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Maximum Balanced Shipments
The Maximum Balanced Shipments problem requires forming the maximum number of balanced shipments from a given set of parcels.
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Medium · Maximum Balanced Shipments core interview pattern
Answer-first summary
The Maximum Balanced Shipments problem requires forming the maximum number of balanced shipments from a given set of parcels.
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The problem involves dividing an array of parcel weights into as many non-overlapping shipments as possible, where each shipment is balanced. A shipment is balanced if the last parcel has a weight strictly less than the maximum weight within that shipment. The approach generally involves identifying valid subarrays using monotonic stacks for efficiency.
Problem Statement
You are given an integer array weight of length n, representing the weights of n parcels arranged in a straight line. A shipment is defined as a contiguous subarray of parcels. A shipment is considered balanced if the weight of the last parcel is strictly less than the maximum weight among all parcels in that shipment.
Select a set of non-overlapping, contiguous, balanced shipments such that each parcel appears in at most one shipment (parcels may remain unshipped). Return the maximum possible number of balanced shipments that can be formed.
Examples
Example 1
Input: weight = [2,5,1,4,3]
Output: 2
We can form the maximum of two balanced shipments as follows: It is impossible to partition the parcels to achieve more than two balanced shipments, so the answer is 2.
Example 2
Input: weight = [4,4]
Output: 0
No balanced shipment can be formed in this case: As there is no way to form even one balanced shipment, the answer is 0.
Constraints
- 2 <= n <= 105
- 1 <= weight[i] <= 109
Solution Approach
Monotonic Stack Approach
To identify the nearest previous index with a weight greater than the current parcel, a monotonic stack can be used. This allows you to efficiently find where a balanced shipment can end and helps in maximizing the number of non-overlapping shipments.
Greedy Selection of Shipments
By iterating through the parcel list and greedily selecting balanced shipments, you can determine the optimal partition of parcels that results in the maximum number of valid shipments. This approach prioritizes maintaining balance in the shipment structure.
Efficient Search for Subarrays
Using a monotonic stack, each parcel can be checked for possible starting points, ensuring that the algorithm runs efficiently even for large inputs, avoiding brute-force checks of all subarrays.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity of the solution depends on the final approach, but using a monotonic stack generally provides an O(n) solution, where n is the number of parcels. Space complexity is also O(n) due to the stack used for tracking indices.
What Interviewers Usually Probe
- The candidate uses a monotonic stack to efficiently find valid subarrays.
- The candidate demonstrates an understanding of greedy algorithms for partitioning.
- The candidate ensures no overlapping parcels in the selected shipments.
Common Pitfalls or Variants
Common pitfalls
- Failing to account for parcels that are unshipped.
- Not ensuring the shipment balance condition holds (last parcel being less than max weight).
- Using an inefficient brute force approach, leading to time limit issues.
Follow-up variants
- Variation 1: Handle edge cases like when there are only two parcels or when all parcels have the same weight.
- Variation 2: Modify the problem to allow a partial shipment, i.e., shipments that do not need to be contiguous.
- Variation 3: Change the weight constraint, making parcels weigh much more or less, affecting the balance condition.
FAQ
What is the pattern used to solve the Maximum Balanced Shipments problem?
The problem can be solved using a monotonic stack to identify valid subarrays and a greedy approach for optimal shipment partitioning.
How can I efficiently find valid shipments in the Maximum Balanced Shipments problem?
Using a monotonic stack to track indices allows you to efficiently find the nearest valid shipment end, helping you partition the array.
What is the time complexity of solving Maximum Balanced Shipments?
The time complexity is O(n), where n is the number of parcels, using a monotonic stack approach.
Can the Maximum Balanced Shipments problem be solved without a monotonic stack?
While a monotonic stack provides an efficient solution, the problem can also be solved with a brute-force approach, though it would not scale well with large inputs.
What are some edge cases to consider for Maximum Balanced Shipments?
Edge cases include arrays where all parcels are the same weight or arrays with only two parcels.
Solution
Solution 1: Greedy
We maintain the maximum value $\text{mx}$ of the currently traversed array, and iterate through each element $x$ in the array. If $x < \text{mx}$, it means the current element can serve as the last parcel of a balanced shipment, so we increment the answer by one and reset $\text{mx}$ to 0. Otherwise, we update $\text{mx}$ to the value of the current element $x$.
class Solution:
def maxBalancedShipments(self, weight: List[int]) -> int:
ans = mx = 0
for x in weight:
mx = max(mx, x)
if x < mx:
ans += 1
mx = 0
return ans