2558. Take Gifts From the Richest Pile

Problem Description

In this problem, you are given an array called gifts, which contains integers representing the number of gifts in different piles. You have a task to perform every second for k seconds, and the task goes as follows:

1. Identify the pile with the most gifts. If there are multiple piles with the same maximum number of gifts, you may choose any one of them.
2. Leave behind the largest integer less than or equal to the square root of the number of gifts in that pile, also known as the floor of the square root. Take the rest of the gifts from the pile.

After performing this task every second for k seconds, you need to determine the total number of gifts left across all piles.

Intuition

To efficiently manage the process of finding and updating the largest pile every second, a max heap data structure is employed. A max heap is a binary tree where the parent node is always greater than its children nodes, making it easy to retrieve and remove the maximum element in the heap.

Here is the intuition behind the solution approach:

1. Convert the list of gifts into a max heap. In Python, a max heap is not provided by default, but a min heap is provided by the heapq library. Therefore, we can simulate a max heap by negating all values in the gifts array when adding them to the heap. This way, the smallest number (after negation, which was actually the largest) comes at the top of the heap.
2. Iterate k times to simulate the seconds passing. During each iteration, pop the maximum element from the heap (which will be returned as a negative number due to our earlier negation), calculate the floor of the square root of that number (negate it again to maintain the heap property), and put the result back into the heap.
3. After doing this for k seconds, the heap contains negative numbers representing the gifts left in each pile. Sum up the negative numbers and negate the result to get the total number of gifts remaining.

This approach lets us update the piles and find out the total number of remaining gifts efficiently.

Learn more about Heap (Priority Queue) patterns.

Solution Approach

The key data structure used in the solution approach is a heap, which is a specialized tree-based data structure that satisfies the heap property. In a max heap, for any given node i, the value of i is greater than or equal to the values of its children, and the maximum element is always at the root node. The algorithm utilizes a min heap with negated values to mimic the behavior of a max heap, as Python's heapq module only provides a min heap implementation.

The process of the algorithm is as follows:

1. We first negate all the values in the gifts array and convert this negated list into a heap using heapify. The heapify function transforms the list into a heap in O(n) time.

   1h = [-v for v in gifts]
   2heapify(h)

2. We then repeatedly perform k iterations to simulate each second. In each iteration, we use the heapreplace function, which first pops the root element of the heap (the smallest element, or, in our negated heap, the pile with the maximum gifts), then calculates the floor of the square root of the negated number (effectively the square root of the original maximum), negates it (to keep it consistent with our negated heap), and finally places this value back into the heap.

   1for _ in range(k):
   2    heapreplace(h, -int(sqrt(-h[0])))

   The heapreplace function consolidates the operations of popping and pushing an element while ensuring the heap property is maintained during the entire iteration. It operates in O(log n) time because it needs to maintain the heap structure after each replacement.

3. Lastly, after all k iterations are completed, we have the negated values of gifts remaining in each pile. The sum of all remaining gifts is the sum of the negated values negative to turn it back into a positive number.

   1return -sum(h)

This solution method is designed to optimize the process of choosing and updating the pile with the maximum gifts efficiently while also ensuring that the overall time complexity is kept to O(k log n), where n is the number of piles, and k is the number of seconds the process is run.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Suppose we have the following array of gifts and we want to perform the task for k = 3 seconds:

1gifts = [9, 7, 4, 1]

Initially, we have four piles with 9, 7, 4, and 1 gifts, respectively.

Step 1: Convert to a Max Heap

Firstly, we negate all the values and convert the gifts array to a heap to effectively create a max heap:

1h = [-9, -7, -4, -1]
2heapify(h)  # h becomes [-9, -7, -4, -1]

Now, our heap (max heap with negated values) represents the piles as [-9, -7, -4, -1].

Step 2: Iterations for k Seconds

Next, we simulate the task for k seconds:

For the first second (k=1):

• The maximum (negated minimum) value is -9.
• We pop this value and calculate the floor of the square root: int(sqrt(9)) = 3.
• We leave behind 3 gifts in the pile and take the rest, which is 6.
• We negate the leftover (-3) and push it back into the heap.

1heapreplace(h, -3)  # h becomes [-7, -3, -4, -1]

For the second second (k=2):

• Now the maximum (negated minimum) value is -7.
• We pop this value and calculate the floor of the square root: int(sqrt(7)) = 2.
• We leave behind 2 gifts and take 5 from this pile.
• We negate the leftover (-2) and push it back into the heap.

1heapreplace(h, -2)  # h becomes [-4, -3, -2, -1]

For the third second (k=3):

• The maximum (negated minimum) value is now -4.
• We pop this value and calculate the floor of the square root: int(sqrt(4)) = 2.
• We leave behind 2 gifts and take 2 from this pile.
• After negating the leftover (-2) we push it back into the heap.

1heapreplace(h, -2)  # h becomes [-3, -2, -2, -1]

Step 3: Calculate Total Gifts Remaining

After k seconds, our heap represents the remaining gifts in each pile as [-3, -2, -2, -1]. To get the total number of gifts remaining, we sum these values and negate the result:

1total_gifts_remaining = -sum(h)  # -(-3 -2 -2 -1) = 8

So, after performing the task for k = 3 seconds, we have a total of 8 gifts remaining across all piles.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is O(n + k * log n). This is because initializing a heap using heapify from a list of n elements has a time complexity of O(n). After heapification, the code performs k operations where each operation involves popping the smallest element from the heap and replacing it with the square root of that element negated, which takes O(log n) time due to the need to maintain the heap structure after each replacement. Therefore, the loop will contribute O(k * log n) to the total time complexity. Combined, we have an overall time complexity of O(n) from the heapify plus O(k * log n) from the loop, yielding O(n + k * log n).

The space complexity of the code is O(n), which accounts for the storage of the heap. No additional data structures that are dependent on the size of the input are used beyond the initial heap h. As such, the space consumed is directly proportional to the size of the input list gifts.

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