1011. Capacity To Ship Packages Within D Days - Detailed Explanation

Problem Statement

You are given an array of positive integers, weights, where each element represents the weight of a package, and an integer D representing the number of days within which you must ship all packages. The packages must be shipped in the given order. Each day, you load as many consecutive packages as possible without exceeding a given ship capacity. The goal is to determine the minimum ship capacity required so that all packages can be shipped within D days.

Example Inputs & Outputs

• Example 1:

  • Input: weights = [1,2,3,4,5,6,7,8,9,10], D = 5
  • Output: 15
  • Explanation:
    With a capacity of 15, you can divide the shipments as follows:
    • Day 1: 1 + 2 + 3 + 4 = 10
    • Day 2: 5 + 6 = 11
    • Day 3: 7
    • Day 4: 8
    • Day 5: 9 + 10 = 19 (Since 9 alone is less than 15, but 9+10=19 exceeds it, adjust grouping appropriately.)
      An appropriate grouping shows that 15 is the minimal capacity needed to satisfy the constraints.

• Example 2:

  • Input: weights = [3,2,2,4,1,4], D = 3
  • Output: 6
  • Explanation:
    One way to group:
    • Day 1: 3, 2 → total = 5
    • Day 2: 2, 4 → total = 6
    • Day 3: 1, 4 → total = 5
      Capacity 6 is enough and is the minimum possible.

• Example 3:

  • Input: weights = [1,2,3,1,1], D = 4
  • Output: 3
  • Explanation:
    Ship the packages in groups:
    • Day 1: 1,2 → total = 3
    • Day 2: 3 → total = 3
    • Day 3: 1 → total = 1
    • Day 4: 1 → total = 1
      The minimum capacity required here is 3.

Constraints

• (1 \leq D \leq \text{weights.length} \leq 50000)
• (1 \leq \text{weights}[i] \leq 500)
• All packages must be shipped in order.

Hints to Get Started

1. Determine the Capacity Range:
   The minimum possible capacity is the weight of the heaviest package (because you cannot split a package), and the maximum possible capacity is the sum of all package weights (which would ship everything in one day).

2. Simulation to Validate a Capacity:
   Write a helper function that, given a candidate capacity, simulates the shipping process by iterating through the weights and counting the number of days required. If you can ship all packages within D days with that capacity, then the candidate is valid.

3. Binary Search Over the Capacity Range:
   Instead of testing every capacity between the minimum and maximum, use binary search to efficiently find the minimum capacity that satisfies the condition.

Approach 1: Brute Force (Simulation for Every Capacity)

Explanation

• Idea:
  Start with the capacity equal to the maximum weight and incrementally test each capacity until you find one that allows you to ship all packages in D days.

• How It Works:
  For each candidate capacity, simulate the shipping process:

  • Iterate over the packages and add weights until the sum exceeds the candidate capacity.
  • Count the days required to ship the packages.
  • If the number of days exceeds D, try a higher capacity.

• Downside:
  This approach is straightforward but inefficient because the capacity range (from max(weight) to sum(weights)) can be large.

Approach 2: Optimal Solution Using Binary Search

Step-by-Step Walkthrough

1. Set the Search Boundaries:

   • Lower Bound: low = max(weights)
     (Any capacity less than the heaviest package will fail.)
   • Upper Bound: high = sum(weights)
     (This is the capacity required to ship all packages in one day.)

2. Binary Search Process:

   • Calculate the mid-point: mid = (low + high) // 2.
   • Simulate Shipping:
     Use a helper function to check if all packages can be shipped within D days with the candidate capacity mid.
     • Initialize days = 1 and current_load = 0.
     • For each package:
       • If adding the package exceeds the capacity, increment the day counter and reset current_load to the package weight.
       • Otherwise, add the package weight to current_load.
   • Update the Search Boundaries:
     • If the candidate capacity is valid (i.e., you can ship within D days), update high = mid (search for a smaller valid capacity).
     • Otherwise, update low = mid + 1 (increase the capacity).

3. Terminate:
   When low meets high, it is the minimum capacity required.

Visual Example (Using Example 1)

Consider weights = [1,2,3,4,5,6,7,8,9,10] and D = 5.

• Lower Bound: low = 10 (max weight)
• Upper Bound: high = 55 (sum of weights)

Iteration 1:

• mid = (10 + 55) // 2 = 32
• Simulate shipping with capacity 32:
  • Load packages until exceeding 32; count days.
  • Result: You might ship in fewer than 5 days.
• Since capacity 32 works, update high = 32.

Iteration 2:

• mid = (10 + 32) // 2 = 21
• Simulate with capacity 21.
• Check the number of days needed.
• Adjust low or high based on simulation.

Repeat until the optimal capacity is found, which is 15 for this example.

Code Examples

Python Code