Introduction

Capacity to Ship Packages means:

  • finding minimum ship capacity
    to transport packages
    within given days

Rule:

Packages must be shipped in the given order.

Goal:

  • minimize ship capacity
    while delivering all packages
    within days

Example:

Weights:
[1,2,3,4,5,6,7,8,9,10]

Days:
5

Output:
15

Explanation:

Capacity 15 allows all packages to ship within 5 days.

This problem is one of the most important applications of:

Binary Search on Answer

Constraints

1 <= weights.length <= 10^5

Approach : Binary Search on Answer

Explanations:

Explanation:

The idea is:

  • search possible ship capacity
    instead of searching indices
  • validate each capacity efficiently

Search Space:

Minimum capacity = max(weights)
Maximum capacity = sum(weights)

Steps:

  1. Pick middle capacity.
  2. Simulate shipping process.
  3. Count required days.
  4. Check if valid.
  5. Reduce answer range.
  6. Continue binary search.

Capacity Validation:

currentWeight + package <= capacity

Conditions:

requiredDays <= days → valid capacity 

requiredDays > days → increase capacity

This approach:

  • optimizes answer searching
  • avoids brute force simulation

Dry Run

Weights:
[1,2,3,4,5,6,7,8,9,10]

Middle capacity:
21

Required days:
3

Valid capacity found.
Try smaller capacity.

Middle capacity:
15

Required days:
5

Minimum valid capacity:
15

Practice :

Complexity Analysis :

Time Complexity:- O(n log m)Explanation :
Binary search runs on answer space while checking all packages.

Space Complexity:- O(1)
Explanation :
Only constant variables are used.

Why This Problem is Important

This problem builds the foundation for:

  • Binary search on answer
  • Search optimization
  • Capacity planning
  • Constraint-based searching
  • Efficient simulation problems

Real-World Applications

Capacity optimization concepts are used in:

  • Shipping systems
  • Logistics platforms
  • Delivery optimization
  • Warehouse planning
  • Resource allocation systems

Common Beginner Mistakes

  • Incorrect search range
  • Wrong day calculation
  • Forgetting package order rule
  • Infinite loop conditions
  • Incorrect boundary updates

Interview Tip

Interviewers often expect:

  • binary search understanding
  • answer-space searching explanation
  • capacity validation discussion
  • monotonic property clarity

Always explain:

  • why answer space is searchable
  • simulation validation logic
  • monotonic capacity condition

Related Questions

  • Koko Eating Bananas
  • Split Array Largest Sum
  • Binary Search
  • Minimum Speed to Arrive on Time
  • Search in Rotated Sorted Array

Final Takeaway

The Capacity to Ship Packages problem is one of the most important intermediate binary search problems.

It teaches:

  • binary search on answer
  • capacity optimization
  • constraint simulation
  • efficient range searching

Understanding this problem builds a strong foundation for:

  •  advanced searching problems
  • optimization techniques
  • interview-level algorithms.