#3266

Hard

auto_awesomeArray plus Heap (Priority Queue)

LeetCode Problem Workspace

Final Array State After K Multiplication Operations II

Optimize the final state of an array after performing k multiplication operations with priority queues.

Problem Statement

You are given an integer array nums, an integer k, and an integer multiplier. You need to perform k operations on nums. In each operation, multiply the smallest element of nums by the multiplier, then apply modulo 10^9 + 7 to the result. Repeat this k times.

After the k operations, the final state of nums should be returned. Note that the problem involves handling large values, so efficient array manipulation and the use of heaps for managing the smallest element are key to solving the problem.

Examples

Example 1

Input: nums = [2,1,3,5,6], k = 5, multiplier = 2

Output: [8,4,6,5,6]

Example 2

Input: nums = [100000,2000], k = 2, multiplier = 1000000

Output: [999999307,999999993]

Constraints

1 <= nums.length <= 104
1 <= nums[i] <= 109
1 <= k <= 109
1 <= multiplier <= 106

Solution Approach

Using a Min-Heap for Efficient Multiplication

To efficiently manage the smallest element in the array during each operation, use a min-heap (priority queue). This allows quick access to the smallest value, which is the target for multiplication in each of the k operations.

Apply the Multiplier and Modulo

For each operation, remove the smallest element from the heap, multiply it by the given multiplier, and apply modulo 10^9 + 7 to the result. Reinsert the updated value back into the heap to maintain the heap property.

Handling Large k Values

If k is very large, consider optimizing the heap operations and using a direct simulation approach where possible, ensuring that each operation can be performed efficiently with respect to both time and space complexity.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the number of operations (k) and the heap operations (log n for each operation). Given the constraints, the solution must handle large k values efficiently. The space complexity is driven by the storage requirements of the heap, which is proportional to the size of the input array (n).

What Interviewers Usually Probe

Candidate demonstrates proficiency in heap operations and optimization techniques.
Candidate's approach handles large values of k efficiently without excessive overhead.
Candidate explains the trade-offs between different heap-based solutions, considering both time and space complexity.

Common Pitfalls or Variants

Common pitfalls

Failing to apply the modulo operation after each multiplication, leading to overflow.
Incorrectly updating the heap, causing inefficient operation due to rebalancing.
Misunderstanding the impact of large k values, leading to inefficiencies in the solution.

Follow-up variants

Consider a variant where the multiplier is a dynamic value, changing between operations.
A version where elements are updated based on other conditions besides the minimum, like the maximum or random elements.
Optimize for cases where k is relatively small, reducing the reliance on heaps.

FAQ

How do heaps help in solving Final Array State After K Multiplication Operations II?

Heaps allow quick access to the smallest element, which is crucial for efficiently multiplying it by the multiplier during each operation.

What is the time complexity of the optimal solution for this problem?

The time complexity is O(k log n), where k is the number of operations and n is the size of the array, due to heap insertions and extractions.

How can I optimize the solution when k is very large?

For large k, focus on optimizing heap operations and consider skipping unnecessary operations if possible, reducing the overall computational load.

What is the role of modulo 10^9 + 7 in this problem?

The modulo operation ensures that the values do not overflow beyond the integer limits, keeping the result manageable and within bounds.

Can the approach be adapted for other array-related problems?

Yes, the heap-based approach is useful in other problems where managing the smallest (or largest) elements efficiently is key, such as in priority scheduling or k-th smallest problems.

terminal

Solution

Solution 1: Priority Queue (Min-Heap) + Simulation

Let the length of the array $\textit{nums}$ be $n$, and the maximum value be $m$.

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class Solution:
    def getFinalState(self, nums: List[int], k: int, multiplier: int) -> List[int]:
        if multiplier == 1:
            return nums
        pq = [(x, i) for i, x in enumerate(nums)]
        heapify(pq)
        m = max(nums)
        while k and pq[0][0] < m:
            x, i = heappop(pq)
            heappush(pq, (x * multiplier, i))
            k -= 1
        n = len(nums)
        mod = 10**9 + 7
        pq.sort()
        for i, (x, j) in enumerate(pq):
            nums[j] = x * pow(multiplier, k // n + int(i < k % n), mod) % mod
        return nums

Continue Practicing

#3264 final array state after k multiplication operations i #3080 mark elements on array by performing queries #3066 minimum operations to exceed threshold value ii