This problem involves simulating k multiplication operations on an array. Each operation applies a multiplier to the smallest element. After k operations, apply modulo 10^9 + 7 to each element. Use priority queues (heaps) to optimize the approach, managing the array's state efficiently.
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.
How GhostInterview Helps
- GhostInterview guides you through understanding and optimizing heap operations to solve this problem efficiently.
- It helps in navigating the trade-offs between simulation and heap-based approaches, especially when dealing with large values of k.
- By providing step-by-step problem-solving, GhostInterview ensures you grasp the key pattern of combining heaps with array manipulation.
Topic Pages
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- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
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.
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Solve the problem of determining the final state of an array after multiple multiplication operations using a priority queue.
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