CF1485E Move and Swap

You are given $ n - 1 $ integers $ a_2, \dots, a_n $ and a tree with $ n $ vertices rooted at vertex $ 1 $ . The leaves are all at the same distance $ d $ from the root. Recall that a tree is a connected undirected graph without cycles. The distance between two vertices is the number of edges on the simple path between them. All non-root vertices with degree $ 1 $ are leaves. If vertices $ s $ and $ f $ are connected by an edge and the distance of $ f $ from the root is greater than the distance of $ s $ from the root, then $ f $ is called a child of $ s $ . Initially, there are a red coin and a blue coin on the vertex $ 1 $ . Let $ r $ be the vertex where the red coin is and let $ b $ be the vertex where the blue coin is. You should make $ d $ moves. A move consists of three steps: - Move the red coin to any child of $ r $ . - Move the blue coin to any vertex $ b' $ such that $ dist(1, b') = dist(1, b) + 1 $ . Here $ dist(x, y) $ indicates the length of the simple path between $ x $ and $ y $ . Note that $ b $ and $ b' $ are not necessarily connected by an edge. - You can optionally swap the two coins (or skip this step). Note that $ r $ and $ b $ can be equal at any time, and there is no number written on the root. After each move, you gain $ |a_r - a_b| $ points. What's the maximum number of points you can gain after $ d $ moves?

The first line contains a single integer $ t $ ( $ 1 \le t \le 10^4 $ ) — the number of test cases. The first line of each test case contains a single integer $ n $ ( $ 2 \leq n \leq 2 \cdot 10^5 $ ) — the number of vertices in the tree. The second line of each test case contains $ n-1 $ integers $ v_2, v_3, \dots, v_n $ ( $ 1 \leq v_i \leq n $ , $ v_i \neq i $ ) — the $ i $ -th of them indicates that there is an edge between vertices $ i $ and $ v_i $ . It is guaranteed, that these edges form a tree. The third line of each test case contains $ n-1 $ integers $ a_2, \dots, a_n $ ( $ 1 \leq a_i \leq 10^9 $ ) — the numbers written on the vertices. It is guaranteed that the sum of $ n $ for all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, print a single integer: the maximum number of points you can gain after $ d $ moves.

In the first test case, an optimal solution is to: - move $ 1 $ : $ r = 4 $ , $ b = 2 $ ; no swap; - move $ 2 $ : $ r = 7 $ , $ b = 6 $ ; swap (after it $ r = 6 $ , $ b = 7 $ ); - move $ 3 $ : $ r = 11 $ , $ b = 9 $ ; no swap. The total number of points is $ |7 - 2| + |6 - 9| + |3 - 9| = 14 $ . In the second test case, an optimal solution is to: - move $ 1 $ : $ r = 2 $ , $ b = 2 $ ; no swap; - move $ 2 $ : $ r = 3 $ , $ b = 4 $ ; no swap; - move $ 3 $ : $ r = 5 $ , $ b = 6 $ ; no swap. The total number of points is $ |32 - 32| + |78 - 69| + |5 - 41| = 45 $ .