Python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：
Python
'''Python program to find maximum value of Sum(i*arr[i])'''
# returns max possible value of Sum(i*arr[i])
def maxSum(arr):
# stores sum of arr[i]
arrSum = 0
# stores sum of i*arr[i]
currVal = 0
n = len(arr)
for i in range(0, n):
arrSum = arrSum + arr[i]
currVal = currVal + (i*arr[i])
# initialize result
maxVal = currVal
# try all rotations one by one and find the maximum
# rotation sum
for j in range(1, n):
currVal = currVal + arrSum-n*arr[n-j]
if currVal > maxVal:
maxVal = currVal
# return result
return maxVal
# test maxsum(arr) function
arr = [10, 1, 2, 3, 4, 5, 6, 7, 8, 9]
print "Max sum is: ", maxSum(arr)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！