Final Array State After K Multiplication Operations II

Final Array State After K Multiplication Operations II

What is this problem about?

The Final Array State After K Multiplication Operations II coding problem is a significantly harder version of the previous problem. The number of operations k can be extremely large (e.g., 10^9), making it impossible to simulate the operations one by one. You need to calculate the final state of the array modulo $10^9 + 7$.

Why is this asked in interviews?

Google and Meta ask this Hard difficulty Math and Heap problem to test your ability to identify a steady-state pattern. It requires understanding Modular Arithmetic and Binary Exponentiation. It evaluates whether you can realize that after some operations, the relative order of elements becomes stable, and the remaining operations can be distributed evenly using a formula.

Algorithmic pattern used

This problem uses Cycle/Stability Analysis and Binary Exponentiation.

1. Initial Phase: Simulate with a heap until the minimum element has been multiplied enough times that it's no longer the absolute minimum even after more multiplications (relative stability).
2. Stable Phase: Once all elements have been "brought up" to a similar magnitude, every $N$ operations will multiply every element by the multiplier exactly once.
3. Formula: Use k / N to find how many full cycles remain. Calculate multiplier^(k/N) % MOD using binary exponentiation.
4. Remainder: Perform the remaining k % N operations manually using the heap.

Example explanation

Array of size 2, k=1,000,001, multiplier=2.

1. After some operations, both elements are "large."
2. One cycle (2 operations) multiplies both by 2.
3. 500,000 cycles multiply both by 2^500,000.
4. The final operation multiplies the current minimum by 2.
5. Use modular arithmetic to keep the numbers within bounds.

Common mistakes candidates make

• Naive Simulation: Trying to run a loop 10^9 times.
• Modular Errors: Applying the modulo incorrectly during the exponentiation or forgetting that the relative comparison in the heap must use the "un-moduloed" values (or their logarithms).
• Ignoring the stability point: Not realizing that you must simulate until the smallest value multiplied by the multiplier is larger than the original largest value.

Interview preparation tip

When k is very large, always look for a way to use k / N and k % N. This usually implies that the process becomes periodic or uniform after a certain point.