Binary TreesBinary Trees36
  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Inorder Traversal of a Binary Tree using Recursion
  4. 4Inorder Traversal of a Binary Tree using Iteration
  5. 5Postorder Traversal of a Binary Tree Using Recursion
  6. 6Postorder Traversal of a Binary Tree using Iteration
  7. 7Level Order Traversal of a Binary Tree using Recursion
  8. 8Level Order Traversal of a Binary Tree using Iteration
  9. 9Reverse Level Order Traversal of a Binary Tree using Iteration
  10. 10Reverse Level Order Traversal of a Binary Tree using Recursion
  11. 11Find Height of a Binary Tree
  12. 12Find Diameter of a Binary Tree
  13. 13Find Mirror of a Binary Tree
  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
GraphsGraphs46
  1. 1Breadth-First Search in Graphs
  2. 2Depth-First Search in Graphs
  3. 3Number of Provinces in an Undirected Graph
  4. 4Connected Components in a Matrix
  5. 5Rotten Oranges Problem - BFS in Matrix
  6. 6Flood Fill Algorithm - Graph Based
  7. 7Detect Cycle in an Undirected Graph using DFS
  8. 8Detect Cycle in an Undirected Graph using BFS
  9. 9Distance of Nearest Cell Having 1 - Grid BFS
  10. 10Surrounded Regions in Matrix using Graph Traversal
  11. 11Number of Enclaves in Grid
  12. 12Word Ladder - Shortest Transformation using Graph
  13. 13Word Ladder II - All Shortest Transformation Sequences
  14. 14Number of Distinct Islands using DFS
  15. 15Check if a Graph is Bipartite using DFS
  16. 16Topological Sort Using DFS
  17. 17Topological Sort using Kahn's Algorithm
  18. 18Cycle Detection in Directed Graph using BFS
  19. 19Course Schedule - Task Ordering with Prerequisites
  20. 20Course Schedule 2 - Task Ordering Using Topological Sort
  21. 21Find Eventual Safe States in a Directed Graph
  22. 22Alien Dictionary Character Order
  23. 23Shortest Path in Undirected Graph with Unit Distance
  24. 24Shortest Path in DAG using Topological Sort
  25. 25Dijkstra's Algorithm Using Set - Shortest Path in Graph
  26. 26Dijkstra’s Algorithm Using Priority Queue
  27. 27Shortest Distance in a Binary Maze using BFS
  28. 28Path With Minimum Effort in Grid using Graphs
  29. 29Cheapest Flights Within K Stops - Graph Problem
  30. 30Number of Ways to Reach Destination in Shortest Time - Graph Problem
  31. 31Minimum Multiplications to Reach End - Graph BFS
  32. 32Bellman-Ford Algorithm for Shortest Paths
  33. 33Floyd Warshall Algorithm for All-Pairs Shortest Path
  34. 34Find the City With the Fewest Reachable Neighbours
  35. 35Minimum Spanning Tree in Graphs
  36. 36Prim's Algorithm for Minimum Spanning Tree
  37. 37Disjoint Set (Union-Find) with Union by Rank and Path Compression
  38. 38Kruskal's Algorithm - Minimum Spanning Tree
  39. 39Minimum Operations to Make Network Connected
  40. 40Most Stones Removed with Same Row or Column
  41. 41Accounts Merge Problem using Disjoint Set Union
  42. 42Number of Islands II - Online Queries using DSU
  43. 43Making a Large Island Using DSU
  44. 44Bridges in Graph using Tarjan's Algorithm
  45. 45Articulation Points in Graphs
  46. 46Strongly Connected Components using Kosaraju's Algorithm

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

Weights Days (D) Minimum Capacity Description
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10] 5 15 One possible allocation: [1,2,3,4,5], [6], [7], [8], [9,10]
[3, 2, 2, 4, 1, 4] 3 6 Minimum capacity needed to fit the packages in 3 days
[10, 50, 50, 10] 2 100 First day: [10, 50]; second day: [50, 10]
[5, 5, 5, 5] 4 5 Each package on a separate day, minimum possible capacity is 5
[5, 5, 5, 5] 1 20 All packages on the same day, total weight = 20
[7, 2, 5, 10, 8] 2 18 Day 1: [7,2,5], Day 2: [10,8]
[1] 1 1 Only one package, one day
[] 1 0 No packages to ship, capacity needed is 0
[4, 8, 5] 5 8 More days than packages, so capacity can be equal to the largest weight

Visualization Player

Solution

Understanding the Problem

We are given an array of package weights and a number D representing the number of days within which all packages must be shipped. The task is to find the minimum ship capacity that allows us to ship all the packages in order, within D days. Each day, the ship can carry a contiguous group of packages without exceeding its maximum weight capacity. Packages cannot be split and must be shipped in the given order.

Step-by-Step Solution with Example

Step 1: Understand the Role of Capacity

Capacity means the maximum total weight the ship can carry in a single day. For instance, if the capacity is 10, then the ship can carry packages like [3,7], but not [6,5] since 6+5 = 11 exceeds the limit.

Step 2: Binary Search Strategy

We aim to find the smallest valid capacity. To do this efficiently, we use binary search between two limits:

  • Lower bound: the maximum weight of any single package (since the ship must at least carry that).
  • Upper bound: the total weight of all packages (meaning all shipped in one day).

Step 3: Check If a Given Capacity Works

For any candidate capacity in our binary search, we simulate the shipping process:

  • Start on day 1 with current day load = 0.
  • Iterate through the packages:
    • Add the current package to the current day's load.
    • If this exceeds the capacity, increment the day count and start a new day with the current package.

If the number of days used exceeds D, the capacity is too small. Otherwise, it's a valid capacity and we try smaller values to minimize it.

Step 4: Example

Input:
weights = [1,2,3,4,5,6,7,8,9,10], D = 5

Step-by-step:
- The max weight is 10 → minimum possible capacity.
- Total weight is 55 → maximum possible capacity.
- We perform binary search between 10 and 55.

First mid = 32:
- Simulated days needed: 3 (valid, try smaller capacity)
Next mid = 21:
- Simulated days needed: 4 (valid, try smaller)
Next mid = 15:
- Simulated days needed: 5 (valid, try smaller)
Next mid = 12:
- Simulated days needed: 6 (too many, try larger)
Next mid = 13:
- Simulated days needed: 5 (valid, try smaller)
...
Eventually, we find that the minimum capacity that allows shipping in 5 days is 15.

Edge Cases

  • Only one package: If there's one package and one day, the capacity must be equal to the package weight.
  • More days than packages: We can ship one package per day, so capacity only needs to handle the largest package.
  • Empty package list: No packages to ship. Return 0 or handle as a special case depending on constraints.
  • Very large weights or days: Use 64-bit integers to avoid overflow when summing weights or calculating mid values in binary search.

Finally

This problem is a classic example of using binary search on the answer space. Instead of iterating through all possible capacities, we strategically test midpoints to narrow the range. By simulating the shipping process for each tested capacity, we ensure the chosen value truly satisfies the D-day constraint. The logic can be extended to similar scheduling and load-balancing problems as well.

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.
Worst 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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