codeforces#P1162A. Zoning Restrictions Again
Zoning Restrictions Again
Description
You are planning to build housing on a street. There are $n$ spots available on the street on which you can build a house. The spots are labeled from $1$ to $n$ from left to right. In each spot, you can build a house with an integer height between $0$ and $h$.
In each spot, if a house has height $a$, you will gain $a^2$ dollars from it.
The city has $m$ zoning restrictions. The $i$-th restriction says that the tallest house from spots $l_i$ to $r_i$ (inclusive) must be at most $x_i$.
You would like to build houses to maximize your profit. Determine the maximum profit possible.
The first line contains three integers $n$, $h$, and $m$ ($1 \leq n,h,m \leq 50$) — the number of spots, the maximum height, and the number of restrictions.
Each of the next $m$ lines contains three integers $l_i$, $r_i$, and $x_i$ ($1 \leq l_i \leq r_i \leq n$, $0 \leq x_i \leq h$) — left and right limits (inclusive) of the $i$-th restriction and the maximum possible height in that range.
Print a single integer, the maximum profit you can make.
Input
The first line contains three integers $n$, $h$, and $m$ ($1 \leq n,h,m \leq 50$) — the number of spots, the maximum height, and the number of restrictions.
Each of the next $m$ lines contains three integers $l_i$, $r_i$, and $x_i$ ($1 \leq l_i \leq r_i \leq n$, $0 \leq x_i \leq h$) — left and right limits (inclusive) of the $i$-th restriction and the maximum possible height in that range.
Output
Print a single integer, the maximum profit you can make.
Samples
3 3 3
1 1 1
2 2 3
3 3 2
14
4 10 2
2 3 8
3 4 7
262
Note
In the first example, there are $3$ houses, the maximum height of a house is $3$, and there are $3$ restrictions. The first restriction says the tallest house between $1$ and $1$ must be at most $1$. The second restriction says the tallest house between $2$ and $2$ must be at most $3$. The third restriction says the tallest house between $3$ and $3$ must be at most $2$.
In this case, it is optimal to build houses with heights $[1, 3, 2]$. This fits within all the restrictions. The total profit in this case is $1^2 + 3^2 + 2^2 = 14$.
In the second example, there are $4$ houses, the maximum height of a house is $10$, and there are $2$ restrictions. The first restriction says the tallest house from $2$ to $3$ must be at most $8$. The second restriction says the tallest house from $3$ to $4$ must be at most $7$.
In this case, it's optimal to build houses with heights $[10, 8, 7, 7]$. We get a profit of $10^2+8^2+7^2+7^2 = 262$. Note that there are two restrictions on house $3$ and both of them must be satisfied. Also, note that even though there isn't any explicit restrictions on house $1$, we must still limit its height to be at most $10$ ($h=10$).