#2551

Hard

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Put Marbles in Bags

The "Put Marbles in Bags" problem challenges you to distribute marbles into bags for maximum score difference using greedy and sorting techniques.

array greedy sorting heap priority queue

Problem Statement

You are given a 0-indexed integer array weights, where each element represents the weight of a marble. You are also given an integer k, which represents the number of bags available for distribution. The goal is to divide the marbles into k bags in such a way that maximizes the difference in scores between the bags.

Each bag's score is the sum of the weights in that bag. The final score of the distribution is the difference between the maximal and minimal scores of any bag combination. Your task is to return this score difference, computed from the best and worst possible marble distributions.

Examples

Example 1

Input: weights = [1,3,5,1], k = 2

Output: 4

The distribution [1],[3,5,1] results in the minimal score of (1+1) + (3+1) = 6. The distribution [1,3],[5,1], results in the maximal score of (1+3) + (5+1) = 10. Thus, we return their difference 10 - 6 = 4.

Example 2

Input: weights = [1, 3], k = 2

Output: 0

The only distribution possible is [1],[3]. Since both the maximal and minimal score are the same, we return 0.

Constraints

1 <= k <= weights.length <= 105
1 <= weights[i] <= 109

Solution Approach

Greedy Choice with Sorting

Sort the array weights. To maximize the score difference, you want to ensure that the lightest and heaviest marbles are placed in separate bags. The strategy is to maximize the difference between the sums of the bags by making sure the most extreme weights (smallest and largest) are separated as much as possible.

Heap Usage for Efficient Distribution

You can use a max-heap or min-heap to efficiently determine how to distribute the weights. By focusing on endpoints of subarrays, you can quickly determine the best possible combinations of weights for the minimal and maximal score scenarios.

Invariant Validation

After sorting, the problem boils down to choosing k-1 split points to divide the sorted weights into k subarrays. By ensuring that the subarrays are balanced in terms of weight extremes, you can verify the correctness of your solution.

Complexity Analysis

Metric	Value
Time	O(n \log n)
Space	O(n + S) \approx O(n)

The time complexity is O(n \log n) due to the sorting step, while space complexity is O(n) for storing the weights and auxiliary heap data structures. These complexities are optimal for the problem's constraints.

What Interviewers Usually Probe

Candidate should demonstrate understanding of greedy algorithms and sorting techniques.
Look for proficiency in heap or priority queue usage.
The solution should efficiently handle large input sizes, leveraging sorting and heap structures.

Common Pitfalls or Variants

Common pitfalls

Failing to correctly identify the need to separate the largest and smallest weights for optimal score distribution.
Overcomplicating the problem by attempting to find all possible subarray combinations instead of leveraging sorting and greedy methods.
Not optimizing for time complexity, leading to inefficient solutions that fail on large inputs.

Follow-up variants

Instead of maximizing the difference in scores, try to minimize the difference using the same approach.
Extend the problem to handle more complex scoring mechanisms, such as considering penalties for imbalances.
Modify the problem to include additional constraints on how marbles can be distributed among bags (e.g., weights must be evenly split).

FAQ

What is the greedy strategy used in "Put Marbles in Bags"?

The greedy strategy involves sorting the marble weights and placing the lightest and heaviest weights in different bags to maximize the score difference.

How do heaps help in solving "Put Marbles in Bags"?

Heaps can be used to efficiently manage and extract the smallest and largest bags when calculating possible score combinations.

What is the main challenge in "Put Marbles in Bags"?

The main challenge is ensuring the correct distribution of weights into k bags while maximizing the score difference between the bags.

How does sorting impact the solution for "Put Marbles in Bags"?

Sorting ensures that the heaviest and lightest weights are correctly positioned for optimal separation, making it easier to compute the maximal and minimal scores.

What are the key trade-offs when solving "Put Marbles in Bags"?

The trade-offs involve balancing between greedy choice selection and maintaining efficient sorting and heap operations to handle large inputs within time limits.

terminal

Solution

Solution 1: Problem Transformation + Sorting

We can transform the problem into: dividing the array `weights` into $k$ consecutive subarrays, that is, we need to find $k-1$ splitting points, each splitting point's cost is the sum of the elements on the left and right of the splitting point. The difference between the sum of the costs of the largest $k-1$ splitting points and the smallest $k-1$ splitting points is the answer.

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class Solution:
    def putMarbles(self, weights: List[int], k: int) -> int:
        arr = sorted(a + b for a, b in pairwise(weights))
        return sum(arr[len(arr) - k + 1 :]) - sum(arr[: k - 1])

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