A12282 | Lunar New Year and Red Envelopes

Lunar New Year is approaching, and Bob is going to receive some red envelopes with countless money! But collecting money from red envelopes is a time-consuming process itself.

Let's describe this problem in a mathematical way. Consider a timeline from time $1$ to $n$ . The $i$ -th red envelope will be available from time $s_i$ to $t_i$ , inclusive, and contain $w_i$ coins. If Bob chooses to collect the coins in the $i$ -th red envelope, he can do it only in an integer point of time between $s_i$ and $t_i$ , inclusive, and he can't collect any more envelopes until time $d_i$ (inclusive) after that. Here $s_i \leq t_i \leq d_i$ holds.

Bob is a greedy man, he collects coins greedily — whenever he can collect coins at some integer time $x$ , he collects the available red envelope with the maximum number of coins. If there are multiple envelopes with the same maximum number of coins, Bob would choose the one whose parameter $d$ is the largest. If there are still multiple choices, Bob will choose one from them randomly.

However, Alice — his daughter — doesn't want her father to get too many coins. She could disturb Bob at no more than $m$ integer time moments. If Alice decides to disturb Bob at time $x$ , he could not do anything at time $x$ and resumes his usual strategy at the time $x + 1$ (inclusive), which may lead to missing some red envelopes.

Calculate the minimum number of coins Bob would get if Alice disturbs him optimally.

The first line contains three non-negative integers $n$ , $m$ and $k$ ( $1 \leq n \leq 10^5$ , $0 \leq m \leq 200$ , $1 \leq k \leq 10^5$ ), denoting the length of the timeline, the number of times Alice can disturb Bob and the total number of red envelopes, respectively.

The following $k$ lines describe those $k$ red envelopes. The $i$ -th line contains four positive integers $s_i$ , $t_i$ , $d_i$ and $w_i$ ( $1 \leq s_i \leq t_i \leq d_i \leq n$ , $1 \leq w_i \leq 10^9$ ) — the time segment when the $i$ -th envelope is available, the time moment Bob can continue collecting after collecting the $i$ -th envelope, and the number of coins in this envelope, respectively.

Output one integer — the minimum number of coins Bob would get if Alice disturbs him optimally.