JOIN

 Problem Statement

Problem Statement for InducedSubgraphs

### Problem Statement

Given an undirected graph G and a subset of its vertices S, the subgraph of G induced by S (denoted G/S) is defined as the subgraph of the graph G such that the following two conditions are satisfied:

• The set of vertices of the subgraph G/S is exactly S.
• For any two vertices x, y in S, there is an edge {x,y} in G/S if and only if there is such an edge in G.
In other words, the induced subgraph always contains all the edges of G it may contain, given its vertices.

In this problem, you are given a tree G containing n vertices and a positive integer k. Initially, the vertices of G are numbered from 0 to n-1, inclusive. The objective is to change this numbering so that all induced subgraphs over {i, i+1, .., i+k-1} are connected, for all 0 <= i <= n-k. How many ways of renumbering are there?

The initial tree G is given as two int[]s edge1 and edge2, each containing n-1 elements. These two int[]s describe the endpoints of edges. For each i, there is an edge between the vertices edge1[i] and edge2[i]. Let C be the number of ways to renumber the vertices that satisfy the condition above. Your method must return (C modulo 1,000,000,009).

### Definition

 Class: InducedSubgraphs Method: getCount Parameters: int[], int[], int Returns: int Method signature: int getCount(int[] edge1, int[] edge2, int k) (be sure your method is public)

### Notes

-A tree is a connected graph with no cycles.

### Constraints

-edge1 will contain between 1 and 40 elements, inclusive.
-edge2 will contain the same number of elements as edge1.
-Each element of edge1 and edge2 will be between 0 and n-1, inclusive, where n is (the number of elements in edge1) + 1.
-A graph represented by edge1 and edge2 will be a tree.
-k will be between 1 and n, inclusive.

### Examples

0)

 `{0, 1}` `{1, 2}` `2`
`Returns: 2`
 Initially, the graph looks as follows: ``` 0-1-2 ``` There are two correct ways to assign the new numbers to its vertices: ``` 0-1-2 ``` and ``` 2-1-0 ```
1)

 `{0, 1, 3}` `{2, 2, 2}` `3`
`Returns: 12`
 The given graph: ``` 0-2-1 | 3 ``` Possible numberings are as follows. ``` 0-1-2 0-1-3 2-1-3 3-1-2 2-1-0 3-1-0 | | | | | | 3 2 0 0 3 2 0-2-3 1-2-3 3-2-1 3-2-0 0-2-1 1-2-0 | | | | | | 1 0 0 1 3 3 ```
2)

 `{5, 0, 1, 2, 2}` `{0, 1, 2, 4, 3}` `3`
`Returns: 4`
 The given graph: ``` 5-0-1-2-4 | 3 ``` Possible ways: ``` 0-1-2-3-4 0-1-2-3-5 | | 5 4 5-4-3-2-1 5-4-3-2-0 | | 0 1 ```
3)

 `{0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6}` `{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}` `11`
`Returns: 481904640`
4)

 ```{5, 9, 4, 10, 10, 0, 7, 6, 2, 1, 11, 8} ``` ```{0, 0, 10, 3, 0, 6, 1, 1, 12, 12, 7, 11} ``` `6`
`Returns: 800`
5)

 ```{0, 5, 1, 0, 2, 3, 5} ``` ```{4, 7, 0, 6, 7, 5, 0} ``` `3`
`Returns: 0`
6)

 `{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}` `{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}` `1`
`Returns: 890964601`

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This problem was used for:
Single Round Match 562 Round 1 - Division I, Level Three