第20天。
今天的题目是 Minimum Moves to Equal Array Elements II ：
Given a non-empty integer array, find the minimum number of moves required to make all array elements equal, where a move is incrementing a selected element by 1 or decrementing a selected element by 1.
You may assume the array’s length is at most 10,000.
Example:
Input:
[1,2,3]
Output:
2
Explanation:
Only two moves are needed (remember each move increments or decrements one element):
[1,2,3] => [2,2,3] => [2,2,2]
这道题需要一些数学推导，它的目标就是：
$$ min_k { \sum_{i=1}^n |n_i - n_k| } $$ 其中 $n_i$ 表示数组排序后中第 $i$ 个元素。
我们将式子展开可以得到： $$ min_k { \sum_{i=1}^n |n_i - n_k| } =
min_k { \sum_{i=1}^k (n_k-n_i) + \sum_{i=k+1}^n(n_i-n_k) } \
= min_k { \sum_{i=1}^k n_k - \sum_{i=1}^k n_i + \sum_{i=k+1}^n n_i - \sum_{i=k+1}^n n_k } \
= min_k { \sum_{i=k+1}^n n_i - \sum_{i=1}^k n_i + (2k - n)n_k } $$ 因此，我们可以写出如下代码：
int minMoves2(vector<int>& nums) {
long long res = LONG_MAX;
sort(nums.begin(), nums.end());
long long rightSum = 0;
for(auto i: nums) rightSum += i;
long long leftSum = 0;
int n = nums.size();
for(int i = 0;i < n; ++i) {
res = min(res, rightSum - leftSum + (2*i - n) * (long long)nums[i]);
rightSum -= nums[i];
leftSum += nums[i];
// cout << res << endl;
}
return res;
}
这样还不是最优解，然而最优解我没看懂（捂脸），为什么用中位数求就是对的呢？：
int minMoves2(vector<int>& nums) {
sort(nums.begin(), nums.end());
int mid;
if (nums.size() % 2 == 0){
mid = (nums[nums.size()/2] + nums[(nums.size()/2) - 1])/2;
}else{
mid = nums[nums.size()/2];
}
int result = 0;
for (int i = 0; i < nums.size(); i++){
result += abs(nums[i] - mid);
}
return result;
}