Boboniu and Banknote Collection

1000ms

262144K

Description:

No matter what trouble you're in, don't be afraid, but face it with a smile.

I've made another billion dollars!

— Boboniu

Boboniu has issued his currencies, named Bobo Yuan. Bobo Yuan (BBY) is a series of currencies. Boboniu gives each of them a positive integer identifier, such as BBY-1, BBY-2, etc.

Boboniu has a BBY collection. His collection looks like a sequence. For example:

$\text{[math]}$

We can use sequence $$$a=[1,2,3,3,2,1,4,4,1]$$$ of length $$$n=9$$$ to denote it.

Now Boboniu wants to fold his collection. You can imagine that Boboniu stick his collection to a long piece of paper and fold it between currencies:

$\text{[math]}$

Boboniu will only fold the same identifier of currencies together. In other words, if $$$a_i$$$ is folded over $$$a_j$$$ ($$$1\le i,j\le n$$$), then $$$a_i=a_j$$$ must hold. Boboniu doesn't care if you follow this rule in the process of folding. But once it is finished, the rule should be obeyed.

A formal definition of fold is described in notes.

According to the picture above, you can fold $$$a$$$ two times. In fact, you can fold $$$a=[1,2,3,3,2,1,4,4,1]$$$ at most two times. So the maximum number of folds of it is $$$2$$$.

As an international fan of Boboniu, you're asked to calculate the maximum number of folds.

You're given a sequence $$$a$$$ of length $$$n$$$, for each $$$i$$$ ($$$1\le i\le n$$$), you need to calculate the maximum number of folds of $$$[a_1,a_2,\ldots,a_i]$$$.

Input:

The first line contains an integer $$$n$$$ ($$$1\le n\le 10^5$$$).

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1\le a_i\le n$$$).

Output:

Print $$$n$$$ integers. The $$$i$$$-th of them should be equal to the maximum number of folds of $$$[a_1,a_2,\ldots,a_i]$$$.

Sample Input:

9
1 2 3 3 2 1 4 4 1

Sample Output:

0 0 0 1 1 1 1 2 2

Sample Input:

9
1 2 2 2 2 1 1 2 2

Sample Output:

0 0 1 2 3 3 4 4 5

Sample Input:

15
1 2 3 4 5 5 4 3 2 2 3 4 4 3 6

Sample Output:

0 0 0 0 0 1 1 1 1 2 2 2 3 3 0

Sample Input:

50
1 2 4 6 6 4 2 1 3 5 5 3 1 2 4 4 2 1 3 3 1 2 2 1 1 1 2 4 6 6 4 2 1 3 5 5 3 1 2 4 4 2 1 3 3 1 2 2 1 1

Sample Output:

0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 6 7 3 3 3 4 4 4 4 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 8

Note:

Formally, for a sequence $$$a$$$ of length $$$n$$$, let's define the folding sequence as a sequence $$$b$$$ of length $$$n$$$ such that:

$$$b_i$$$ ($$$1\le i\le n$$$) is either $$$1$$$ or $$$-1$$$.
Let $$$p(i)=[b_i=1]+\sum_{j=1}^{i-1}b_j$$$. For all $$$1\le i<j\le n$$$, if $$$p(i)=p(j)$$$, then $$$a_i$$$ should be equal to $$$a_j$$$.

($$$[A]$$$ is the value of boolean expression $$$A$$$. i. e. $$$[A]=1$$$ if $$$A$$$ is true, else $$$[A]=0$$$).

Now we define the number of folds of $$$b$$$ as $$$f(b)=\sum_{i=1}^{n-1}[b_i\ne b_{i+1}]$$$.

The maximum number of folds of $$$a$$$ is $$$F(a)=\max\{ f(b)\mid b \text{ is a folding sequence of }a \}$$$.

Informação

Provedor Codeforces

Código CF1394E