<p>There are some robots and factories on the X-axis. You are given an integer array <code>robot</code> where <code>robot[i]</code> is the position of the <code>i<sup>th</sup></code> robot. You are also given a 2D integer array <code>factory</code> where <code>factory[j] = [position<sub>j</sub>, limit<sub>j</sub>]</code> indicates that <code>position<sub>j</sub></code> is the position of the <code>j<sup>th</sup></code> factory and that the <code>j<sup>th</sup></code> factory can repair at most <code>limit<sub>j</sub></code> robots.</p>

<p>The positions of each robot are <strong>unique</strong>. The positions of each factory are also <strong>unique</strong>. Note that a robot can be <strong>in the same position</strong> as a factory initially.</p>

<p>All the robots are initially broken; they keep moving in one direction. The direction could be the negative or the positive direction of the X-axis. When a robot reaches a factory that did not reach its limit, the factory repairs the robot, and it stops moving.</p>

<p><strong>At any moment</strong>, you can set the initial direction of moving for <strong>some</strong> robot. Your target is to minimize the total distance traveled by all the robots.</p>

<p>Return <em>the minimum total distance traveled by all the robots</em>. The test cases are generated such that all the robots can be repaired.</p>

<p><strong>Note that</strong></p>

<ul>
	<li>All robots move at the same speed.</li>
	<li>If two robots move in the same direction, they will never collide.</li>
	<li>If two robots move in opposite directions and they meet at some point, they do not collide. They cross each other.</li>
	<li>If a robot passes by a factory that reached its limits, it crosses it as if it does not exist.</li>
	<li>If the robot moved from a position <code>x</code> to a position <code>y</code>, the distance it moved is <code>|y - x|</code>.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/09/15/example1.jpg" style="width: 500px; height: 320px;" />
<pre>
<strong>Input:</strong> robot = [0,4,6], factory = [[2,2],[6,2]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> As shown in the figure:
- The first robot at position 0 moves in the positive direction. It will be repaired at the first factory.
- The second robot at position 4 moves in the negative direction. It will be repaired at the first factory.
- The third robot at position 6 will be repaired at the second factory. It does not need to move.
The limit of the first factory is 2, and it fixed 2 robots.
The limit of the second factory is 2, and it fixed 1 robot.
The total distance is |2 - 0| + |2 - 4| + |6 - 6| = 4. It can be shown that we cannot achieve a better total distance than 4.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/09/15/example-2.jpg" style="width: 500px; height: 329px;" />
<pre>
<strong>Input:</strong> robot = [1,-1], factory = [[-2,1],[2,1]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> As shown in the figure:
- The first robot at position 1 moves in the positive direction. It will be repaired at the second factory.
- The second robot at position -1 moves in the negative direction. It will be repaired at the first factory.
The limit of the first factory is 1, and it fixed 1 robot.
The limit of the second factory is 1, and it fixed 1 robot.
The total distance is |2 - 1| + |(-2) - (-1)| = 2. It can be shown that we cannot achieve a better total distance than 2.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= robot.length, factory.length &lt;= 100</code></li>
	<li><code>factory[j].length == 2</code></li>
	<li><code>-10<sup>9</sup> &lt;= robot[i], position<sub>j</sub> &lt;= 10<sup>9</sup></code></li>
	<li><code>0 &lt;= limit<sub>j</sub> &lt;= robot.length</code></li>
	<li>The input will be generated such that it is always possible to repair every robot.</li>
</ul>

from heapq import heappush, heappop


class edge:
    def __init__(self, to, cap, cost, rev):
        self.to, self.cap, self.cost, self.rev = to, cap, cost, rev


class MinCostFlow:

    def __init__(self, n):
        self.n = n
        self.graph = [[] for _ in range(n)]

    def add_edge(self, src, to, cap, cost):
        self.graph[src].append(edge(to, cap, cost, len(self.graph[to])))
        self.graph[to].append(edge(src, 0, -cost, len(self.graph[src]) - 1))
    
    def solve(self, s, t, f):
        h = [0] * self.n
        prevv, preve = [0] * self.n, [0] * self.n
        ret = 0
        while f:
            dis = [float('inf')] * self.n
            dis[s] = 0
            hq = [(0, s)]
            while hq:
                d, v = heappop(hq)
                if dis[v] < d: continue
                for i, e in enumerate(self.graph[v]):
                    if e.cap > 0 and dis[e.to] > dis[v] + e.cost + h[v] - h[e.to]:
                        dis[e.to] = dis[v] + e.cost + h[v] - h[e.to]
                        prevv[e.to], preve[e.to] = v, i
                        heappush(hq, (dis[e.to], e.to))
            if dis[t] == float('inf'): return -1
            for v in range(self.n): h[v] += dis[v]
            d, v = f, t
            while v != s:
                d = min(d, self.graph[prevv[v]][preve[v]].cap)
                v = prevv[v]
            f -= d
            ret += d * h[t]
            v = t
            while v != s:
                self.graph[prevv[v]][preve[v]].cap -= d
                self.graph[v][self.graph[prevv[v]][preve[v]].rev].cap += d
                v = prevv[v]
        return ret


class Solution:
    def minimumTotalDistance(self, robot: List[int], factory: List[List[int]]) -> int:
        m, n = len(robot), len(factory)
        s, t = m + n, m + n + 1
        mcf = MinCostFlow(t + 1)
        for i in range(m): mcf.add_edge(s, i, 1, 0)
        for j in range(n): mcf.add_edge(j + m, t, factory[j][1], 0)
        for i in range(m):
            for j in range(n): mcf.add_edge(i, j + m, 1, abs(robot[i] - factory[j][0]))
        return mcf.solve(s, t, len(robot))
20400