F. Christmas Game
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Alice and Bob are going to celebrate Christmas by playing a game with a tree of presents. The tree has $$$n$$$ nodes (numbered $$$1$$$ to $$$n$$$, with some node $$$r$$$ as its root). There are $$$a_i$$$ presents are hanging from the $$$i$$$-th node.

Before beginning the game, a special integer $$$k$$$ is chosen. The game proceeds as follows:

  • Alice begins the game, with moves alternating each turn;
  • in any move, the current player may choose some node (for example, $$$i$$$) which has depth at least $$$k$$$. Then, the player picks some positive number of presents hanging from that node, let's call it $$$m$$$ $$$(1 \le m \le a_i)$$$;
  • the player then places these $$$m$$$ presents on the $$$k$$$-th ancestor (let's call it $$$j$$$) of the $$$i$$$-th node (the $$$k$$$-th ancestor of vertex $$$i$$$ is a vertex $$$j$$$ such that $$$i$$$ is a descendant of $$$j$$$, and the difference between the depth of $$$j$$$ and the depth of $$$i$$$ is exactly $$$k$$$). Now, the number of presents of the $$$i$$$-th node $$$(a_i)$$$ is decreased by $$$m$$$, and, correspondingly, $$$a_j$$$ is increased by $$$m$$$;
  • Alice and Bob both play optimally. The player unable to make a move loses the game.

For each possible root of the tree, find who among Alice or Bob wins the game.

Note: The depth of a node $$$i$$$ in a tree with root $$$r$$$ is defined as the number of edges on the simple path from node $$$r$$$ to node $$$i$$$. The depth of root $$$r$$$ itself is zero.

Input

The first line contains two space-separated integers $$$n$$$ and $$$k$$$ $$$(3 \le n \le 10^5, 1 \le k \le 20)$$$.

The next $$$n-1$$$ lines each contain two integers $$$x$$$ and $$$y$$$ $$$(1 \le x, y \le n, x \neq y)$$$, denoting an undirected edge between the two nodes $$$x$$$ and $$$y$$$. These edges form a tree of $$$n$$$ nodes.

The next line contains $$$n$$$ space-separated integers denoting the array $$$a$$$ $$$(0 \le a_i \le 10^9)$$$.

Output

Output $$$n$$$ integers, where the $$$i$$$-th integer is $$$1$$$ if Alice wins the game when the tree is rooted at node $$$i$$$, or $$$0$$$ otherwise.

Example
Input
5 1
1 2
1 3
5 2
4 3
0 3 2 4 4
Output
1 0 0 1 1
Note

Let us calculate the answer for sample input with root node as 1 and as 2.

Root node 1

Alice always wins in this case. One possible gameplay between Alice and Bob is:

  • Alice moves one present from node 4 to node 3.
  • Bob moves four presents from node 5 to node 2.
  • Alice moves four presents from node 2 to node 1.
  • Bob moves three presents from node 2 to node 1.
  • Alice moves three presents from node 3 to node 1.
  • Bob moves three presents from node 4 to node 3.
  • Alice moves three presents from node 3 to node 1.

Bob is now unable to make a move and hence loses.

Root node 2

Bob always wins in this case. One such gameplay is:

  • Alice moves four presents from node 4 to node 3.
  • Bob moves four presents from node 5 to node 2.
  • Alice moves six presents from node 3 to node 1.
  • Bob moves six presents from node 1 to node 2.

Alice is now unable to make a move and hence loses.