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.
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