# Compute maximum of the function efficiently over all sub-arrays

Given an array, arr[] and a function F(i, j). The task is to compute max{F(i, j)} over all sub-arrays [i..j].

The fucntion F() is defined as: Examples:

Input : arr[] = { 1, 5, 4, 7 }
Output : 6
Values of F(i, j) for all the sub-arrays:
{ 1, 5 } = |1 – 5| * (1) = 4
{ 1, 5, 4 } = |1 – 5| * (1) + |5 – 4| * (-1) = 3
{ 1, 5, 4, 7 } = |1 – 5| * (1) + |5 – 4| * (-1) + |4 – 7| * (1) = 6
{ 5, 4 } = |5 – 4| * (1) = 1
{ 5, 4, 7 } = |5 – 4| * (1) + |4 – 7| * (-1) = -2
{ 4, 7 } = |4 – 7| * (1) = 3

Max of all the above values = 6.

Input : arr[] = { 1, 4, 2, 3, 1 }
Output : 3

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive Approach: A naive approach is to traverse over all sub-arrays and calculate the maximum of function F over all the sub-arrays.

Efficient Approach: A better approach is to consider segments in F(l, r) with odd and even l separately. Two different arrays B[] and C[] can be constructed for this purpose such that:

B[i] = |arr[i] - arr[i + 1]| * (-1)i
C[i] = |arr[i] - arr[i + 1]| * (-1)i + 1


Now if we observe closely, we just need to find the maximum sum subarray of the arrays B[] and C[] and final answer of the function will be the maximum among both the arrays.

Below is the implementation of the above approach:

## C++

 // C++ implementation of the above approach  #include     #define MAX 100005     using namespace std;     // Function to return maximum sum of a sub-array  int kadaneAlgorithm(const int* ar, int n)  {      int sum = 0, maxSum = 0;         for (int i = 0; i < n; i++) {             sum += ar[i];             if (sum < 0)              sum = 0;             maxSum = max(maxSum, sum);      }         return maxSum;  }     // Function to return maximum value of function F  int maxFunction(const int* arr, int n)  {         int b[MAX], c[MAX];         // Compute arrays B[] and C[]      for (int i = 0; i < n - 1; i++) {          if (i & 1) {              b[i] = abs(arr[i + 1] - arr[i]);              c[i] = -b[i];          }          else {              c[i] = abs(arr[i + 1] - arr[i]);              b[i] = -c[i];          }      }         // Find maximum sum sub-array of both of the      // arrays and take maximum among them      int ans = kadaneAlgorithm(b, n - 1);      ans = max(ans, kadaneAlgorithm(c, n - 1));         return ans;  }     // Driver code  int main()  {      int arr[] = { 1, 5, 4, 7 };      int n = sizeof(arr) / sizeof(arr);         cout << maxFunction(arr, n);         return 0;  }

## Java

 // Java implementation of the approach  import java.util.*;     class GFG  {  static int MAX = 100005;        // Function to return maximum sum of a sub-array   static int kadaneAlgorithm(int[] ar, int n)   {       int sum = 0, maxSum = 0;          for (int i = 0; i < n; i++)       {           sum += ar[i];              if (sum < 0)               sum = 0;              maxSum = Math.max(maxSum, sum);       }          return maxSum;   }      // Function to return maximum value   // of function F   static int maxFunction(int[] arr, int n)   {          int []b = new int[MAX];      int []c = new int[MAX];         // Compute arrays B[] and C[]       for (int i = 0; i < n - 1; i++)      {           if (i % 2 == 1)          {               b[i] = Math.abs(arr[i + 1] - arr[i]);               c[i] = -b[i];           }           else          {               c[i] = Math.abs(arr[i + 1] - arr[i]);               b[i] = -c[i];           }       }          // Find maximum sum sub-array of both of the       // arrays and take maximum among them       int ans = kadaneAlgorithm(b, n - 1);       ans = Math.max(ans, kadaneAlgorithm(c, n - 1));          return ans;   }      // Driver code   public static void main(String[] args)   {      int arr[] = { 1, 5, 4, 7 };       int n = arr.length;       System.out.println(maxFunction(arr, n));  }  }     // This code is contributed by PrinciRaj1992

## Python3

 # Python3 implementation of the above approach   MAX = 100005;     # Function to return maximum   # sum of a sub-array   def kadaneAlgorithm(ar, n) :          sum = 0; maxSum = 0;          for i in range(n) :              sum += ar[i];              if (sum < 0) :              sum = 0;              maxSum = max(maxSum, sum);          return maxSum;      # Function to return maximum   # value of function F   def maxFunction(arr, n) :          b =  * MAX;      c =  * MAX;          # Compute arrays B[] and C[]       for i in range(n - 1) :           if (i & 1) :              b[i] = abs(arr[i + 1] - arr[i]);               c[i] = -b[i];                      else :              c[i] = abs(arr[i + 1] - arr[i]);               b[i] = -c[i];          # Find maximum sum sub-array of both of the       # arrays and take maximum among them       ans = kadaneAlgorithm(b, n - 1);       ans = max(ans, kadaneAlgorithm(c, n - 1));          return ans;      # Driver code   if __name__ == "__main__" :          arr = [ 1, 5, 4, 7 ];       n = len(arr)         print(maxFunction(arr, n));      # This code is contributed by Ryuga

## C#

 // C# implementation of the approach  using System;         class GFG  {  static int MAX = 100005;     // Function to return maximum sum of a sub-array   static int kadaneAlgorithm(int[] ar, int n)   {       int sum = 0, maxSum = 0;          for (int i = 0; i < n; i++)       {           sum += ar[i];              if (sum < 0)               sum = 0;              maxSum = Math.Max(maxSum, sum);       }          return maxSum;   }      // Function to return maximum value   // of function F   static int maxFunction(int[] arr, int n)   {       int []b = new int[MAX];      int []c = new int[MAX];         // Compute arrays B[] and C[]       for (int i = 0; i < n - 1; i++)      {           if (i % 2 == 1)          {               b[i] = Math.Abs(arr[i + 1] - arr[i]);               c[i] = -b[i];           }           else         {               c[i] = Math.Abs(arr[i + 1] - arr[i]);               b[i] = -c[i];           }       }          // Find maximum sum sub-array of both of the       // arrays and take maximum among them       int ans = kadaneAlgorithm(b, n - 1);       ans = Math.Max(ans, kadaneAlgorithm(c, n - 1));          return ans;   }      // Driver code   public static void Main(String[] args)   {      int []arr = { 1, 5, 4, 7 };       int n = arr.Length;       Console.WriteLine(maxFunction(arr, n));  }  }     // This code is contributed by PrinciRaj1992

Output:

6


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