1551: Bear and Tree Jumps
Time Limit: 2 Sec Memory Limit: 256 MBSubmit: 0 Solved: 0
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Description
A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.
Limak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.
Limak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.
For a pair of vertices (s,t) we define f(s,t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s,t) over all pairs of vertices (s,t) such that s<t.
The first line of the input contains two integers n and k (2≤n≤200000, 1≤k≤5)− the number of vertices in the tree and the maximum allowed jump distance respectively.
The next n-1 lines describe edges in the tree. The i-th of those lines contains two integers ai and bi (1≤ai,bi≤n)− the indices on vertices connected with i-th edge.
It's guaranteed that the given edges form a tree.
Print one integer, denoting the sum of f(s,t) over all pairs of vertices (s,t) such that s<t.
6 2
1 2
1 3
2 4
2 5
4 6
20
13 3
1 2
3 2
4 2
5 2
3 6
10 6
6 7
6 13
5 8
5 9
9 11
11 12
114
3 5
2 1
3 1
3
In the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself).
There are 
pairs of vertices (s,t) such that s<t. For 5 of those pairs Limak would need two jumps: (1,6),(3,4),(3,5),(3,6),(5,6). For other 10 pairs one jump is enough. So, the answer is 5·2+10·1=20.
In the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s<t), so the answer is 3·1=3.