Given a pack of 2^N cards (0 … 2^N – 1), shuffle it in N steps. At step k (0 < k < N) we divide the deck into 2k equal-sized decks. Each one of those decks is reordered by having all the cards that lie on even positions first, followed by all cards that lie on odd positions (the order is preserved in each one of the two subsequences). Now, we are given a key (index). We have to answer the card on that position (0-based indexing). Examples:
Input : N = 3 (Size = 2^N), Key = 3 Output : 6 Explanation : Pack : 0 1 2 3 4 5 6 7 Shuffle 1 : 0 2 4 6|1 3 5 7 Shuffle 2 : 0 4|2 6|1 5|3 7 Card at index 3 : 6
Method 1 : We can simply simulate the whole process and find the exact order of the cards after all the N shuffles are done.
Time Complexity : O(N * 2^N)
Method 2 :
Let us try to find the binary representation of Key and the final answer and try to spot some observations based on it.
Let N = 3
Below is the table :
It is clearly visible that the answer is the reverse of binary representation of Key.
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