G1. Spinning Round (Easy Version)
divide and conquer; dp; trees
This is the easy version of the problem. The only difference between the two versions are the allowed characters in \(s\). In the easy version, \(s\) only contains the character ?. You can make hacks only if both versions of the problem are solved.You are given a permutation \(p\) of length \(n\). You are also given a string \(s\) of length \(n\), consisting only of the character ?.For each \(i\) from \(1\) to \(n\): Define \(l_i\) as the largest index \(j < i\) such that \(p_j > p_i\). If there is no such index, \(l_i := i\). Define \(r_i\) as the smallest index \(j > i\) such that \(p_j > p_i\). If there is no such index, \(r_i := i\). Initially, you have an undirected graph with \(n\) vertices (numbered from \(1\) to \(n\)) and no edges. Then, for each \(i\) from \(1\) to \(n\), add one edge to the graph: If \(s_i =\) L, add the edge \((i, l_i)\) to the graph. If \(s_i =\) R, add the edge \((i, r_i)\) to the graph. If \(s_i =\) ?, either add the edge \((i, l_i)\) or the edge \((i, r_i)\) to the graph at your choice. Find the maximum possible diameter\(^{\text{∗}}\) over all connected graphs that you can form. Output \(-1\) if it is not possible to form any connected graphs.\(^{\text{∗}}\) Let \(d(s, t)\) denote the smallest number of edges on any path between \(s\) and \(t\).The diameter of the graph is defined as the maximum value of \(d(s, t)\) over all pairs of vertices \(s\) and \(t\).
Each test contains multiple test cases. The first line of input contains a single integer \(t\) (\(1 \le t \le 2 \cdot 10^4\)) — the number of test cases. The description of the test cases follows.The first line of each test case contains a single integer \(n\) (\(2 \le n \le 4 \cdot 10^5\)) — the length of the permutation \(p\). The second line of each test case contains \(n\) integers \(p_1,p_2,\ldots, p_n\) (\(1 \le p_i \le n\)) — the elements of \(p\), which are guaranteed to form a permutation.The third line of each test case contains a string \(s\) of length \(n\). It is guaranteed that it consists only of the character ?.It is guaranteed that the sum of \(n\) over all test cases does not exceed \(4 \cdot 10^5\).
For each test case, output the maximum possible diameter over all connected graphs that you form, or \(-1\) if it is not possible to form any connected graphs.
In the first test case, here are some possible connected graphs that you can form (the labels are indices): In the second test case, the only connected graph has a diameter of \(1\).