Problem Description

You have an array of tasks where each task is represented as [actual, minimum]:

• actual is the energy consumed when completing the task
• minimum is the minimum energy level required to start the task

To complete a task, your current energy must be at least equal to the minimum value. After completing the task, your energy decreases by the actual amount.

For example, with task [10, 12]:

• If your current energy is 11, you cannot start the task (11 < 12)
• If your current energy is 13, you can start the task, and after completion you'll have 13 - 10 = 3 energy remaining

You can complete the tasks in any order you choose. The goal is to find the minimum initial energy needed to complete all tasks.

The solution uses a greedy approach with custom sorting. The key insight is that to minimize the initial energy, we should complete tasks in a specific order based on the difference actual - minimum. By sorting tasks in ascending order of actual[i] - minimum[i], we ensure that tasks requiring more "buffer" energy (where minimum is much larger than actual) are done later when we have more flexibility.

The algorithm tracks the current energy level and whenever it falls below a task's minimum requirement, it adds just enough energy to meet that requirement. This ensures we use the absolute minimum initial energy needed to complete all tasks in the optimal order.

Intuition

Let's think about what happens when we complete tasks in different orders. The key observation is that the total energy consumed (sum of all actual values) is fixed regardless of order, but the initial energy requirement changes based on the sequence.

Consider two adjacent tasks in our sequence. If we have tasks A and B, should we do A first or B first? Let's analyze what initial energy we need for each ordering:

If we do A then B:

• We need at least minimum_A energy to start
• After A, we have energy reduced by actual_A
• To do B next, we need the remaining energy to be at least minimum_B
• So initial energy must be at least max(minimum_A, actual_A + minimum_B)

If we do B then A:

• Initial energy must be at least max(minimum_B, actual_B + minimum_A)

To minimize initial energy, we want to choose the ordering that gives us the smaller requirement. Through mathematical analysis, we find that we should do A before B when actual_A - minimum_A < actual_B - minimum_B.

This leads to the insight: tasks with smaller actual - minimum difference should be done first. Why? Because these tasks give us less "energy buffer" after completion. Tasks where minimum is much larger than actual are like "energy gates" - they require a lot to start but don't consume much. It's better to save these for last when we've already accumulated enough energy.

By sorting tasks by actual - minimum in ascending order, we ensure we're always making the optimal local choice, which leads to the global minimum initial energy. As we traverse the sorted tasks, we only add energy when absolutely necessary (when current energy falls below a task's minimum), ensuring we use the least possible initial energy.

Learn more about Greedy and Sorting patterns.

Solution Approach

The implementation follows the greedy strategy with custom sorting that we identified from our intuition.

Step 1: Sort the tasks

We sort the tasks based on actual - minimum in ascending order using a lambda function:

sorted(tasks, key=lambda x: x[0] - x[1])

This ensures tasks with smaller difference (less energy buffer) are processed first.

Step 2: Initialize variables

• ans: tracks the minimum initial energy required (this is what we'll return)
• cur: tracks the current energy level as we process tasks

Step 3: Process tasks in sorted order

For each task [a, m] in the sorted list:

1. Check if we have enough energy to start: If cur < m, we don't have enough energy to begin this task. We need to add energy to our initial amount:

   if cur < m:
       ans += m - cur  # Add just enough to meet the minimum
       cur = m         # Update current energy to the minimum required

2. Complete the task: After ensuring we have enough energy to start, we complete the task and reduce our current energy by the actual amount consumed:

   cur -= a

Why this works:

As proven in the reference approach, for any sequence of tasks, the minimum initial energy needed is: $E \geq \sum_{i=1}^{n} a_i + (m_n - a_n)$

Where the last task determines the additional buffer needed beyond the total energy consumed. By sorting tasks to minimize m_n - a_n (equivalently, maximize a_n - m_n), we minimize the required initial energy.

The algorithm cleverly tracks this by only adding energy when absolutely necessary, ensuring the final ans represents the minimum initial energy needed to complete all tasks in the optimal order.

