Problem 2. Election Queries

USACO 2025 US Open Contest, Gold

Note: The time limit for this problem is 3s, 1.5x the default.

Farmer John has $N$ ($2 \leq N \leq 2 \cdot 10^5$) cows numbered from $1$ to $N$. An election is being held in FJ's farm to determine the two new head cows in his farm. Initially, it is known that cow $i$ will vote for cow $a_i$ ($1 \leq a_i \leq N$).

To determine the two head cows, FJ will hold his election in the following process:

• Choose an arbitrary subset $S$ of his cows that contains at least one cow but not all cows. FJ is able to choose cow $x$ as the first head cow if its votes appear most frequently among all votes cast by cows in $S$.
• FJ is able to choose cow $y$ as the second head cow if its votes appear most frequently among votes cast by cows not in $S$.
• For a fixed subset $S$, FJ denotes the diversity between his head cows as $|x - y|$. As FJ does not like having leaders with similar numbers, he wants to choose $S$ such that the diversity is maximized. Note that if FJ is not able to choose two distinct head cows, then the diversity is $0$.

However, some cows keep changing their minds, and FJ may have to rerun the election many times! Therefore, he asks you $Q$ ($1 \leq Q \leq 10^5$) queries. In each query, a cow changes their vote. After each query, he asks you for the maximum possible

diversity

among his new head cows.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $N$ and $Q$.

The following line contains $a_1, a_2, \ldots, a_N$.

The following $Q$ lines contain two integers $i$ and $x$, representing the update $a_i = x$ ($1 \leq i, x \leq N$).

OUTPUT FORMAT (print output to the terminal / stdout):

Output $Q$ lines, the $i$'th of which is the maximum possible diversity after the first $i$ queries.

SAMPLE INPUT:

5 3
1 2 3 4 5
3 4
1 2
5 2

SAMPLE OUTPUT:

4
3
2

After the first query, $a = [1, 2, 4, 4, 5]$. At the first step of the election, FJ can make $S = \{1, 3\}$. Here, cow $1$ receives one vote and cow $4$ receives one vote. Therefore, FJ can choose either cow $1$ or cow $4$ as its first head cow.

For all cows not in the election, cow $2$ receives one vote, cow $4$ receives one vote, and cow $5$ also receives one vote. Therefore, FJ can choose any one of cows $2$, $4$, or $5$ to be its second head cow.

To obtain the maximum diversity, FJ can choose cow $1$ as the first head cow and cow $5$ as the second head cow. Therefore, the diversity is $|1-5| = 4$.

After the second query, $a=[2,2,4,4,5]$ and FJ can make $S = \{4, 5\}$. Then, he can choose $5$ as the first head cow and cow $2$ as the second head cow. The maximum possible diversity is $|5 - 2| = 3$.

SAMPLE INPUT:

8 5
8 1 4 2 5 4 2 3
7 4
8 4
4 1
5 8
8 4

SAMPLE OUTPUT:

4
4
4
7
7

SCORING: Inputs 3-4: $N, Q \leq 100$Inputs 5-7: $N, Q \leq 3000$Inputs 8-15: No additional constraints.

Problem credits: Chongtian Ma and Haokai Ma