
Problem Statement
You are provided with two arrays, boxes
and warehouse
, each consisting of positive integers representing the heights of boxes and the heights of various rooms in a warehouse, respectively. The rooms in the warehouse are sequentially labeled from 0 to n - 1
, where n
is the length of the warehouse array. Each box has a unit width, and the height of a room is defined as warehouse[i]
, with i
denoting the position of the room.
The challenge lies in determining the maximum number of these boxes that can be fittingly stored in the warehouse under specific conditions. Boxes cannot be stacked atop one another, but you can change the order in which they are to be inserted. You can also decide whether to start placing the boxes from the left or right end of the warehouse. Importantly, if a box is taller than a room, it blocks the placement of other boxes behind it relative to that room. The goal is to optimize the arrangements to maximize the number of boxes stored in the warehouse.
Examples
Example 1
Input:
boxes = [1,2,2,3,4], warehouse = [3,4,1,2]
Output:
4
Explanation:
We can store the boxes in the following order: 1- Put the yellow box in room 2 from either the left or right side. 2- Put the orange box in room 3 from the right side. 3- Put the green box in room 1 from the left side. 4- Put the red box in room 0 from the left side. Notice that there are other valid ways to put 4 boxes such as swapping the red and green boxes or the red and orange boxes.
Example 2
Input:
boxes = [3,5,5,2], warehouse = [2,1,3,4,5]
Output:
3
Explanation:
It is not possible to put the two boxes of height 5 in the warehouse since there's only 1 room of height >= 5. Other valid solutions are to put the green box in room 2 or to put the orange box first in room 2 before putting the green and red boxes.
Constraints
n == warehouse.length
1 <= boxes.length, warehouse.length <= 105
1 <= boxes[i], warehouse[i] <= 109
Approach and Intuition
The essence of solving this problem efficiently hinges on a strategic reordering and positioning of the boxes based on their height relative to the warehouse rooms’ dimensions. Given the rules, understanding and following a logical order of operations is crucial:
Preprocess the warehouse array to find the maximal box height that each room can accommodate by moving from both left to right and vice versa. The smallest height encountered in any direction essentially restricts that path's capacity. This preprocessed information can be used to make informed choices about from which direction to insert the boxes.
Sort the array of boxes based on their heights. This helps in systematically attempting to fit the smallest available boxes first, thereby maximizing the chances of placing more boxes inside the warehouse.
Use a greedy approach to fit boxes from either end (left or right):
- From the left, attempt to place each box in the next available room that can accommodate it, moving sequentially through the rooms.
- From the right, similarly, try to fit each box into the rooms, starting from the last room backwards.
- The logic is to fill in the easier (bigger rooms per the preprocessed limits) spaces first to ensure maximum utilization.
Compare the number of boxes fitted using both leftward and rightward approaches, then return the greater number as the result. This ensures that you account for different room height variations across the warehouse.
By following the steps outlined, one can efficiently determine the maximum number of boxes that can be accommodated in the warehouse, taking into account the possibility of entering from either end and the constraint of non-stackable boxes.
Solutions
- C++
class Solution {
public:
int maxBoxesInWarehouse(vector<int>& boxes, vector<int>& warehouse) {
int n = warehouse.size();
// Sorting boxes by their sizes
sort(boxes.begin(), boxes.end());
int start = 0;
int end = n - 1;
int totalBoxes = 0;
int boxIter = boxes.size() - 1;
// Going through each box from largest to smallest
while (start <= end && boxIter >= 0) {
// Put the current largest box if possible in the leftmost part
if (boxes[boxIter] <= warehouse[start]) {
totalBoxes++;
start++;
// Otherwise try to place it in the rightmost part
} else if (boxes[boxIter] <= warehouse[end]) {
totalBoxes++;
end--;
}
boxIter--;
}
return totalBoxes;
}
};
The problem involves placing as many of the given boxes as possible into a warehouse with spatial constraints given by two entry points. Boxes and spaces in the warehouse have definite sizes and the goal is to maximize the number of boxes stored. The solution employs a two-pointer technique and involves sorting the array of box sizes for strategic placement from largest to smallest.
To understand the approach:
- Sort the list of box sizes in ascending order.
- Use two pointers to represent the leftmost and rightmost available positions in the warehouse.
- Iterate over the sorted list of boxes from the largest to the smallest.
- Check if the largest box can fit into the leftmost position. If it fits, move the left pointer to the right, indicating that this position is now taken, and increment the count of boxes placed.
- If the box does not fit on the left, check the rightmost position. If it fits, move the right pointer to the left, and increment the count of boxes placed.
