Longest Non-Increasing Subsequence in a Binary String

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Given a binary string S of size N, the task is to find the length of the longest non-increasing subsequence in the given string S.

Examples:

Input: S = “0101110110100001011”
Output: 12
Explanation: The longest non-increasing subsequence is “111111100000”, having length equal to 12.

Input: S = 10101
Output: 3

Approach: The given problem can be solved based on the observation that the string S is a binary string, so a non-increasing subsequence will always consist of 0 with more consecutive 1s or 1 with more consecutive 0s. Follow the steps below to solve the problem:

Initialize an array, say pre[], that stores the number of 1s till each index i for i is over the range [0, N – 1].
Initialize an array, say post[], that stores the number of 0s till each index i to the end of the string for i over the range [0, N – 1].
Initialize a variable, say ans that stores the length of the longest non-increasing subsequence in the given string S.
Iterate over the range [0, N – 1] and update the value of ans to the maximum of ans and (pre[i] + post[i]).
After completing the above steps, print the value of ans as the result.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the length of the
// longest non-increasing subsequence
int findLength(string str, int n)
{
    // Stores the prefix and suffix
    // count of 1s and 0s respectively
    int pre[n], post[n];
 
    // Initialize the array
    memset(pre, 0, sizeof(pre));
    memset(post, 0, sizeof(post));
 
    // Store the number of '1's
    // up to current index i in pre
    for (int i = 0; i < n; i++) {
 
        // Find the prefix sum
        if (i != 0) {
            pre[i] += pre[i - 1];
        }
 
        // If the current element
        // is '1', update the pre[i]
        if (str[i] == '1') {
            pre[i] += 1;
        }
    }
 
    // Store the number of '0's over
    // the range [i, N - 1]
    for (int i = n - 1; i >= 0; i--) {
 
        // Find the suffix sum
        if (i != n - 1)
            post[i] += post[i + 1];
 
        // If the current element
        // is '0', update post[i]
        if (str[i] == '0')
            post[i] += 1;
    }
 
    // Stores the maximum length
    int ans = 0;
 
    // Find the maximum value of
    // pre[i] + post[i]
    for (int i = 0; i < n; i++) {
        ans = max(ans, pre[i] + post[i]);
    }
 
    // Return the answer
    return ans;
}
 
// Driver Code
int main()
{
    string S = "0101110110100001011";
    cout << findLength(S, S.length());
 
    return 0;
}

Java

// Java program for the above approach
class GFG{
 
// Function to find the length of the
// longest non-increasing subsequence
static int findLength(String str, int n)
{
     
    // Stores the prefix and suffix
    // count of 1s and 0s respectively
    int pre[] = new int[n];
    int post[] = new int[n];
 
    // Initialize the array
    for(int i = 0; i < n; i++) 
    {
        pre[i] = 0;
        post[i] = 0;
    }
 
    // Store the number of '1's
    // up to current index i in pre
    for(int i = 0; i < n; i++) 
    {
         
        // Find the prefix sum
        if (i != 0) 
        {
            pre[i] += pre[i - 1];
        }
 
        // If the current element
        // is '1', update the pre[i]
        if (str.charAt(i) == '1')
        {
            pre[i] += 1;
        }
    }
 
    // Store the number of '0's over
    // the range [i, N - 1]
    for(int i = n - 1; i >= 0; i--) 
    {
         
        // Find the suffix sum
        if (i != n - 1)
            post[i] += post[i + 1];
 
        // If the current element
        // is '0', update post[i]
        if (str.charAt(i) == '0')
            post[i] += 1;
    }
 
    // Stores the maximum length
    int ans = 0;
 
    // Find the maximum value of
    // pre[i] + post[i]
    for(int i = 0; i < n; i++)
    {
        ans = Math.max(ans, pre[i] + post[i]);
    }
 
    // Return the answer
    return ans;
}
 
// Driver Code
public static void main(String[] args)
{
    String S = "0101110110100001011";
    System.out.println(findLength(S, S.length()));
}
}
 
