Java程序查找 Sum(i*arr[i]) 的最大值，仅允许在给定数组上进行旋转

给定一个数组，只允许对数组进行旋转操作。我们可以根据需要多次旋转数组。返回 i*arr[i] 的最大可能总和。

例子 ：

Input: arr[] = {1, 20, 2, 10}
Output: 72
We can get 72 by rotating array twice.
{2, 10, 1, 20}
20*3 + 1*2 + 10*1 + 2*0 = 72

Input: arr[] = {10, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Output: 330
We can get 330 by rotating array 9 times.
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
0*1 + 1*2 + 2*3 ... 9*10 = 330

我们强烈建议您最小化您的浏览器并首先自己尝试。
一个简单的解决方案是逐个找到所有旋转，检查每个旋转的总和并返回最大总和。该解决方案的时间复杂度为 O(n ² )。

我们可以使用Efficient Solution在 O(n) 时间内解决这个问题。
令 R _j为 i*arr[i] 的值，旋转 j 次。这个想法是从前一个旋转计算下一个旋转值，即从 R _j-1计算 R _j 。我们可以将结果的初始值计算为 R ₀ ，然后继续计算下一个旋转值。

如何从 R _j-1有效地计算 R _j ？
这可以在 O(1) 时间内完成。以下是详细信息。

Let us calculate initial value of i*arr[i] with no rotation
R0 = 0*arr[0] + 1*arr[1] +...+ (n-1)*arr[n-1]

After 1 rotation arr[n-1], becomes first element of array, 
arr[0] becomes second element, arr[1] becomes third element
and so on.
R1 = 0*arr[n-1] + 1*arr[0] +...+ (n-1)*arr[n-2]

R1 - R0 = arr[0] + arr[1] + ... + arr[n-2] - (n-1)*arr[n-1]

After 2 rotations arr[n-2], becomes first element of array, 
arr[n-1] becomes second element, arr[0] becomes third element
and so on.
R2 = 0*arr[n-2] + 1*arr[n-1] +...+ (n-1)*arr[n-3]

R2 - R1 = arr[0] + arr[1] + ... + arr[n-3] - (n-1)*arr[n-2] + arr[n-1]

If we take a closer look at above values, we can observe 
below pattern

Rj - Rj-1 = arrSum - n * arr[n-j]

Where arrSum is sum of all array elements, i.e., 

arrSum = ∑ arr[i]
        0<=i<=n-1 

下面是完整的算法：

1) Compute sum of all array elements. Let this sum be 'arrSum'.

2) Compute R0 by doing i*arr[i] for given array. 
   Let this value be currVal.

3) Initialize result: maxVal = currVal // maxVal is result.

// This loop computes Rj from  Rj-1 
4) Do following for j = 1 to n-1
......a) currVal = currVal + arrSum-n*arr[n-j];
......b) If (currVal > maxVal)
            maxVal = currVal   

5) Return maxVal

以下是上述想法的实现：

Java

// Java program to find max value of i*arr[i]
  
import java.util.Arrays;
  
class Test
{
    static int arr[] = new int[]{10, 1, 2, 3, 4, 5, 6, 7, 8, 9};
      
    // Returns max possible value of i*arr[i]
    static int maxSum()
    {
        // Find array sum and i*arr[i] with no rotation
        int arrSum = 0;  // Stores sum of arr[i]
        int currVal = 0;  // Stores sum of i*arr[i]
        for (int i=0; i maxVal)
                maxVal = currVal;
        }
       
        // Return result
        return maxVal;
    }
      
    // Driver method to test the above function
    public static void main(String[] args) 
    {
        System.out.println("Max sum is " + maxSum());
    }
}

输出 ：

Max sum is 330

时间复杂度： O(n)
辅助空间： O(1)

请参阅完整的文章 Find maximum of Sum(i*arr[i]) with only rotation on given array allowed for more details！