Example Walkthrough

Let's walk through a concrete example with tasks: [[1, 3], [2, 4], [10, 11], [10, 12], [8, 9]]

Step 1: Sort by (actual - minimum)

Calculate the differences:

• Task [1, 3]: difference = 1 - 3 = -2
• Task [2, 4]: difference = 2 - 4 = -2
• Task [10, 11]: difference = 10 - 11 = -1
• Task [10, 12]: difference = 10 - 12 = -2
• Task [8, 9]: difference = 8 - 9 = -1

After sorting in ascending order: [[1, 3], [2, 4], [10, 12], [10, 11], [8, 9]]

Step 2: Initialize tracking variables

• ans = 0 (minimum initial energy needed)
• cur = 0 (current energy level)

Step 3: Process each task in sorted order

Task 1: [1, 3]

• Current energy: 0, Required minimum: 3
• Since 0 < 3, we need more energy
• Add to initial: ans = 0 + (3 - 0) = 3
• Update current: cur = 3
• Complete task: cur = 3 - 1 = 2

Task 2: [2, 4]

• Current energy: 2, Required minimum: 4
• Since 2 < 4, we need more energy
• Add to initial: ans = 3 + (4 - 2) = 5
• Update current: cur = 4
• Complete task: cur = 4 - 2 = 2

Task 3: [10, 12]

• Current energy: 2, Required minimum: 12
• Since 2 < 12, we need more energy
• Add to initial: ans = 5 + (12 - 2) = 15
• Update current: cur = 12
• Complete task: cur = 12 - 10 = 2

Task 4: [10, 11]

• Current energy: 2, Required minimum: 11
• Since 2 < 11, we need more energy
• Add to initial: ans = 15 + (11 - 2) = 24
• Update current: cur = 11
• Complete task: cur = 11 - 10 = 1

Task 5: [8, 9]

• Current energy: 1, Required minimum: 9
• Since 1 < 9, we need more energy
• Add to initial: ans = 24 + (9 - 1) = 32
• Update current: cur = 9
• Complete task: cur = 9 - 8 = 1

Result: The minimum initial energy needed is 32.

This example demonstrates how the algorithm greedily adds just enough energy whenever needed, ensuring we find the absolute minimum initial energy required to complete all tasks in the optimal order.

Solution Implementation

Time and Space Complexity

The time complexity is O(n × log n), where n is the number of tasks. This is dominated by the sorting operation sorted(tasks, key=lambda x: x[0] - x[1]), which uses a comparison-based sorting algorithm (typically Timsort in Python) that has O(n × log n) time complexity. The subsequent for loop iterates through all tasks once, contributing O(n) time, but this is absorbed by the larger sorting complexity.

The space complexity is O(n). While the reference answer mentions O(1) when ignoring the space overhead of sorting, the sorted() function in Python creates a new sorted list rather than sorting in-place, which requires O(n) additional space to store the sorted copy of the tasks list. If we were to ignore this sorting overhead as suggested in the reference, then the remaining operations use only a constant amount of extra space (ans, cur, and loop variables), resulting in O(1) space complexity.

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Incorrect Sorting Logic - Sorting by Wrong Criteria

The Pitfall: A common mistake is sorting by the wrong criteria, such as:

• Sorting by minimum alone (ascending or descending)
• Sorting by actual alone
• Sorting by minimum - actual instead of actual - minimum

Why it fails: Consider tasks [[10, 10], [1, 20]]:

• If sorted by minimum ascending: [[10, 10], [1, 20]] → requires 30 initial energy
• If sorted by actual - minimum: [[10, 10], [1, 20]] → requires 21 initial energy (optimal)

Sorting by the wrong criteria leads to suboptimal task ordering and higher initial energy requirements.

Solution: Always sort by actual - minimum in ascending order (or equivalently, minimum - actual in descending order):

sorted_tasks = sorted(tasks, key=lambda task: task[0] - task[1])

2. Not Understanding the Energy Update Logic

The Pitfall: Incorrectly updating the current energy when insufficient, such as:

# Wrong approach 1: Adding the minimum required instead of the deficit
if cur < m:
    ans += m  # Wrong! This adds the full minimum, not the deficit
    cur = m
  
# Wrong approach 2: Setting ans to maximum seen so far
if cur < m:
    ans = max(ans, m)  # Wrong! This doesn't accumulate properly

Why it fails: The algorithm needs to track the cumulative energy additions, not individual minimums. Each time we're short on energy, we need to add exactly the deficit to our initial energy pool.

Solution: Add only the deficit (shortage) to the initial energy:

if cur < m:
    ans += m - cur  # Add only what's missing
    cur = m         # Update current to minimum required

3. Confusing Initial Energy vs Current Energy

The Pitfall: Mixing up what ans (initial energy) and cur (current energy) represent:

# Wrong: Thinking ans represents current energy at any point
for a, m in sorted_tasks:
    if ans < m:  # Wrong! Should check 'cur' not 'ans'
        ans += m - ans

Why it fails:

• ans accumulates the total initial energy we need to start with
• cur tracks our energy level as we progress through tasks
• Checking ans < m doesn't tell us if we can currently start a task

Solution: Keep the roles clear:

• Use cur to track current energy state
• Use ans only to accumulate the initial energy requirement
• Always check cur < minimum to determine if we can start a task