How To Find the minimum value of the expression below:
SUM(|x - A[i]|*B[i]), i >= 0 and i < N, B[i] > 0
When B[i] = 1 I know the answer is given by x = Median(A).How to calculate x when B itself varies.
How To Find the minimum value of the expression below:
SUM(|x - A[i]|*B[i]), i >= 0 and i < N, B[i] > 0
When B[i] = 1 I know the answer is given by x = Median(A).How to calculate x when B itself varies.
| № | Пользователь | Рейтинг |
|---|---|---|
| 1 | Benq | 3676 |
| 2 | Kevin114514 | 3659 |
| 3 | jiangly | 3640 |
| 4 | turmax | 3450 |
| 5 | tourist | 3382 |
| 6 | strapple | 3360 |
| 7 | Um_nik | 3354 |
| 8 | squareOf105 | 3348 |
| 9 | maroonrk | 3312 |
| 10 | idontwannadothis…e | 3191 |
| Страны | Города | Организации | Всё → |
| № | Пользователь | Вклад |
|---|---|---|
| 1 | Qingyu | 155 |
| 2 | nik_exists | 151 |
| 3 | maspy | 149 |
| 4 | AmShZ | 145 |
| 5 | Um_nik | 143 |
| 6 | Errichto | 139 |
| 7 | BledDest | 136 |
| 8 | adamant | 134 |
| 9 | maroonrk | 132 |
| 10 | cry | 130 |
Another way to think about this is to imagine you have a new array $$$A'$$$ such that it has element $$$A_i$$$ repeated $$$B_i$$$ times in it. Now it is easy to see that the answer to your problem for $$$A'$$$ and $$$B'_i = 1$$$ will be the same as the one for $$$(A, B)$$$. This means that we can simply find the median of $$$A'$$$.
To do this fast, we sort the pairs $$$(A_i, B_i)$$$ by $$$A$$$, then we find the sum of all $$$B_i$$$ and go from the beginning while the prefix sum of $$$2 * B_i \lt SumB$$$. The last $$$A$$$ we have went through will be our answer.
Thanks for the explanation :)
Can anybody prove the median thing if B[i] = 1?
The function $$$f(x) = \sum_{i = 1}^{n} |x - a_i| $$$ is a piecwise linear function, meaning that it can be broken into straight lines, and changes slope at breakpoints $$$a_i$$$'s, which confirms that minimum is achieved at one of $$$a_i$$$'s. Now as you move from $$$-\infty$$$ to $$$+\infty$$$ slope changes from $$$-n$$$ to $$$n$$$, and is $$$0$$$ at median (when exactly half terms contribute +1 to slope and other half -1)
How can you assume the slope changes from -n to n? A[i] values can be anything
Ok thanks