Score : 800 points
There are N rabbits on a number line. The rabbits are conveniently numbered 1 through N. The coordinate of the initial position of rabbit i is x_i.
The rabbits will now take exercise on the number line, by performing sets described below. A set consists of M jumps. The j-th jump of a set is performed by rabbit a_j (2≤a_j≤N-1). For this jump, either rabbit a_j-1 or rabbit a_j+1 is chosen with equal probability (let the chosen rabbit be rabbit x), then rabbit a_j will jump to the symmetric point of its current position with respect to rabbit x.
The rabbits will perform K sets in succession. For each rabbit, find the expected value of the coordinate of its eventual position after K sets are performed.
The input is given from Standard Input in the following format:
N x_1 x_2 ... x_N M K a_1 a_2 ... a_M
Print N lines. The i-th line should contain the expected value of the coordinate of the eventual position of rabbit i after K sets are performed. The output is considered correct if the absolute or relative error is at most 10^{-9}.
3 -1 0 2 1 1 2
-1.0 1.0 2.0
Rabbit 2 will perform the jump. If rabbit 1 is chosen, the coordinate of the destination will be -2. If rabbit 3 is chosen, the coordinate of the destination will be 4. Thus, the expected value of the coordinate of the eventual position of rabbit 2 is 0.5×(-2)+0.5×4=1.0.
3 1 -1 1 2 2 2 2
1.0 -1.0 1.0
x_i may not be distinct.
5 0 1 3 6 10 3 10 2 3 4
0.0 3.0 7.0 8.0 10.0