2558. Take Gifts From the Richest Pile
Difficulty: Easy
Topics: Array, Heap (Priority Queue), Simulation
You are given an integer array gifts denoting the number of gifts in various piles. Every second, you do the following:
- Choose the pile with the maximum number of gifts.
- If there is more than one pile with the maximum number of gifts, choose any.
- Leave behind the floor of the square root of the number of gifts in the pile. Take the rest of the gifts.
Return the number of gifts remaining after k seconds.
Example 1:
- Input: gifts = [25,64,9,4,100], k = 4
- Output: 29
-
Explanation: The gifts are taken in the following way:
- In the first second, the last pile is chosen and 10 gifts are left behind.
- Then the second pile is chosen and 8 gifts are left behind.
- After that the first pile is chosen and 5 gifts are left behind.
- Finally, the last pile is chosen again and 3 gifts are left behind.
- The final remaining gifts are [5,8,9,4,3], so the total number of gifts remaining is 29.
Example 2:
- Input: gifts = [1,1,1,1], k = 4
- Output: 4
-
Explanation: In this case, regardless which pile you choose, you have to leave behind 1 gift in each pile.
- That is, you can't take any pile with you.
- So, the total gifts remaining are 4.
Constraints:
- 1 3
- 1 9
- 1 3
Hint:
- How can you keep track of the largest gifts in the array
- What is an efficient way to find the square root of a number?
- Can you keep adding up the values of the gifts while ensuring they are in a certain order?
- Can we use a priority queue or heap here?
Solution:
We can utilize a max-heap (priority queue) since we need to repeatedly pick the pile with the maximum number of gifts. A max-heap will allow us to efficiently access the largest pile in constant time and update the heap after taking gifts from the pile.
Approach:
-
Use a Max-Heap:
- Since we need to repeatedly get the pile with the maximum number of gifts, a max-heap (priority queue) is ideal. In PHP, we can use SplPriorityQueue, which is a priority queue that by default works as a max-heap.
- To simulate a max-heap, we will insert the number of gifts as a negative value, since SplPriorityQueue is a min-heap by default. By inserting negative values, the smallest negative value will represent the largest original number.
-
Process Each Second:
- In each second, pop the pile with the maximum number of gifts from the heap.
- Take all the gifts except the floor of the square root of the number of gifts in that pile.
- Push the modified pile back into the heap.
-
Termination:
- We stop after k seconds or once we've processed all the seconds.
Let's implement this solution in PHP: 2558. Take Gifts From the Richest Pile
<?php /** * @param Integer[] $gifts * @param Integer $k * @return Integer */ function pickGifts($gifts, $k) { ... ... ... /** * go to ./solution.php */ } // Example usage: $gifts1 = [25, 64, 9, 4, 100]; $k1 = 4; echo pickGifts($gifts1, $k1) . "\n"; // Output: 29 $gifts2 = [1, 1, 1, 1]; $k2 = 4; echo pickGifts($gifts2, $k2) . "\n"; // Output: 4 ?>
Explanation:
-
Heap Initialization:
- SplPriorityQueue is used to simulate a max-heap. The insert method allows us to push elements into the heap with a priority.
-
Processing the Largest Pile:
- For k iterations, the largest pile is extracted using extract.
- The number of gifts left behind is calculated as the floor of the square root of the current largest pile using floor(sqrt(...)).
- The reduced pile is re-inserted into the heap.
-
Summing Remaining Gifts:
- After the k operations, all elements in the heap are extracted and summed up to get the total number of remaining gifts.
-
Edge Cases:
- If gifts is empty, the result is 0.
- If k is larger than the number of operations possible, the algorithm gracefully handles it.
Time Complexity:
- Heap operations (insert and extract): Each heap operation (insertion and extraction) takes O(log n), where n is the number of piles.
- Looping through k operations: We perform k operations, each involving heap extraction and insertion, both taking O(log n).
Thus, the total time complexity is O(k log n), where n is the number of piles and k is the number of seconds.
Example Walkthrough:
For the input:
<?php /** * @param Integer[] $gifts * @param Integer $k * @return Integer */ function pickGifts($gifts, $k) { ... ... ... /** * go to ./solution.php */ } // Example usage: $gifts1 = [25, 64, 9, 4, 100]; $k1 = 4; echo pickGifts($gifts1, $k1) . "\n"; // Output: 29 $gifts2 = [1, 1, 1, 1]; $k2 = 4; echo pickGifts($gifts2, $k2) . "\n"; // Output: 4 ?>
- Initially, the priority queue has the piles: [25, 64, 9, 4, 100].
- After 1 second: Choose 100, leave 10 behind. The remaining gifts are: [25, 64, 9, 4, 10].
- After 2 seconds: Choose 64, leave 8 behind. The remaining gifts are: [25, 8, 9, 4, 10].
- After 3 seconds: Choose 25, leave 5 behind. The remaining gifts are: [5, 8, 9, 4, 10].
- After 4 seconds: Choose 10, leave 3 behind. The remaining gifts are: [5, 8, 9, 4, 3].
The sum of the remaining gifts is 5 8 9 4 3 = 29.
This approach efficiently solves the problem using a max-heap and performs well within the given constraints.
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