USACO 2024 December Contest, Gold

Problem 1. Cowdependence


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Farmer John's $N$ $(1 \leq N \leq 10^5)$ cows have been arranged into a line. The $i$th cow has label $a_i$ ($1 \leq a_i \leq N$). A group of cows can form a friendship group if they all have the same label and each cow is within $x$ cows of all the others in the group, where $x$ is an integer in the range $[1,N]$. Every cow must be in exactly one friendship group.

For each $x$ from $1$ to $N$, calculate the minimum number of friendship groups that could have formed.

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

The first line consists of an integer $N$.

The next line contains $a_1 ... a_N$, the labels of each cow.

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

For each $x$ from $1$ to $N$, output the minimum number of friendship groups for that $x$ on a new line.

SAMPLE INPUT:

9
1 1 1 9 2 1 2 1 1

SAMPLE OUTPUT:

7
5
4
4
4
4
4
3
3

Here are examples of how to assign cows to friendship groups for $x=1$ and $x=2$ in a way that minimizes the number of groups. Each letter corresponds to a different group. 

Example:

       1 1 1 9 2 1 2 1 1
x = 1: A B B C D E F G G (7 groups)
x = 1: A A B C D E F G G (7 groups, alternative grouping)
x = 2: A A A B C D C E E (5 groups)
x = 2: A A A B C D C D E (5 groups, alternative grouping)

SCORING:

  • Inputs 2-3: $N\le 5000$
  • Inputs 4-7: $a_i\le 10$ for all $i$
  • Inputs 8-11: No label appears more than $10$ times.
  • Inputs 12-20: No additional constraints.

Problem credits: Chongtian Ma

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