Triple Inversions CodeChef Solution | CodeChef Problem Solution 2022

Triple Inversions CodeChef Solution | CodeChef Problem Solution 2022

For a permutation P$P$ of the integers 1$1$ to N$N$, we define a new array AP$A_P$ of length N−2$N-2$ as follows:

• For 1≤i≤N−2$1\le i\le N-2$, (AP)i$(A_P{)}_i$ denotes the number of inversions in the subarray P[i:i+2]$P[i:i+2]$, i.e, the number of inversions in the array [Pi,Pi+1,Pi+2]$[P_i,P_{i+1},P_{i+2}]$.

You are given an array A$A$ of length N$N$, all of whose elements lie between 0$0$ and 3$3$. Does there exist a permutation P$P$ of length N+2$N+2$ such that AP=A$A_P=A$?

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Input Format

• The first line of input will contain one integer T$T$, the number of test cases. The description of T$T$ test cases follows.

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Triple Inversions CodeChef Solution | CodeChef Problem Solution 2022

• Each test case consists of two lines of input.
• The first line of each test case contains a single integer N$N$, the size of A$A$.
• The second line of each test case contains N$N$ space-separated integers — the values of A1,A2,…,AN$A_1,A_2,\dots ,A_N$.

Output Format

For each test case, output in a single line the answer — 𝚈𝙴𝚂$\mathtt{\text{YES}}$ if a permutation that satisfies the given conditions exists, and 𝙽𝙾$\mathtt{\text{NO}}$ otherwise.

The output is not case sensitive, so for example the strings 𝚈𝙴𝚂, 𝚈𝚎𝚜, 𝚢𝙴𝚂$\mathtt{\text{YES, Yes, yES}}$, etc. will all be treated as correct.

Constraints

• 1≤T≤105$1\le T\le {10}^5$
• 1≤N≤105$1\le N\le {10}^5$
• 0≤Ai≤3$0\le A_i\le 3$
• The sum of N$N$ over all test cases doesn't exceed 105${10}^5$

Sample Input 1 

4
4
0 1 3 2
4
0 1 2 3
4
0 0 0 0
5
0 1 2 1 0

Sample Output 1 

YES
NO
YES
NO

Explanation

Test case 1$1$: Consider P=[1,2,6,5,3,4]$P=[1,2,6,5,3,4]$. It can be verified that AP=[0,1,3,2]$A_P=[0,1,3,2]$. There are other permutations which give the same array — for example [2,3,6,5,1,4]$[2,3,6,5,1,4]$ and [3,4,6,5,1,2]$[3,4,6,5,1,2]$.

Test case 2$2$: It can be verified that no permutation P$P$ of length 6$6$ has AP=[0,1,2,3]$A_P=[0,1,2,3]$.

Test case 3$3$: The only permutation that satisfies the condition is P=[1,2,3,4,5,6]$P=[1,2,3,4,5,6]$.

Test case 4$4$: It can be verified that no permutation P$P$ of length 7$7$ has AP=[0,1,2,1,0]

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