Drone Delivery Optimization Problem

Problem Description

A delivery drone starts at a depot with a fixed battery capacity and must make a sequence of deliveries. The goal is to determine the maximum number of deliveries the drone can complete in a single trip without recharging.

Problem Details

• The drone starts at a depot with battery capacity B (integer)
• Each delivery involves:
  • Flying from depot to customer location at distance d
  • Returning to the depot
  • Total battery consumption: 2*d units
• After each delivery, the drone can start the next delivery if it has sufficient battery
• The drone stops when its battery is insufficient for any additional round trip

Input Format

• Line 1: Integer B (0 ≤ B ≤ 10^9) - total available battery units
• Line 2: N space-separated positive integers (0 ≤ N ≤ 10^5, each distance ≤ 10^6) - distances to delivery locations

Output Format

• A single integer: the maximum number of deliveries the drone can make without exceeding its battery capacity

Solution Approach

Algorithm: Greedy Strategy with Sorting

1. Sort distances in ascending order to prioritize closest locations first
2. Iterate through sorted distances and calculate round-trip cost (2 × distance)
3. Check battery constraint for each delivery:
   • If sufficient battery: make delivery and update battery usage
   • If insufficient battery: stop (remaining deliveries will cost even more)
4. Return the count of completed deliveries

Why This Works

The greedy approach is optimal because:

• To maximize the number of deliveries, we should minimize battery usage per delivery
• Choosing the closest available location first always uses the least battery
• Once we can't afford a delivery, we can't afford any remaining deliveries (since they're farther)

Complexity Analysis

• Time Complexity: O(N log N) due to sorting
• Space Complexity: O(1) additional space (in-place sorting)

Edge Cases Handled

• Empty array of distances → return 0
• Zero battery capacity → return 0
• Battery sufficient for all deliveries → return total count
• Large numbers within constraint limits

Example Walkthrough

Input:
B = 10
distances = [3, 1, 4, 2]

Step 1: Sort distances → [1, 2, 3, 4]
Step 2: Calculate round-trip costs → [2, 4, 6, 8]
Step 3: Check deliveries:
  - Delivery 1 (distance 1): cost 2, total used = 2 ≤ 10 ✓
  - Delivery 2 (distance 2): cost 4, total used = 6 ≤ 10 ✓
  - Delivery 3 (distance 3): cost 6, total used = 12 > 10 ✗

Output: 2 deliveries

Performance Expectations

• Handles up to 10^5 delivery locations efficiently
• Works with battery capacities up to 10^9
• Optimized for competitive programming constraints
• Memory efficient with minimal additional space usage

Usage

The solution can be used in two ways:

1. Function call: max_deliveries(B, distances) - returns the maximum number of deliveries
2. Input parsing: solve_from_input() - reads from standard input in the specified format

Perfect for competitive programming problems involving optimization with resource constraints.