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| <h2><a href="https://leetcode.com/problems/minimum-total-distance-traveled/?envType=daily-question&envId=2024-10-31">2463. Minimum Total Distance Traveled</a></h2><h3>Hard</h3><hr><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> | ||
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| <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> | ||
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| <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> | ||
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| <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> | ||
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| <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> | ||
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| <p><strong>Note that</strong></p> | ||
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| <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> | ||
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| <p> </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> | ||
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| <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> | ||
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| <p> </p> | ||
| <p><strong>Constraints:</strong></p> | ||
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| <ul> | ||
| <li><code>1 <= robot.length, factory.length <= 100</code></li> | ||
| <li><code>factory[j].length == 2</code></li> | ||
| <li><code>-10<sup>9</sup> <= robot[i], position<sub>j</sub> <= 10<sup>9</sup></code></li> | ||
| <li><code>0 <= limit<sub>j</sub> <= robot.length</code></li> | ||
| <li>The input will be generated such that it is always possible to repair every robot.</li> | ||
| </ul> |