E. Kia Bakes a Caketime limit per test6 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard output You are given a binary string $$$s$$$ of length $$$n$$$ and a tree $$$T$$$ with $$$n$$$ vertices. Let $$$k$$$ be the number of 1s in $$$s$$$. We will construct a complete undirected weighted graph with $$$k$$$ vertices as follows: For each $$$1\le i\le n$$$ with $$$s_i = \mathtt{1}$$$, create a vertex labeled $$$i$$$. For any two vertices labeled $$$u$$$ and $$$v$$$ that are created in the above step, define the edge weight between them $$$w(u, v)$$$ as the distance$$$^{\text{∗}}$$$ between vertex $$$u$$$ and vertex $$$v$$$ in the tree $$$T$$$. A simple path$$$^{\text{†}}$$$ that visits vertices labeled $$$v_1, v_2, \ldots, v_m$$$ in this order is nice if for all $$$1\le i\le m - 2$$$, the condition $$$2\cdot w(v_i, v_{i + 1})\le w(v_{i + 1}, v_{i + 2})$$$ holds. In other words, the weight of each edge in the path must be at least twice the weight of the previous edge. Note that $$$s_{v_i} = \mathtt{1}$$$ has to be satisfied for all $$$1\le i\le m$$$, as otherwise, there would be no vertex with the corresponding label.For each vertex labeled $$$i$$$ ($$$1\le i\le n$$$ and $$$s_i = \mathtt{1}$$$) in the complete undirected weighted graph, determine the maximum number of vertices in any nice simple path starting from the vertex labeled $$$i$$$.$$$^{\text{∗}}$$$The distance between two vertices $$$a$$$ and $$$b$$$ in a tree is equal to the number of edges on the unique simple path between vertex $$$a$$$ and vertex $$$b$$$.$$$^{\text{†}}$$$A path is a sequence of vertices $$$v_1, v_2, \ldots, v_m$$$ such that there is an edge between $$$v_i$$$ and $$$v_{i + 1}$$$ for all $$$1\le i\le m - 1$$$. A simple path is a path with no repeated vertices, i.e., $$$v_i\neq v_j$$$ for all $$$1\le i < j\le m$$$.InputEach test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows. The first line of each test case contains a single integer $$$n$$$ ($$$1\le n\le 7\cdot10^4$$$) — the length of the binary string $$$s$$$ and the number of vertices in the tree $$$T$$$.The second line of each test case contains a binary string with $$$n$$$ characters $$$s_1s_2\ldots s_n$$$ ($$$s_i\in \{\mathtt{0}, \mathtt{1}\}$$$) — the string representing the vertices to be constructed in the complete undirected weighted graph.Each of the next $$$n - 1$$$ lines contains two integers $$$u$$$ and $$$v$$$ ($$$1\le u, v\le n$$$) — the endpoints of the edges of the tree $$$T$$$.It is guaranteed that the given edges form a tree.It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$7\cdot10^4$$$. OutputFor each test case, output $$$n$$$ integers, the $$$i$$$-th integer representing the maximum number of vertices in any nice simple path starting from the vertex labeled $$$i$$$. If there is no vertex labeled $$$i$$$, i.e., $$$s_i = \mathtt{0}$$$, output $$$-1$$$ instead.ExampleInput35011111 22 33 44 517011010111101011011 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 172011 2Output-1 3 3 3 3
-1 5 4 -1 4 -1 5 5 5 5 -1 4 -1 5 5 -1 3
-1 1
NoteIn the first test case, the tree $$$T$$$ and the constructed graph are as follows: Left side is the tree $$$T$$$ with selected nodes colored yellow. The right side is the constructed complete graph. The nice path shown in the diagram is $$$3\rightarrow 4\rightarrow 2$$$. The path is nice as $$$w(4, 2) = 2$$$ is at least twice of $$$w(3, 4) = 1$$$. Extending the path using $$$2\rightarrow 5$$$ is not possible as $$$w(2, 5) = 3$$$ is less than twice of $$$w(4, 2) = 2$$$.In the second test case, the tree $$$T$$$ is a simple path of length $$$17$$$. An example of a nice path starting from the vertex labeled $$$2$$$ is $$$2\rightarrow 3\rightarrow 5\rightarrow 9\rightarrow 17$$$, which has edge weights of $$$1, 2, 4, 8$$$ doubling each time.