201491: [AtCoder]ARC149 B - Two LIS Sum
Description
Score : $500$ points
Problem Statement
For a sequence $P = (P_1, \ldots, P_N)$, let $\mathrm{LIS}(P)$ denote the length of a longest increasing subsequence.
You are given permutations $A = (A_1, \ldots, A_N)$ and $B = (B_1, \ldots, B_N)$ of integers from $1$ through $N$. You may perform the following operation on these sequences any number of times (possibly zero).
- Choose an integer $i$ such that $1\leq i\leq N-1$. Swap $A_i$ and $A_{i+1}$, and swap $B_i$ and $B_{i+1}$.
Find the maximum possible final value of $\mathrm{LIS}(A) + \mathrm{LIS}(B)$.
What is a longest increasing subsequence?
A subsequence of a sequence is a sequence obtained by removing zero or more elements from the original sequence and then concatenating the remaining elements without changing the order. For instance, $(10,30)$ is a subsequence of $(10,20,30)$, but $(20,10)$ is not a subsequence of $(10,20,30)$.
A longest increasing subsequence of a sequence is a subsequence of that sequence with the greatest length among its subsequences that are strictly increasing.
Constraints
- $2\leq N\leq 3\times 10^5$
- $1\leq A_i\leq N$
- $1\leq B_i\leq N$
- $A_i\neq A_j$ and $B_i\neq B_j$, if $i\neq j$.
Input
The input is given from Standard Input in the following format:
$N$ $A_1$ $\ldots$ $A_N$ $B_1$ $\ldots$ $B_N$
Output
Print the maximum possible final value of $\mathrm{LIS}(A) + \mathrm{LIS}(B)$.
Sample Input 1
5 5 2 1 4 3 3 1 2 5 4
Sample Output 1
8
Here is one way to achieve $\mathrm{LIS}(A) + \mathrm{LIS}(B) = 8$.
- Do the operation with $i = 2$. Now, $A = (5,1,2,4,3)$, $B = (3,2,1,5,4)$.
- Do the operation with $i = 1$. Now, $A = (1,5,2,4,3)$, $B = (2,3,1,5,4)$.
- Do the operation with $i = 4$. Now, $A = (1,5,2,3,4)$, $B = (2,3,1,4,5)$.
Here, $A$ has a longest increasing subsequence $(1,2,3,4)$, so $\mathrm{LIS}(A)=4$, and $B$ has a longest increasing subsequence $(2,3,4,5)$, so $\mathrm{LIS}(B)=4$.
Sample Input 2
5 1 2 3 4 5 1 2 3 4 5
Sample Output 2
10
You can decide not to perform the operation at all to achieve $\mathrm{LIS}(A) + \mathrm{LIS}(B) = 10$.