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Problem Name: Circuit
Description:
You are given a circuit board consisting of N component slots and R external switches.
Each slot contains either an AND gate (&) or an OR gate (|). The actual type of each slot is hidden.
Your task is to determine the true type of every slot by interacting with the judge.
Specifications of the Circuit Board:
- The board consists of 2N+1 switches numbered 0..2N, each with an ON/OFF state, outputting 0 or 1.
- The board also consists of N component slots numbered 0..N-1, each outputting 0 or 1.
- The outputs are determined from the highest index down to 0, according to the following rules:
* For j = 2N, 2N-1, ..., N:
- If switch j is OFF, it outputs 0.
- If switch j is ON, it outputs 1.
* For j = N-1, N-2, ..., 0:
- Let the output of slot j be x.
- If switch j is OFF, it outputs x.
- If switch j is ON, it outputs 1-x.
* For i = N-1, N-2, ..., 0:
- Slot i is connected to two switches Ui, Vi (i < Ui < Vi ≤ 2N).
- If slot i is AND, it outputs min(output(Ui), output(Vi)).
- If slot i is OR, it outputs max(output(Ui), output(Vi)).
- For each j = 1..2N, exactly one slot i satisfies Ui=j or Vi=j.
- The output of the circuit is defined as the output of switch 0.
Interaction:
At the beginning of the interaction, the judge prints N and R, followed by N lines each containing Ui Vi.
This describes the wiring of the board. Then your program must interact as follows:
1. Query:
You may print a line of the form: "? s"
where s is a binary string of length 2N+1. Each character sets the ON/OFF state of the corresponding switch.
The judge replies with a single line containing 0 or 1, the output of the circuit (switch 0).
2. Answer:
When you have determined the types of all slots, you must print a line of the form: "! t"
where t is a string of length N, each character being & or |, representing your guess of the hidden circuit.
If your guess is completely correct, the judge accepts. Otherwise, you receive Wrong Answer.
Constraints:
- 1 ≤ N ≤ 8000
- 1 ≤ R ≤ min(N, 120)
- i < Ui < Vi ≤ 2N for all 0 ≤ i < N
- For each j=1..2N, exactly one i satisfies Ui=j or Vi=j
- At most 5000 queries may be made. Exceeding this limit results in Wrong Answer.
Special Requirement:
- Your program must be written in C++.
Scoring:
- If the number of queries ≤ 900, you receive full score (100 points).
- If the number of queries ≥ 5000, you receive 0 points.
- Otherwise, the score is computed by linear interpolation:
Score = (5000 - queries) / (5000 - 900) * 100
Sample Interaction:
Suppose the input file contains:
3 2
3 4
2 4
0 1
&&|
The judge first outputs:
3 2
3 4
2 4
0 1
(Here the hidden true circuit is T = "&&|".)
Then the interaction may proceed as follows (lines beginning with '#' are explanations):
? 01001
# Player asks a query with binary string s.
1
# Judge replies that the circuit output is 1.
? 11111
# Another query.
0
# Judge replies with 0.
! &&|
# Player outputs the final answer.
# The hidden circuit matches the answer, so the judge accepts.
|