#2463

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Total Distance Traveled

Optimize the total distance traveled by robots to factories using dynamic programming, sorting, and state transitions.

Problem Statement

You are given an integer array robot where robot[i] represents the position of the ith robot on the X-axis. You are also given a 2D integer array factory where factory[j] = [positionj, limitj] indicates that the jth factory is located at positionj and can repair at most limitj robots.

All robots are initially broken and they move in one direction. When a robot reaches a factory that hasn’t reached its limit, it gets repaired. You must minimize the total distance traveled by all robots to reach their assigned factories, taking into account the limit of repairs each factory can provide.

Examples

Example 1

Input: robot = [0,4,6], factory = [[2,2],[6,2]]

Output: 4

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.

Example 2

Input: robot = [1,-1], factory = [[-2,1],[2,1]]

Output: 2

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.

Constraints

1 <= robot.length, factory.length <= 100
factory[j].length == 2
-109 <= robot[i], positionj <= 109
0 <= limitj <= robot.length
The input will be generated such that it is always possible to repair every robot.

Solution Approach

Sort robots and factories

First, sort both the robot array and the factory array based on their positions. Sorting helps in optimizing the match between robots and factories to minimize travel distance. This reduces unnecessary calculations and helps us focus on the most efficient repair assignments.

Dynamic programming approach

Use dynamic programming to track the minimum total distance. Define a state that represents the number of robots repaired and calculate the best possible total distance for each state. Transition between states based on the number of robots repaired by each factory.

Greedy robot allocation

For each factory, greedily allocate robots that are closest to the factory's position, ensuring the factory’s limit is respected. This step helps in reducing the total distance by repairing robots that require minimal movement.

Complexity Analysis

Metric	Value
Time	O(m \cdot n^2)
Space	O(m + S)

The time complexity is O(m * n^2) due to the dynamic programming approach, where m is the number of factories and n is the number of robots. The space complexity is O(m + S), where S is the total number of robots that can be repaired, as we store state transitions for each possible combination of repaired robots.

What Interviewers Usually Probe

Ensure the candidate understands state transition dynamic programming and how it applies to minimizing the total distance.
Look for clear reasoning behind sorting robots and factories before applying dynamic programming.
Assess the candidate's ability to handle limits in factory repair capacity effectively.

Common Pitfalls or Variants

Common pitfalls

Not sorting the robots and factories before applying dynamic programming, which may result in suboptimal solutions.
Overcomplicating the problem by not recognizing the greedy approach to allocate robots to factories based on proximity.
Failing to handle edge cases where factories have a repair limit of 0 or where robots are initially at the factory positions.

Follow-up variants

Consider variations where each robot has its own repair time, introducing a more complex allocation strategy.
Change the problem to a multidimensional dynamic programming approach where there are multiple repair factories along a multi-dimensional axis.
Introduce constraints where robots must be repaired in a specific order, adding a dependency to the dynamic programming approach.

FAQ

What is the primary algorithm used in the Minimum Total Distance Traveled problem?

The primary algorithm used is state transition dynamic programming, where we optimize the robot assignments to factories based on sorted positions and factory limits.

How do I minimize the total distance in this problem?

By sorting the robots and factories by position and using dynamic programming to track the optimal distance based on factory repair limits.

What are the key challenges in solving this problem?

The main challenges are handling the factory repair limits, ensuring robots are assigned efficiently to factories, and managing the state transitions in the dynamic programming solution.

What does the term 'state transition dynamic programming' mean in this context?

It refers to a dynamic programming approach where we transition between states representing the number of robots repaired and calculate the minimum total distance at each step.

How does GhostInterview help with the Minimum Total Distance Traveled problem?

GhostInterview helps by providing a structured breakdown of the problem, guiding you through the sorting and dynamic programming steps, and offering insights into common pitfalls and optimization strategies.

terminal

Solution

Solution 1: Memoization Search

First, we sort the robots and factories in ascending order. Then we define a function $dfs(i, j)$ to represent the minimum total moving distance starting from the $i$-th robot and the $j$-th factory.

class Solution:
    def minimumTotalDistance(self, robot: List[int], factory: List[List[int]]) -> int:
        @cache
        def dfs(i, j):
            if i == len(robot):
                return 0
            if j == len(factory):
                return inf
            ans = dfs(i, j + 1)
            t = 0
            for k in range(factory[j][1]):
                if i + k == len(robot):
                    break
                t += abs(robot[i + k] - factory[j][0])
                ans = min(ans, t + dfs(i + k + 1, j + 1))
            return ans

        robot.sort()
        factory.sort()
        ans = dfs(0, 0)
        dfs.cache_clear()
        return ans

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