The “Put Marbles in Bags” problem is a fascinating challenge in algorithm design that tests your ability to optimize distributions under specific constraints. In this article, we will explore the problem in detail, discuss various approaches to solve it, and provide a C++ implementation for both brute-force and optimized solutions. Whether you’re a beginner or an experienced programmer, this guide will help you understand the intricacies of this problem.

Problem Overview

In the “Put Marbles in Bags” problem, you are given an array of weights representing marbles and an integer k, which indicates the number of bags you need to fill. The goal is to distribute the marbles into these bags while adhering to specific rules and calculating the difference between the maximum and minimum possible scores based on the distribution.

Key Points

• Input Details: You have an array called weights, where each element represents the weight of a marble. The integer k indicates the number of bags.
• Distribution Rules:
  • Each bag must contain at least one marble (no bag can be empty).
  • If you place two marbles in the same bag, all marbles between them in the array must also be included in that bag.
  • The cost of a bag is calculated as the sum of the weights of the first and last marbles in that bag.
• Score Calculation: The total score is the sum of the costs of all k bags.

Example Walkthrough

Let’s consider an example to illustrate the problem:

Example Input

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

Minimum Score Calculation

One possible distribution is:

• Bag 1: [1] (Cost: 1 + 1 = 2)
• Bag 2: [3, 5, 1] (Cost: 3 + 1 = 4)

Total Minimum Score: 2 + 4 = 6

Maximum Score Calculation

Another distribution could be:

• Bag 1: [1, 3] (Cost: 1 + 3 = 4)
• Bag 2: [5, 1] (Cost: 5 + 1 = 6)

Total Maximum Score: 4 + 6 = 10

Final Calculation

The difference between the maximum score (10) and the minimum score (6) is:

10 - 6 = 4

Approaches to Solve the Problem

Brute-Force Approach

The brute-force approach involves generating all possible ways to partition the array of weights into k bags and calculating the scores for each partition. While straightforward, this method can be computationally expensive, especially for larger arrays or higher values of k.

Complexity Analysis

• Time Complexity: Approximately \(O(n^k)\) in the worst case.
• Space Complexity: Significant due to recursion stack and storage of partitions.

Optimized Approach Using Dynamic Programming

To improve efficiency, we can use a dynamic programming approach. This method significantly reduces the time complexity by avoiding the need to generate all possible partitions explicitly.

Dynamic Programming Table

• dp_min[i][j]: Minimum score achievable by distributing the first i marbles into j bags.
• dp_max[i][j]: Maximum score achievable by distributing the first i marbles into j bags.

Complexity Analysis

• Time Complexity: \(O(n^2 \cdot k)\)
• Space Complexity: \(O(n \cdot k)\)

C++ Implementation

Here’s a C++ implementation of the optimized approach:

#include <iostream>
#include <vector>
#include <climits>
using namespace std;

int putMarblesInBags(vector<int>& weights, int k) {
    int n = weights.size();
    if (k > n) return 0;

    vector<vector<int>> dp_min(n + 1, vector<int>(k + 1, INT_MAX));
    vector<vector<int>> dp_max(n + 1, vector<int>(k + 1, INT_MIN));

    for (int i = 1; i <= n; ++i) {
        dp_min[i][1] = weights[0] + weights[i - 1];
        dp_max[i][1] = weights[0] + weights[i - 1];
    }

    for (int j = 2; j <= k; ++j) {
        for (int i = j; i <= n; ++i) {
            for (int p = j - 1; p < i; ++p) {
                int cost = weights[p] + weights[i - 1];
                dp_min[i][j] = min(dp_min[i][j], dp_min[p][j - 1] + cost);
                dp_max[i][j] = max(dp_max[i][j], dp_max[p][j - 1] + cost);
            }
        }
    }

    return dp_max[n][k] - dp_min[n][k];
}

int main() {
    vector<int> weights = {1, 3, 5, 1};
    int k = 2;
    int result = putMarblesInBags(weights, k);
    cout << "Difference between max and min scores: " << result << endl; // Output: 4
    return 0;
}

Conclusion

The “Put Marbles in Bags” problem is an excellent exercise in algorithm design, challenging you to think critically about how to optimally group items under specific constraints. By understanding both the brute-force and optimized approaches, you