- Continue this process until all boxes are either placed or no suitable positions are left.
Key concepts utilized:
- Sorting
- Two-pointer technique
- Iteration
- Conditional logic
This method ensures that the largest boxes are considered first for placement, optimizing the space they can occupy either from the leftmost or rightmost sides of the warehouse, which maximizes the total number of boxes stored.
- Java
class Solution {
public int maximumBoxesInWarehouse(int[] boxes, int[] warehouse) {
int sizeOfWarehouse = warehouse.length;
// Sorting the boxes array
Arrays.sort(boxes);
int minIndex = 0;
int maxIndex = sizeOfWarehouse - 1;
int numOfBoxes = 0;
int indexOfBox = boxes.length - 1;
// Descending through the boxes array
while (minIndex <= maxIndex && indexOfBox >= 0) {
// Fitting the box in the leftmost space if possible
if (boxes[indexOfBox] <= warehouse[minIndex]) {
numOfBoxes++;
minIndex++;
// Fitting the box in the rightmost space if possible
} else if (boxes[indexOfBox] <= warehouse[maxIndex]) {
numOfBoxes++;
maxIndex--;
}
indexOfBox--;
}
return numOfBoxes;
}
}
This code defines a method in Java for determining the maximum number of boxes that can be placed inside a warehouse given certain constraints. The sizes of the boxes and the maximum heights available in parts of the warehouse are provided.
- Begin by sorting the
boxes
array to organize them by size. - Initialize pointers
minIndex
andmaxIndex
to track the smallest and largest available positions in thewarehouse
array, respectively. numOfBoxes
, which starts at zero, tracks the count of boxes successfully placed.- Iterate from the largest box in the sorted
boxes
array towards the smallest, checking each box against the available heights atminIndex
andmaxIndex
in thewarehouse
.- If the current box fits at the
minIndex
, incrementnumOfBoxes
and move theminIndex
pointer to the right. - If it doesn't fit at the
minIndex
but fits at themaxIndex
, incrementnumOfBoxes
and move themaxIndex
pointer to the left.
- If the current box fits at the
- The loop continues until either all boxes are considered or no fitting spaces are available in the warehouse.
- Lastly, return the total number of
numOfBoxes
that fits inside the warehouse.
This approach closely utilizes a two-pointer technique to effectively and efficiently match boxes to the available space in the warehouse from both ends.
- Python
class Solution:
def countBoxes(self, boxes: List[int], rooms: List[int]) -> int:
# Sorting the list of boxes
boxes.sort()
total_rooms = len(rooms)
min_idx = 0
max_idx = total_rooms - 1
entered_boxes = 0
current_box = len(boxes) - 1
# Assign the tallest boxes to either the smallest room they can fit into
while min_idx <= max_idx and current_box >= 0:
if boxes[current_box] <= rooms[min_idx]:
entered_boxes += 1
min_idx += 1
elif boxes[current_box] <= rooms[max_idx]:
entered_boxes += 1
max_idx -= 1
current_box -= 1
return entered_boxes
The Python solution for the problem of placing boxes into warehouses with entrance limits on height from both ends employs a two-pointer strategy. The solution class is defined with a method countBoxes
, which accepts a list of box heights and a list of available room or doorway heights. Follow these detailed steps for the provided approach:
Start by sorting the array of box heights. This arrangement aids in efficiently determining the fitability of the tallest boxes into the available spaces.
Initialize pointers for both ends of the warehouse room list (
min_idx
for the leftmost, andmax_idx
for the rightmost) and a pointer (current_box
) for the boxes starting from the last index.Use a while loop to traverse from the tallest box down to the smallest, checking against both the smallest and largest available room sizes from the warehouse list:
- If the current box fits into the room at the
min_idx
position, increment theentered_boxes
counter and shift themin_idx
pointer to the right, indicating that this room is now occupied. - If it doesn't fit there but fits in the
max_idx
room, increment theentered_boxes
counter as well and shift themax_idx
pointer to the left. - If the box doesn't fit in either, simply move to the next smaller box by decrementing
current_box
.
- If the current box fits into the room at the
The loop terminates when either all boxes have been tried (
current_box
becomes -1) or the warehouse room pointers overlap (min_idx
exceedsmax_idx
), meaning no more rooms are available.The function then returns the total count of
entered_boxes
, representing how many boxes were successfully placed into the warehouse.
This approach is efficient and ensures the maximum number of boxes are placed into the warehouse, given the constraints of varying room sizes available from both sides of the warehouse.
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