Minimize the maximum subarray sum with 1s and -2s Last Updated : 05 Mar, 2024 Summarize Comments Improve Suggest changes Share Like Article Like Report Given two integers X and Y. X and Y represent the frequency of elements 1 and -2 you have. You have to arrange all elements such that the maximum sum over all subarrays is minimized, and then find the maximum subarray sum of such rearrangement. Examples: Input: X = 1, Y = 1Output: 1Explanation: X = 1 and Y = 1 means we have one 1 and one -2. There can be only two possible arrangements with these elements as {1, -2} or {-2, 1}. In both of the cases the maximum subarray sum will be 1. Input: X = 5, Y = 1Output: 3Explanation: The optimal arrangement of elements will be {1, 1, -2, 1, 1, 1}. Then, the maximum subarray sum will be 3. Approach: To solve the problem, follow the below idea: In case the number of 1's are too great, the -2s won't affect the subarray sum too much. Hence, we also calculate the sum of the whole array, which is (X - 2*Y) (Since the whole array is also its own subarray). If there are Y number of -2s then there will be (Y+1) groups of 1s in arrangement. For example, X = 5 and Y = 2, then the arrangement will be as follows: {}, -2, {}, -2, {}. Where "{}" shows the collective group of ones (equal to (Y+1) = 3). We need to distribute X number of 1s in (Y+1) spaces equally. So that the distribution will be as Ceil (X/ (Y+1)). Then, the maximum subarray sum will be max of both of these as max((X - 2*Y), Ceil (X/ (Y+1))). The problem is observation based. The main idea is, for any subarray to have its sum minimized, we need to distribute the 1s among the -2s such that the count of 1s in between -2s are smallest. This can only be done by optimally dividing the number of 1's between the 2's. Illustration: Let us understand it with some examples and find some observations: Example 1: X = 5, Y = 2We have five 1s and two -2s.We need to distribute 1s in between -2s, then the optimal arrangement will be as follows: {1, 1, -2, 1, -2, 1, 1}. The maximum subarray sum will be 2 in this case. Max((X - 2*Y), Ciel (X/ (Y+1))) = 2. Example 2: X = 5, Y = 1We have five 1s and one -2s.The optimal arrangement will be as follows: {1, 1, -2, 1, 1, 1}. The maximum subarray sum will be 3 in this case. Ceil(5/(1+1)) = 3. Max((X - 2*Y), Ciel (X/ (Y+1))) = 3.Example 3: X = 1, Y = 3We have one 1 and three -2s.The optimal arrangement will be as follows: {-2, -2, -2, 1}. The maximum subarray sum will be 1. Max((X - 2*Y), Ciel (X/ (Y+1))) = 1. Step-by-step algorithm: Declare two variables let say A and B.Initialize A with Ciel (X/ (Y+1)).Initialize B with sum of all elements as (X - 2*Y).Output Max (A, B).Below is the implementation of the algorithm: C++ #include <iostream> #include <cmath> // Function to output the maximum subarray sum after // optimal arrangement void FindMaxSum(double X, double Y) { double A = std::ceil(X / (Y + 1.0)); double B = (X - 2 * Y); std::cout << std::max(A, B) << std::endl; } // Driver Function int main() { // Inputs double X = 5, Y = 2; // Function_call FindMaxSum(X, Y); return 0; } // This code is contributed by akshitaguprzj3 Java // Java code to implement the approach import java.util.*; // Driver Class public class Main { // Driver Function public static void main(String[] args) { // Inputs double X = 5, Y = 2; // Function_call FindMaxSum(X, Y); } // Function to output the maximum subarray sum after // optimal arrangement public static void FindMaxSum(Double X, Double Y) { double A = Math.ceil(X / (Y + 1.0)); double B = (X - 2 * Y); System.out.println(Math.max(A, B)); } } Python3 # Python Implementation import math def find_max_sum(X, Y): A = math.ceil(X / (Y + 1.0)) B = X - 2 * Y return max(A, B) X = 5 Y = 2 max_sum = find_max_sum(X, Y) print(max_sum) # This code is contributed by Sakshi C# // C# code to implement the approach using System; // Driver Class public class GFG { // Driver Function public static void Main(string[] args) { // Inputs double X = 5, Y = 2; // Function_call FindMaxSum(X, Y); } // Function to output the maximum subarray sum after // optimal arrangement public static void FindMaxSum(double X, double Y) { double A = Math.Ceiling(X / (Y + 1.0)); double B = (X - 2 * Y); Console.WriteLine(Math.Max(A, B)); } } JavaScript // Function to output the maximum subarray sum after optimal arrangement function findMaxSum(X, Y) { let A = Math.ceil(X / (Y + 1)); let B = X - 2 * Y; console.log(Math.max(A, B)); } // Driver Function function main() { // Inputs let X = 5, Y = 2; // Function call findMaxSum(X, Y); } // Invoke main function main(); Output2.0Time Complexity: O(1)Auxiliary Space: O(1) Comment More infoAdvertise with us Next Article Minimize the maximum subarray sum with 1s and -2s A ayami Follow Improve Article Tags : Greedy Geeks Premier League DSA Arrays Geeks Premier League 2023 +1 More Practice Tags : ArraysGreedy Similar Reads DSA Tutorial - Learn Data Structures and Algorithms DSA (Data Structures and Algorithms) is the study of organizing data efficiently using data structures like arrays, stacks, and trees, paired with step-by-step procedures (or algorithms) to solve problems effectively. 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