Binary TreesBinary Trees36

  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Postorder Traversal of a Binary Tree Using Recursion
  4. 4Postorder Traversal of a Binary Tree using Iteration
  5. 5Level Order Traversal of a Binary Tree using Recursion
  6. 6Level Order Traversal of a Binary Tree using Iteration
  7. 7Reverse Level Order Traversal of a Binary Tree using Iteration
  8. 8Reverse Level Order Traversal of a Binary Tree using Recursion
  9. 9Find Height of a Binary Tree
  10. 10Find Diameter of a Binary Tree
  11. 11Find Mirror of a Binary Tree - Todo
  12. 12Inorder Traversal of a Binary Tree using Recursion
  13. 13Inorder Traversal of a Binary Tree using Iteration
  14. 14Left View of a Binary Tree
  15. 15Right View of a Binary Tree
  16. 16Top View of a Binary Tree
  17. 17Bottom View of a Binary Tree
  18. 18Zigzag Traversal of a Binary Tree
  19. 19Check if a Binary Tree is Balanced
  20. 20Diagonal Traversal of a Binary Tree
  21. 21Boundary Traversal of a Binary Tree
  22. 22Construct a Binary Tree from a String with Bracket Representation
  23. 23Convert a Binary Tree into a Doubly Linked List
  24. 24Convert a Binary Tree into a Sum Tree
  25. 25Find Minimum Swaps Required to Convert a Binary Tree into a BST
  26. 26Check if a Binary Tree is a Sum Tree
  27. 27Check if All Leaf Nodes are at the Same Level in a Binary Tree
  28. 28Lowest Common Ancestor (LCA) in a Binary Tree
  29. 29Solve the Tree Isomorphism Problem
  30. 30Check if a Binary Tree Contains Duplicate Subtrees of Size 2 or More
  31. 31Check if Two Binary Trees are Mirror Images
  32. 32Calculate the Sum of Nodes on the Longest Path from Root to Leaf in a Binary Tree
  33. 33Print All Paths in a Binary Tree with a Given Sum
  34. 34Find the Distance Between Two Nodes in a Binary Tree
  35. 35Find the kth Ancestor of a Node in a Binary Tree
  36. 36Find All Duplicate Subtrees in a Binary Tree

Find Minimum Ship Capacity to Ship Packages in D Days
Binary Search Approach



Problem Statement

You are given an array of package weights and a number D representing the number of days within which all the packages must be delivered. Each day, you can ship packages in the given order, but the total weight of the packages shipped on a day cannot exceed a certain ship capacity.

Your task is to determine the minimum weight capacity of the ship so that all packages can be shipped within D days.

Note: Packages must be shipped in order and cannot be split.

Examples

WeightsDays (D)Minimum CapacityDescription
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]515One possible allocation: [1,2,3,4,5], [6], [7], [8], [9,10]
[3, 2, 2, 4, 1, 4]36Minimum capacity needed to fit the packages in 3 days
[10, 50, 50, 10]2100First day: [10, 50]; second day: [50, 10]
[5, 5, 5, 5]45Each package on a separate day, minimum possible capacity is 5
[5, 5, 5, 5]120All packages on the same day, total weight = 20
[7, 2, 5, 10, 8]218Day 1: [7,2,5], Day 2: [10,8]
[1]11Only one package, one day
[]10No packages to ship, capacity needed is 0
[4, 8, 5]58More days than packages, so capacity can be equal to the largest weight

Solution

To solve this problem, we need to find the minimum capacity of a ship such that all packages can be shipped within D days. Since the packages must be shipped in order and cannot be split, the key is to determine how to divide them across D days while respecting the ship's capacity.

What does 'capacity' mean? It’s the maximum weight the ship can carry in a single day. If a ship has capacity 10, it can carry packages like [4,6] or [5,5], but not [4,7] since 4+7 = 11 exceeds the limit.

We want the smallest such capacity that still allows us to ship all packages in exactly D or fewer days. To find this efficiently, we can use binary search.

Understanding the Search Range

  • The lowest possible capacity is the weight of the heaviest package. For example, if the largest package weighs 10, the ship must at least carry 10.
  • The highest possible capacity is the total weight of all packages, meaning we ship everything in one day.

How to Check If a Capacity Works

For any capacity we try, we simulate how many days it would take to ship all packages using that capacity:

  • Start with day 1 and a sum = 0
  • For each package:
    • Add its weight to the sum for the current day
    • If the sum exceeds the capacity, increment the day count and reset the sum to that package’s weight

If the total number of days required exceeds D, it means the capacity is too small. Otherwise, it’s a valid capacity and we try a smaller one to see if we can do better.

What Happens in Special Cases?

  • Single package: If there's only one package and one day, capacity equals its weight.
  • More days than packages: We can ship one package per day, so the capacity only needs to match the largest package weight.
  • Empty array: If there are no packages, no capacity is needed. Return 0.

By using binary search between the lowest and highest possible capacities, and checking how many days each candidate capacity would require, we can efficiently find the minimum ship capacity needed to meet the D day requirement.

Visualization

Algorithm Steps

  1. Set the search range: low = max(weights) and high = sum(weights).
  2. While low ≤ high:
  3. → Compute mid = (low + high) / 2.
  4. → Check if it’s possible to ship all packages within D days using mid as the capacity.
  5. → If possible, store mid as answer and try to minimize (high = mid - 1).
  6. → Otherwise, increase capacity (low = mid + 1).
  7. Return the minimum valid capacity found.

Code

Python
Java
JavaScript
C
C++
C#
Kotlin
Swift
Php
def shipWithinDays(weights, D):
    def canShip(capacity):
        days = 1
        total = 0
        for weight in weights:
            if total + weight > capacity:
                days += 1  # Need an extra day
                total = 0  # Start new day
            total += weight
        return days <= D

    low = max(weights)  # Minimum capacity must be at least the max weight
    high = sum(weights) # Maximum capacity is sum of all weights (one day)
    result = high

    while low <= high:
        mid = (low + high) // 2
        if canShip(mid):
            result = mid
            high = mid - 1
        else:
            low = mid + 1

    return result

weights = [1,2,3,4,5,6,7,8,9,10]
D = 5
print("Minimum capacity:", shipWithinDays(weights, D))

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(n)When the optimal capacity is found early in the binary search range and only a single pass over weights is needed to validate.
Average CaseO(n * log(sum - max))Binary search runs over the capacity range and for each capacity, we do a full scan of weights.
Average CaseO(n * log(sum - max))In the worst case, binary search tries nearly all capacity values between max(weights) and sum(weights).

Space Complexity

O(1)

Explanation: No extra space is used except a few variables for computation.



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