1655. Minimum Initial Energy to Finish Tasks

1655. Minimum Initial Energy to Finish Tasks

Difficulty: Hard

Topics: Senior Staff, Array, Greedy, Sorting, Weekly Contest 216

You are given an array tasks where tasks[i] = [actuali, minimumi]:

• actuali is the actual amount of energy you spend to finish the iᵗʰ task.
• minimumi is the minimum amount of energy you require to begin the iᵗʰ task.

For example, if the task is [10, 12] and your current energy is 11, you cannot start this task. However, if your current energy is 13, you can complete this task, and your energy will be 3 after finishing it.

You can finish the tasks in any order you like.

Return the minimum initial amount of energy you will need to finish all the tasks.

Example 1:

• Input: tasks = [[1,2],[2,4],[4,8]]
• Output: 8
• Explanation:
  • Starting with 8 energy, we finish the tasks in the following order:
    • 3rd task. Now energy = 8 - 4 = 4.
    • 2nd task. Now energy = 4 - 2 = 2.
    • 1st task. Now energy = 2 - 1 = 1.
  • Notice that even though we have leftover energy, starting with 7 energy does not work because we cannot do the 3rd task.

Example 2:

• Input: tasks = [[1,3],[2,4],[10,11],[10,12],[8,9]]
• Output: 32
• Explanation:
  • Starting with 32 energy, we finish the tasks in the following order:
    • 1st task. Now energy = 32 - 1 = 31.
    • 2nd task. Now energy = 31 - 2 = 29.
    • 3rd task. Now energy = 29 - 10 = 19.
    • 4th task. Now energy = 19 - 10 = 9.
    • 5th task. Now energy = 9 - 8 = 1.

Example 3:

• Input: tasks = [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
• Output: 27
• Explanation:
  • Starting with 27 energy, we finish the tasks in the following order:
    • 5th task. Now energy = 27 - 5 = 22.
    • 2nd task. Now energy = 22 - 2 = 20.
    • 3rd task. Now energy = 20 - 3 = 17.
    • 1st task. Now energy = 17 - 1 = 16.
    • 4th task. Now energy = 16 - 4 = 12.
    • 6th task. Now energy = 12 - 6 = 6.

Constraints:

• 1 <= tasks.length <= 10⁵
• 1 <= actualᵢ <= minimumᵢ <= 10⁴

Hint:

1. We can easily figure that the f(x) : does x solve this array is monotonic so binary Search is doable
2. Figure a sorting pattern

Solution:

The solution determines the minimum initial energy required to complete all tasks in any order, where each task has an actual energy cost and a minimum required energy to start.

It uses greedy sorting by (minimum - actual) in descending order to ensure tasks that are "harder to start" are done earlier, then calculates the needed starting energy.

Approach:

• Sorting Strategy Sort tasks by (minimum - actual) in descending order. This prioritizes tasks where the gap between required start energy and actual energy is largest.
• Simulation & Energy Tracking Iterate through tasks in sorted order, keeping track of:
  • sumActual = total actual energy spent so far
  • initial = maximum starting energy needed to reach the current point
• Formula Updating For each task (actual, minimum), the energy needed before this task is at least minimum + sumActual. Update initial to the maximum such value across all tasks, then add actual to sumActual.
• Result After processing all tasks, initial contains the minimum starting energy.

Let's implement this solution in PHP: 1655. Minimum Initial Energy to Finish Tasks

<?php
/**
 * @param Integer[][] $tasks
 * @return Integer
 */
function minimumEffort(array $tasks): int
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo minimumEffort([[1,2],[2,4],[4,8]]) . "\n";                             // Output: 8
echo minimumEffort([[1,3],[2,4],[10,11],[10,12],[8,9]]) . "\n";             // Output: 32
echo minimumEffort([[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]) . "\n";        // Output: 27
?>

Explanation:

• Sorting by minimum - actual descending ensures tasks with high minimum requirement relative to actual cost are attempted first. This is optimal because starting with less energy fails for tasks that require a high threshold, so earlier energy "buffer" is more valuable for them.
• The greedy choice is proven optimal by exchange arguments (typical for scheduling with constraints).
• The formula initial = max(initial, minimum + sumActual) ensures that before doing the current task, we have enough energy to meet its minimum requirement, considering all previous tasks have been done (cost added to sumActual).
• No binary search needed because sorting + linear scan directly gives the minimal starting energy.

Complexity

• Time Complexity: O(n log n) due to sorting.
• Space Complexity: O(1) extra space (ignoring input storage).
• Sorting Comparison: Each comparison is O(1).
• Iteration: O(n) after sorting.

Contact Links

If you found this series helpful, please consider giving the repository a star on GitHub or sharing the post on your favorite social networks 😍. Your support would mean a lot to me!

If you want more helpful content like this, feel free to follow me:

• LinkedIn
• GitHub