// This code is contributed by abhinavjain194

Python3

# Python3 program for the above approach
 
# Function to find the length of the
# longest non-increasing subsequence
def findLength(str, n):
     
    # Stores the prefix and suffix
    # count of 1s and 0s respectively
    pre = [0] * n
    post  = [0] * n
 
    # Store the number of '1's
    # up to current index i in pre
    for i in range(n):
 
        # Find the prefix sum
        if (i != 0):
            pre[i] += pre[i - 1]
     
        # If the current element
        # is '1', update the pre[i]
        if (str[i] == '1'):
            pre[i] += 1
         
    # Store the number of '0's over
    # the range [i, N - 1]
    for i in range(n - 1, -1, -1): 
 
        # Find the suffix sum
        if (i != (n - 1)):
            post[i] += post[i + 1]
 
        # If the current element
        # is '0', update post[i]
        if (str[i] == '0'):
            post[i] += 1
     
    # Stores the maximum length
    ans = 0
 
    # Find the maximum value of
    # pre[i] + post[i]
    for i in range(n):
        ans = max(ans, pre[i] + post[i])
     
    # Return the answer
    return ans
 
# Driver Code
S = "0101110110100001011"
n = len(S)
 
print(findLength(S, n))
 
# This code is contributed by susmitakundugoaldanga

C#

// C# program for the above approach
using System;
 
class GFG{
 
// Function to find the length of the
// longest non-increasing subsequence
static int findLength(String str, int n)
{
     
    // Stores the prefix and suffix
    // count of 1s and 0s respectively
    int []pre = new int[n];
    int []post = new int[n];
 
    // Initialize the array
    for(int i = 0; i < n; i++) 
    {
        pre[i] = 0;
        post[i] = 0;
    }
 
    // Store the number of '1's
    // up to current index i in pre
    for(int i = 0; i < n; i++) 
    {
         
        // Find the prefix sum
        if (i != 0) 
        {
            pre[i] += pre[i - 1];
        }
 
        // If the current element
        // is '1', update the pre[i]
        if (str[i] == '1')
        {
            pre[i] += 1;
        }
    }
 
    // Store the number of '0's over
    // the range [i, N - 1]
    for(int i = n - 1; i >= 0; i--) 
    {
         
        // Find the suffix sum
        if (i != n - 1)
            post[i] += post[i + 1];
 
        // If the current element
        // is '0', update post[i]
        if (str[i] == '0')
            post[i] += 1;
    }
 
    // Stores the maximum length
    int ans = 0;
 
    // Find the maximum value of
    // pre[i] + post[i]
    for(int i = 0; i < n; i++)
    {
        ans = Math.Max(ans, pre[i] + post[i]);
    }
 
    // Return the answer
    return ans;
}
 
// Driver Code
public static void Main(String[] args)
{
    String S = "0101110110100001011";
    Console.WriteLine(findLength(S, S.Length));
}
}
 
// This code is contributed by Princi Singh

Javascript

<script>
 
// Javascript program for the above approach
 
// Function to find the length of the
// longest non-increasing subsequence
function findLength(str, n)
{
     
    // Stores the prefix and suffix
    // count of 1s and 0s respectively
    let pre = Array.from({length: n}, (_, i) => 0);
    let post = Array.from({length: n}, (_, i) => 0);
 
    // Initialize the array
    for(let i = 0; i < n; i++) 
    {
        pre[i] = 0;
        post[i] = 0;
    }
 
    // Store the number of '1's
    // up to current index i in pre
    for(let i = 0; i < n; i++) 
    {
         
        // Find the prefix sum
        if (i != 0) 
        {
            pre[i] += pre[i - 1];
        }
 
        // If the current element
        // is '1', update the pre[i]
        if (str[i] == '1')
        {
            pre[i] += 1;
        }
    }
 
    // Store the number of '0's over
    // the range [i, N - 1]
    for(let i = n - 1; i >= 0; i--) 
    {
         
        // Find the suffix sum
        if (i != n - 1)
            post[i] += post[i + 1];
 
        // If the current element
        // is '0', update post[i]
        if (str[i] == '0')
            post[i] += 1;
    }
 
    // Stores the maximum length
    let ans = 0;
 
    // Find the maximum value of
    // pre[i] + post[i]
    for(let i = 0; i < n; i++)
    {
        ans = Math.max(ans, pre[i] + post[i]);
    }
 
    // Return the answer
    return ans;
}
 
// Driver Code
     
    let S = "0101110110100001011";
    document.write(findLength(S, S.length));
       
</script>

Output:

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

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