party | votes |
A | 7 |
B | 4 |
C | 2 |
D | 6 |
E | 6 |
Coalition {A, B} has 7 + 4 = 11 votes, which is not a majority. When party C joins coalition {A, B}, however, {A, B, C} becomes a winning coalition with 7+4+2 = 13 votes. So even though C is a small party, it can play an important role.
As a measure of a party's power in a block voting system, John F. Banzhaf III proposed to use the power index.^{1} The key idea is that a party's power is determined by the number of minority coalitions that it can join and turn into a (winning) majority coalition. Note that the empty coalition is also a minority coalition and that a coalition only forms a majority when it has more than half of the total number of votes. In the example just given, a majority coalition must have at least 13 votes.
In an ideal system, a party's power index is proportional to the number of members of that party.
Your task is to write a program that, given an input as shown above, computes for each party its power index.
The first number on a line contains an integer P in [1...20] which equals the number of parties for that test case. This integer is followed by P positive integers, separated by spaces. Each of these integers represents the number of members of a party in the electoral system. The i-th number represents party number i. No electoral system has more than 1000 votes.
where I is the power index of party i.party i has power index I
3 5 7 4 2 6 6 6 12 9 7 3 1 1 3 2 1 1
party 1 has power index 10 party 2 has power index 2 party 3 has power index 2 party 4 has power index 6 party 5 has power index 6 party 1 has power index 18 party 2 has power index 14 party 3 has power index 14 party 4 has power index 2 party 5 has power index 2 party 6 has power index 2 party 1 has power index 3 party 2 has power index 1 party 3 has power index 1