Problem Link - Messages
Problem Statement:
Chef, now an elite spy, is in trouble! He needs to quickly send a message to you, who is in his Headquarters. Chef’s message is a sequence of N non-negative integers.
To avoid the risk of the message being intercepted by the enemy, you and Chef have decided on an elaborate plan. First, Chef will construct his message, which can be represented as the array A0: A0,0, A0,1, A0,2, ..., A0,N-1. Then, he will construct 2^N - 1 fake messages, each of which is also a sequence of N non-negative integers. The fake messages are numbered from 1 to 2^N - 1, with the i-th fake message represented by the array Ai. The i-th fake message is constructed via the following process:
-
Let the binary representation of
ibe denoted asBi. -
If the
j-thbit ofBiis0,Ai,j = A0,j. -
Otherwise,
Ai,jcan be any non-negative integer exceptA0,j.
In other words, if Bi,j = 0, then Ai,j = A0,j, and if Bi,j = 1, then Ai,j ≠ A0,j.
After the real message and the 2^N - 1 fake messages (for a total of M = 2^N) are constructed, they are all put together and randomly shuffled so that it is unknown which was the real message.
Your task is this: given the final randomly shuffled sequence of messages, determine which message(s) could have been the real message. In particular, in some cases, it may be possible that more than one of the messages could have been the real message. (For example, in the preceding example, apart from message 2, message 1 also could have been the real message.) It is also possible that Chef made a mistake when generating the messages and so none of the messages could have been the real message. In this case, report this.
Approach:
The goal is to identify which of the shuffled messages could have been the real one. Start by analyzing the frequencies of each integer at every position across all messages. The real message must have integers at positions where the majority of messages ( specifically M/2 ) agree, as fake messages only alter values based on the binary representation of their index. For each position in the messages, determine the integer that appears M/2 times (if any) and consider it as a possible value for the corresponding position in the real message. Then, verify the integrity of the identified message by checking whether all fake messages satisfy the rules of generation from this candidate. If any violations occur, the candidate is invalid, and no real message exists. Finally, if a valid real message is found, identify all messages that match the candidate and output their indices.
Time Complexity:
The time complexity is O(M \times N), where M is the number of messages and N is the length of each message.
Space Complexity:
The space complexity is O(M \times N), primarily for storing the messages and auxiliary data structures.