Hello Codeforces! Today I want to share a detailed breakdown of the problem D. diss_quack and Array Game. This problem initially looks like a complex game theory simulation, but it elegantly reduces to bitwise math and offline precomputation.

Here is a step-by-step approach to solving it.

Prerequisites

• Bitwise Operations (Popcount, Leading Zeros)
• Greedy Algorithms
• Suffix Minimums / Precomputation

Observation 1: Bob's "Stopper" Strategy

Let's analyze the game from Bob's perspective. Bob wants to maximize Alice's moves. Alice's most powerful move is dividing the entire array by 2 (which she can only do if $$$a_1$$$ and all elements in the prefix are even).

To stop this, Bob's optimal strategy is to find an odd number and swap it to index 2. This acts as a "stopper," completely breaking Alice's even prefix and forcing her to only divide $$$a_1$$$, completely isolating her moves to one element at a time.

If Bob can do this, Alice is forced to eliminate the set bits (via subtraction) and the trailing zeros (via division) of every single element individually. The standalone cost to reduce any isolated number $$$x$$$ to 0 under Bob's optimal defense is:

Observation 2: The Global Divide Exception

Bob's strategy has one fatal flaw: It requires an odd number. If Alice can ensure that every single number in the array is even, Bob has no "stopper" to swap. Alice will get a free "Global Divide," halving the entire array in a single move.

If Alice can force the entire array to survive $$$m$$$ consecutive global divides, every number in the array must have at least $$$m$$$ trailing zeros. In other words, every element $$$a_i$$$ must be bumped up to a multiple of $$$2^m$$$.

If she successfully executes $$$m$$$ global divides, she saves $$$(n - 1) \cdot m$$$ moves overall!

The Strategy & Cost Function

Before the game starts, Alice can increment elements. Suppose she decides on a global strategy to execute exactly $$$m$$$ global divides. She must push every single element $$$a_i$$$ up to a target $$$x \ge a_i$$$, where $$$x$$$ is a multiple of $$$2^m$$$.

For a fixed $$$m$$$ and a chosen target $$$x$$$, the total moves spent on a specific element is the cost to increment it plus the cost to clear it later:

Alice's goal is to find the target multiple $$$x$$$ that minimizes this sum. Since the maximum value in the array is 100,000, we can safely cap our maximum target search at roughly 300,000 (incrementing beyond this costs too much to be optimal).

The Core Optimization: Suffix Minimums

We cannot afford to search for the best multiple for every element on the fly. Instead, we precompute!

Sometimes, pushing an element to the next multiple is cheaper because it has far fewer set bits (e.g., pushing 1023 to 1024 drops the popcount from 10 to 1). To capture this, we use a backward-iterating suffix minimum array:

1. For every target $$$m \in [0, 17]$$$, iterate through all valid multiples $$$k$$$ of $$$2^m$$$.
2. Calculate the base cost $$$x + F(x)$$$.
3. Iterate backwards and store the minimum cost between the current multiple $$$k$$$ and the next multiple $$$k+1$$$.

This transforms our array into a "cheapest cost from the $$$k$$$-th multiple onward" lookup table. During the game simulation, we can find the absolute cheapest optimal target for any $$$a_i$$$ in $$$O(1)$$$ time!