Minimum Cost to Connect Two Groups of Points

You can solve this problem using a recursive approach with memoization. This problem can be considered as a type of bipartite graph matching problem, where you’re trying to find a matching with a minimum cost.

Here’s the code to do it:

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class Solution:
    def connectTwoGroups(self, cost: List[List[int]]) -> int:
        size1, size2 = len(cost), len(cost[0])
        min_cost_2 = [min(cost[i][j] for i in range(size1)) for j in range(size2)]
        @lru_cache(None)
        def dp(i: int, mask: int) -> int:
            # If all first group's points are connected, connect remaining second group's points
            if i == size1:
                return sum(min_cost_2[j] for j in range(size2) if not (mask & (1 << j)))

            # Try every second group's point for the i-th first group's point
            result = float('inf')
            for j in range(size2):
                result = min(result, cost[i][j] + dp(i + 1, mask | (1 << j)))

            return result

        return dp(0, 0)

Explanation:

  1. Base Case: If all points in the first group have been considered, connect remaining points in the second group with minimum cost using min_cost_2.

  2. Recursive Step: For each point i in the first group, try to connect it with every point j in the second group. Use a bitmask mask to represent which points in the second group have already been connected.

  3. Memoization: The lru_cache decorator caches the results of recursive calls to avoid redundant computations.

  4. Return Value: The result is the minimum cost to connect both groups.

The time complexity of this solution is (O(2^{n} \cdot m \cdot n)), where (n) is size2 and (m) is size1. The space complexity is (O(2^{n} \cdot m)).

10 Prerequisite LeetCode Problems

“1595. Minimum Cost to Connect Two Groups of Points” combines dynamic programming, bit manipulation, and graph concepts. Here are some simpler problems to prepare for this:

  1. LeetCode 64. Minimum Path Sum

    • This problem introduces the basic concept of finding the minimum path sum in a grid.

  2. LeetCode 787. Cheapest Flights Within K Stops

    • This problem involves finding a path with minimum cost, which is similar to connecting groups of points with minimum cost.

  3. LeetCode 1048. Longest String Chain

    • This problem will help you understand the concept of dynamic programming over sequences and chains.

  4. LeetCode 1262. Greatest Sum Divisible by Three

    • This problem involves selecting elements to maximize a quantity, similar to the “Minimum Cost to Connect Two Groups of Points” problem.

  5. LeetCode 322. Coin Change

    • This problem is about making choices to minimize cost, which can help you understand how to set up dynamic programming states and transitions.

  6. LeetCode 198. House Robber

    • This problem introduces the concept of dynamic programming for making decisions to maximize a quantity.

  7. LeetCode 1365. How Many Numbers Are Smaller Than the Current Number

    • This problem helps you understand how to use a comparison mechanism in a problem, similar to comparing costs in this problem.

  8. LeetCode 518. Coin Change 2

    • This problem is a classic dynamic programming problem where you need to make decisions at each step.

  9. LeetCode 78. Subsets

    • This problem introduces the concept of creating subsets, which can help you understand bit manipulation used in the “Minimum Cost to Connect Two Groups of Points” problem.

  10. LeetCode 338. Counting Bits

  • This problem involves bit manipulation, which is a concept used in the “Minimum Cost to Connect Two Groups of Points” problem.

These cover how to use dynamic programming to solve problems involving cost minimization, bit manipulation, and making decisions, which are key to solving the “1595. Minimum Cost to Connect Two Groups of Points” problem.

Problem Classification

Problem Statement:

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Language Agnostic Coding Drills

Consider the following piece of complex software code.

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.