complexity: O(N) where N is the length of array.

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