IOI '23 P4 - Beech Tree

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Points: 35 (partial)
Time limit: 1.5s
Memory limit: 1G

Problem types
Allowed languages
C++

Vétyem Woods is a famous woodland with lots of colorful trees. One of the oldest and tallest beech trees is called Ős Vezér.

The tree Ős Vezér can be modeled as a set of N nodes and N1 edges. Nodes are numbered from 0 to N1 and edges are numbered from 1 to N1. Each edge connects two distinct nodes of the tree. Specifically, edge i (1i<N) connects node i to node P[i], where 0P[i]<i. Node P[i] is called the parent of node i, and node i is called a child of node P[i].

Each edge has a color. There are M possible edge colors numbered from 1 to M. The color of edge i is C[i]. Different edges may have the same color.

Note that in the definitions above, the case i=0 does not correspond to an edge of the tree. For convenience, we let P[0]=1 and C[0]=0.

For example, suppose that Ős Vezér has N=18 nodes and M=3 possible edge colors, with 17 edges described by connections P=[1,0,0,0,1,1,1,2,2,3,3,3,4,4,5,10,11,11] and colors C=[0,1,2,3,1,2,3,1,3,3,2,1,1,2,2,1,2,3]. The tree is displayed in the following figure:

Árpád is a talented forester who likes to study specific parts of the tree called subtrees. For each r such that 0r<N, the subtree of node r is the set T(r) of nodes with the following properties:

  • Node r belongs to T(r).
  • Whenever a node x belongs to T(r), all children of x also belong to T(r).
  • No other nodes belong to T(r).

The size of the set T(r) is denoted as |T(r)|.

Árpád recently discovered a complicated but interesting subtree property. Árpád's discovery involved a lot of playing with pen and paper, and he suspects you might need to do the same to understand it. He will also show you multiple examples you can then analyze in detail.

Suppose we have a fixed r and a permutation v0,v1,,v|T(r)|1 of the nodes in the subtree T(r).

For each i such that 1i<|T(r)|, let f(i) be the number of times the color C[vi] appears in the following sequence of i1 colors: C[v0],C[v1],,C[vi1].

(Note that f(1) is always 0 because the sequence of colors in its definition is empty.)

The permutation v0,v1,,v|T(r)|1 is a beautiful permutation if and only if all the following properties hold:

  • v0=r.
  • For each i such that 1i<|T(r)|, the parent of node vi is node vf(i).

For any r such that 0r<N, the subtree T(r) is a beautiful subtree if and only if there exists a beautiful permutation of the nodes in T(r). Note that according to the definition every subtree which consists of a single node is beautiful.

Consider the example tree above. It can be shown that the subtrees T(0) and T(3) of this tree are not beautiful. The subtree T(14) is beautiful, as it consists of a single node. Below, we will show that the subtree T(1) is also beautiful.

Consider the sequence of distinct integers [v0,v1,v2,v3,v4,v5,v6]=[1,4,5,12,13,6,14]. This sequence is a permutation of the nodes in T(1). The figure below depicts this permutation. The labels attached to the nodes are the indices at which those nodes appear in the permutation.

We will now verify that this is a beautiful permutation.

  • v0=1.
  • f(1)=0 since C[v1]=C[4]=1 appears 0 times in the sequence [].
    • Correspondingly, the parent of v1isv0. That is, the parent of node 4 is node 1. (Formally, P[4]=1.)
  • f(2)=0 since C[v2]=C[5]=2 appears 0 times in the sequence [1].
    • Correspondingly, the parent of v2isv0. That is, the parent of 5 is 1.
  • f(3)=1 since C[v3]=C[12]=1 appears 1 time in the sequence [1,2].
    • Correspondingly, the parent of v3 is v1. That is, the parent of 12 is 4.
  • f(4)=1 since C[v4]=C[13]=2 appears 1 time in the sequence [1,2,1].
    • Correspondingly, the parent of v4 is v1. That is, the parent of 13 is 4.
  • f(5)=0 since C[v5]=C[6]=3 appears 0 times in the sequence [1,2,1,2].
    • Correspondingly, the parent of v5 is v0. That is, the parent of 6 is 1.
  • f(6)=2 since C[v6]=C[14]=2 appears 2 times in the sequence [1,2,1,2,3].
    • Correspondingly, the parent of v6 is v2. That is, the parent of 14 is 5.

As we could find a beautiful permutation of the nodes in T(1), the subtree T(1) is a beautiful subtree.

Your task is to help Árpád decide for every subtree of Ős Vezér whether it is beautiful.

Implementation Details

You should implement the following procedure.

Copy
std::vector<int> beechtree(int N, int M, std::vector<int> P, std::vector<int> C)
  • N: the number of nodes in the tree.
  • M: the number of possible edge colors.
  • P,C: arrays of length N describing the edges of the tree.
  • This procedure should return an array b of length N. For each r such that 0r<N, b[r] should be 1 if T(r) is beautiful, and 0 otherwise.
  • This procedure is called exactly once for each test case.

Examples

Example 1

Consider the following call:

Copy
beechtree(4, 2, [-1, 0, 0, 0], [0, 1, 1, 2])

The tree is displayed in the following figure:

T(1), T(2), and T(3) each consist of a single node and are therefore beautiful. T(0) is not beautiful. Therefore, the procedure should return [0,1,1,1].

Example 2

Consider the following call:

Copy
beechtree(18, 3, [-1, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 10, 11, 11], [0, 1, 2, 3, 1, 2, 3, 1, 3, 3, 2, 1, 1, 2, 2, 1, 2, 3])

This example is illustrated in the task description above.

The procedure should return [0,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1].

Example 3

Consider the following call:

Copy
beechtree(7, 2, [-1, 0, 1, 1, 0, 4, 5], [0, 1, 1, 2, 2, 1, 1])

This example is illustrated in the following figure.

T(0) is the only subtree that is not beautiful. The procedure should return [0,1,1,1,1,1,1].

Constraints

  • 3N200 000
  • 2M200 000
  • 0P[i]<i (for each i such that 1i<N)
  • 1C[i]M (for each i such that 1i<N)
  • P[0]=1 and C[0]=0

Subtasks

  1. (9 points) N8 and M500
  2. (5 points) Edge i connects node i to node i1. That is, for each i such that 1i<N, P[i]=i1.
  3. (9 points) Each node other than node 0 is either connected to node 0, or is connected to a node which is connected to node 0. That is, for each i such that 1i<N, either P[i]=0 or P[P[i]]=0.
  4. (8 points) For each c such that 1cM, there are at most two edges of color c.
  5. (14 points) N200 and M500
  6. (14 points) N2 000 and M=2
  7. (12 points) N2 000
  8. (17 points) M=2
  9. (12 points) No additional constraints.

Sample Grader

The sample grader reads the input in the following format:

  • line 1: N M
  • line 2: P[0] P[1]  P[N1]
  • line 3: C[0] C[1]  C[N1]

Let b[0],b[1], denote the elements of the array returned by beechtree. The sample grader prints your answer in a single line, in the following format:

  • line 1: b[0] b[1]

Attachment Package

The sample grader along with sample test cases are available here.


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