Problem Description

The problem involves a basket with a maximum weight limit of 5000 units and a list of integers representing the weights of several apples. The task is to find the maximum number of apples that can be placed in the basket without exceeding the weight limit. This is an optimization problem where you must maximize the quantity of items selected under a certain constraint, in this case, the weight capacity of the basket.

Intuition

The intuition behind the solution is to use a greedy algorithm approach. It makes intuitive sense to start by adding the lightest apples to the basket first, as this would allow us to fit more apples into the basket. By sorting the weights of the apples, we ensure we always consider the lightest apple available for placement in the basket next.

Once we have the sorted list, we iterate through the list, keeping track of the total weight added to the basket so far. We continue to add apples one at a time until adding another apple would exceed the weight limit of 5000 units. When we reach an apple that we cannot add without exceeding the limit, we stop and return the number of apples we have added up to that point.

Learn more about Greedy and Sorting patterns.

Solution Approach

The solution follows a straightforward approach aligned with greedy algorithms, which often provide an optimal solution to optimization problems under certain conditions. Here is how it's implemented:

1. Sorting: We sort the weight array to ensure that we pick apples starting from the lightest to the heaviest. Sorting is a common preparatory step in greedy algorithms, facilitating a sequential choice of items based on some criteria—in this case, the weight.

2. Iteration with Accumulation: We initialize a variable s to zero, representing the current total weight of apples in the basket.

3. Iterative Weight Addition: Sequentially, we iterate over the sorted apple weights. For each apple, we add its weight to the current total s. As we do this, we check whether the running total exceeds the maximum allowed weight of 5000 units.

4. Condition Checking: If the addition of the current apple's weight causes the total weight s to exceed the limit, the loop breaks, and we return the number of apples that have been successfully added so far, which is given by the current index i.

5. Returning the Result: If we complete the iteration without exceeding the weight limit, it means all apples in the list have been added to the basket. Hence, we return the total count of apples, which can be obtained using len(weight).

Using this method, the algorithm reaches an efficient solution that maximizes the number of apples within the allowed weight limit. The use of sorting followed by a linear scan of the list is characteristic of many greedy strategies and is effectively employed in this implementation.

Example Walkthrough

To illustrate the solution approach, let's consider a small example. Suppose we have the following weights for the apples: [600, 1000, 950, 200, 400]. We aim to place as many of these apples into a basket with a weight limit of 5000 units.

1. Sorting: First, we sort the apples by their weight, resulting in the sorted list [200, 400, 600, 950, 1000].

2. Iteration with Accumulation: We start with a total current weight s of zero and begin iterating over the list.

3. Iterative Weight Addition: As we scan the list, we add the weight of each apple to s:

   • Add 200 (s = 200)
   • Add 400 (s = 600)
   • Add 600 (s = 1200)
   • Add 950 (s = 2150)
   • Add 1000 (s = 3150)

   We successfully add all the apples in the sorted list to the basket since the total weight does not exceed 5000 units.

4. Condition Checking: At each step of adding an apple, we check if the next addition will exceed the weight limit. In this case, we never reach a point where s exceeds 5000, so the condition check always passes.

5. Returning the Result: Since we were able to add all apples without surpassing the weight limit, we return the total number of apples, which is 5.

Therefore, using our greedy approach, we concluded that we can fit all five apples in the basket because their combined weight is 3150 units, which is within the 5000-unit limit.

Solution Implementation

Time and Space Complexity

The given Python code snippet sorts the input list weight and then iterates through the sorted list to calculate the cumulative weight until it exceeds the limit of 5000, or until all the apples are counted.

Time Complexity:

The time complexity of the algorithm is dominated by the sorting operation at the beginning. Sorting a list of n elements typically has a time complexity of O(n log n) when using the built-in sorted function in Python (Timsort algorithm). After sorting, there is a single pass over the input list to calculate the cumulative weight. The single pass has a time complexity of O(n), where n is the number of elements in weight.

Since the sorting cost is higher than the single pass, the overall time complexity of the code is O(n log n).

Space Complexity:

The space complexity is O(1) (constant space) aside from the space required to store the input, because the sorting is done in-place (altering the input list itself) and only a few extra variables (i, x, s) are used to keep track of the running total and index. Therefore, the space required does not depend on the size of the input list.

Learn more about how to find time and space complexity quickly using problem constraints.