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Minimum Initial Energy to Finish Tasks

Difficulty: Hard

Problem Description

You are given an array tasks where tasks[i] = [actual_i, minimum_i]: actual_i is the actual amount of energy you spend to finish the i-th task, and minimum_i is the minimum amount of energy you require to begin the i-th task. The goal is to determine the minimum initial amount of energy you will need to finish all the tasks, given that you can complete the tasks in any order.

Key Insights

  • You need to ensure that your current energy is at least the minimum required to start each task.
  • The order of tasks matters because completing tasks that require more energy first may help conserve energy for subsequent tasks.
  • Sorting the tasks based on their minimum requirement can help establish a strategy for task completion.
  • The total energy spent and the required energy must be balanced to determine the initial energy required.

Space and Time Complexity

Time Complexity: O(n log n) - due to the sorting of tasks. Space Complexity: O(1) - if we only use variables for calculations.

Solution

To solve this problem, we can follow these steps:

  1. Sort the tasks by their minimum energy requirement in ascending order.
  2. Iterate through the sorted list and keep track of the total energy spent and the current energy level.
  3. If at any point our current energy is less than the minimum required for the next task, we calculate how much more initial energy we need to start that task.
  4. The answer will be the maximum of the calculated initial energy needed throughout the iterations.

This approach ensures that we always have enough energy to start and complete each task.

Code Solutions

def minimum_initial_energy(tasks):
    # Sort tasks based on the minimum energy required
    tasks.sort(key=lambda x: x[1])
    initial_energy_needed = 0
    current_energy = 0

    for actual, minimum in tasks:
        # If current energy is less than the minimum required, we need more initial energy
        if current_energy < minimum:
            initial_energy_needed += minimum - current_energy
            current_energy = minimum
        # Deduct the actual energy spent for the task
        current_energy -= actual

    return initial_energy_needed

# Example usage
print(minimum_initial_energy([[1,2],[2,4],[4,8]]))  # Output: 8
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