Problem Description

In this task, we are given an array called tasks, with each element being a pair of values [actual_i, minimum_i]. Each pair represents a task where actual_i is the amount of energy needed to complete the i-th task and minimum_i is the minimum amount of energy required to start the task. The challenge is to figure out the least amount of energy we need initially to be able to complete all tasks, but what's unique is that we can tackle the tasks in any sequence we prefer.

Imagine having some tasks that need a significant amount of energy to start but consume less upon completion, while others are easy to start but require a lot of energy to complete. The optimal sequence to complete these tasks will ensure that you never find yourself short of energy before starting a task. Given this, we are asked to calculate the minimum initial energy.

Intuition

To arrive at an intuitive solution, consider that a task requiring a lot of energy to begin but not as much to finish should be tackled later than a task that consumes more energy than it requires to start. This is because finishing tasks that spend more energy than the minimum required will deplete the energy reserve, potentially preventing the start of tasks with high initial energy requirements.

Hence, the solution sorts the tasks by the difference between actual_i and minimum_i. The tasks with the smallest difference (even potentially negative) are placed first, since completing these tasks either depletes the least energy from the reserve or can even recharge it.

While iterating over the sorted tasks, the algorithm checks if the current energy level (cur) is sufficient to start the current task (m). If it's not enough, we need only increase our initial energy just enough to start the task (m - cur). We then deduct the actual energy spent to complete the task from cur. Throughout this process, ans tracks the total additional energy required, if any, which ultimately will be the minimum initial energy needed to complete all tasks.

Learn more about Greedy and Sorting patterns.

Solution Approach

The solution makes use of the sort function, a basic algorithm in Python, to arrange the tasks such that the task with the smallest difference between the actual energy and the minimum required energy needed to start comes first. This is done to prioritize tasks that will either decrease our energy the least or could potentially increase it (if actual_i < minimum_i).

To understand this, let's assume we have a list of tasks, each represented as [actual_i, minimum_i]. We apply the sorted function with a custom sort key — the difference x[0] - x[1] (actual_i - minimum_i), which helps us to process tasks that are less energy-consuming first.

The data structure used here is a simple list of lists, a common choice for representing a sequence of elements where each element itself contains multiple pieces of data.

The algorithm follows these steps:

1. Initialize ans and cur to zero. ans will hold our final answer, and cur tracks the current energy level.
2. Sort the tasks by the difference (x[0] - x[1]), which is calculated for each task using a lambda function as the key in the sorted method.
3. Iterate through the sorted tasks list.
   • For each task, check if our current energy cur is less than the minimum required energy m.
   • If so, we update ans with the difference m - cur, as we need to boost our energy level up to the minimum required to start the task. We also update cur to reflect this new level of energy.
   • Deduct the actual amount of energy spent on the task a from cur.
4. The loop exits after all tasks have been processed, and ans now holds the minimum initial amount of energy we need to finish all the tasks without running out of energy at any point.

Using this approach allows us to guarantee that we always have enough energy to start each task by incrementing our initial energy only when necessary and by the smallest amount needed. After all tasks are completed, ans tells us the minimum energy we would need to reserve initially to accomplish this.

Example Walkthrough

Let's say we're given the following array of tasks: [[3,1], [2,2], [1,2]]. Each task is represented as [actual_i, minimum_i].

1. We start by initializing ans and cur to zero. These variables will help us keep track of the additional energy needed (ans) and the current energy level (cur).

2. We need to sort the tasks by the difference between actual_i and minimum_i. Using a lambda function as the key in the sorted method, the tasks are sorted as follows:

   [[1,2], [2,2], [3,1]] → The task [1,2] has a difference of -1 (it actually gives us energy), [2,2] breaks even, and [3,1] requires 2 units more energy than it gives back.

3. We iterate through the sorted tasks:

   • The first task is [1, 2]. cur is 0 and is less than the minimum required energy 2 for the task. We need to increase it by 2-0=2, so ans becomes 2 and cur becomes 2. After completing the task that consumes 1 unit of energy, cur is now 2-1=1.

   • The next task is [2, 2]. cur is now 1 which is less than the 2 required to start the task. We must increase cur by 2-1=1, which means ans is updated from 2 to 2+1=3. After finishing the task that uses 2 units of energy, cur becomes 1-2=-1. (This indicates that we finished the task using not just the current energy we had, but also some of the initial energy we reserved.)

   • The final task is [3, 1]. cur is -1, and since the minimum energy to start is 1, we will boost cur by 1-(-1)=2, making ans now 3+2=5. We then complete the task by spending 3 units of energy, and cur is updated to -1-3=-4.

4. Having processed all tasks, ans holds the minimum initial amount of energy we need to complete all of them. In this case, we must start with at least 5 units of energy to ensure we can complete the tasks in this specific order.

By carefully selecting the sequence of the tasks and calculating the increments in our energy reserve only as needed, we find that starting with 5 units of energy allows us to complete all tasks successfully.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the solution is dominated by the sorting of the tasks array and the subsequent iteration through the sorted tasks.

The sorting function uses the Timsort algorithm in Python, which has a time complexity of O(n log n) in the average and worst case, where n is the number of tasks. After sorting, the function goes through the tasks one by one, which takes O(n) time.

Therefore, combining both steps, the total time complexity of the function is O(n log n) due to the sorting step being the most significant factor.

Space Complexity

The space complexity of the solution can be considered as O(1) because the sorting is done in-place and no additional data structures that grow with the input size are used. The variables ans, cur, a, and m are all constant in space, requiring a fixed amount of memory regardless of the number of tasks.

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