1011. Capacity To Ship Packages Within D Days

Problem

A conveyor belt has packages that must be shipped from one port to another within days days.

The ith package on the conveyor belt has a weight of weights[i]. Each day, we load the ship with packages on the conveyor belt (in the order given by weights). We may not load more weight than the maximum weight capacity of the ship.

Return the least weight capacity of the ship that will result in all the packages on the conveyor belt being shipped within days days. (link)

Example 1:

Input: weights = [1,2,3,4,5,6,7,8,9,10], days = 5
Output: 15
Explanation: A ship capacity of 15 is the minimum to ship all the packages in 5 days like this:
1st day: 1, 2, 3, 4, 5
2nd day: 6, 7
3rd day: 8
4th day: 9
5th day: 10

Note that the cargo must be shipped in the order given, 
so using a ship of capacity 14 and splitting the packages into parts like 
(2, 3, 4, 5), (1, 6, 7), (8), (9), (10) is not allowed.

Example 2:

Input: weights = [3,2,2,4,1,4], days = 3
Output: 6
Explanation: A ship capacity of 6 is the minimum to ship all the packages in 3 days like this:
1st day: 3, 2
2nd day: 2, 4
3rd day: 1, 4

Example 3:

Input: weights = [1,2,3,1,1], days = 4
Output: 3
Explanation:
1st day: 1
2nd day: 2
3rd day: 3
4th day: 1, 1

Constraints:

• 1 <= days <= weights.length <= 5 * 10^4

• 1 <= weights[i] <= 500

Solution

Brute Force Approach

We initiate the process by determining the number of days required for shipping based on a given capacity. Once we ascertain the number of days, we set the ship's capacity to the minimum possible value and gradually increase it to its maximum potential. Starting from the minimum, we calculate the number of days, halting at the capacity where the calculated days equal or fall below the provided threshold, then return this value.

The ship's minimum capacity corresponds to the maximum weight among the weights array. If the capacity is lower than this maximum weight, it becomes impractical to transport all the weights. On the other hand, the ship's maximum capacity aligns with the total weight of all items. This ensures that if the shipping duration is just one day, the ship can accommodate all the weights.

To compute the number of days required to ship weights for a given capacity, we continually add weights to the ship until it reaches or exceeds the designated capacity. If it surpasses the capacity, we schedule the remaining weights for the following day of shipping.

class Solution {
    public int calculateDays(int[] weights, int capacity){
        int load = 0;
        int days = 1;
        for(int weight: weights){
            if(load+weight<=capacity){
                load+=weight;
            }
            else{
                load = weight;
                days+=1;
            }
        }
        return days;
    }
    public int shipWithinDays(int[] weights, int days) {
        int minCapacity = Arrays.stream(weights).max().getAsInt();
        int maxCapacity = Arrays.stream(weights).sum();

        for(int capacity=minCapacity; capacity<=maxCapacity; capacity++){
            if(calculateDays(weights, capacity)<=days){
                return capacity;
            }
        }
        return -1;
    }
}

Optimal Approach

We implement an optimization technique by incorporating binary search into the brute force approach. In this strategy, we apply binary search to the potential answers. Thus, we set the lower bound (low) to the minimum ship capacity and the upper bound (high) to the maximum capacity. As we increase the ship's capacity, we ensure that all the weights can be transported within the specified timeframe. Consequently, beyond a certain capacity threshold, every subsequent value satisfies the condition of being able to ship all the weights.

X X X X Y Y Y Y Y Y  Y
1 2 3 4 5 6 7 8 9 10 11

// From 5, each number satisfies condition (total days <= days)

class Solution {
    public int calculateDays(int[] weights, int capacity){
        int load = 0;
        int days = 1;
        for(int weight: weights){
            if(load+weight<=capacity){
                load+=weight;
            }
            else{
                load = weight;
                days+=1;
            }
        }
        return days;
    }
    public int shipWithinDays(int[] weights, int days) {
        int minCapacity = Arrays.stream(weights).max().getAsInt();
        int maxCapacity = Arrays.stream(weights).sum();

        int low = minCapacity;
        int high = maxCapacity;

        while(low<=high){
            int mid = (low+high)/2;
            if(calculateDays(weights, mid) > days){
                low = mid + 1;
            }
            else{
                high = mid - 1;
            }
        }

        return low;
    }
}