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The Cows have constructed a randomized stink bomb for the purpose of driving away the Piggies. The Piggy civilization consists of N (2 <= N <= 300) Piggy cities conveniently numbered 1..N connected by M (1 <= M <= 44,850) bidirectional roads specified by their distinct endpoints A_j and B_j (1 <= A_j <= N; 1 <= B_j <= N). Piggy city 1 is always connected to at least one other city.
The stink bomb is deployed in Piggy city 1. Each hour (including the first one), it has a P/Q (1 <= P <= 1,000,000; 1 <= Q <= 1,000,000; P <= Q) chance of polluting the city it occupies. If it does not go off, it chooses a random road out of the city and follows it until it reaches a new city. All roads out of a city are equally likely to be chosen.
Because of the random nature of the stink bomb, the Cows are wondering which cities are most likely to be polluted. Given a map of the Piggy civilization and the probability that the stink bomb detonates in a given hour, compute for each city the probability that it will be polluted.
For example, suppose that the Piggie civilization consists of two cities connected together and that the stink bomb, which starts in city 1, has a probability of 1/2 of detonating each time it enters a city:
We have the following possible paths for the stink bomb (where the last entry is the ending city):
1: 1 2: 1-2 3: 1-2-1 4: 1-2-1-2 5: 1-2-1-2-1 etc.
To find the probability that the stink bomb ends at city 1, we can add up the probabilities of taking the 1st, 3rd, 5th, ... paths above (specifically, every odd-numbered path in the above list). The probability of taking path number k is exactly (1/2)^k - the bomb must not remain in its city for k - 1 turns (each time with a probability of 1 - 1/2 = 1/2) and then land in the last city (probability 1/2).
So our probability of ending in city 1 is represented by the sum 1/2 + (1/2)^3 + (1/2)^5 + ... . When we sum these terms infinitely, we will end up with exactly 2/3 as our probability, approximately 0.666666667. This means the probability of landing in city 2 is 1/3, approximately 0.333333333.
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