AMR15C - Editorial



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Problem Setter: Suhash
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Editorialist: Sunny Aggarwal
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Constructive Algorithm, Implementation, Observation, Simulation.


Given 2 integers N and K and we are asked to find the lexographically smallest permutation of first N natural number such that abs(i - position_i) \ge K for every i \in [1, N] if such a permutation exists. Print -1 otherwise.


We can claim that when K \le \lfloor{N/2}\rfloor, lexographical smallest solution will have the following form for each 2 * K size block i.e lexographically smallest answer will have first 2 * K elements present in the following order and similar order for each set of 2 * K elements.

K+1, K+2, K+3... ,2*K, ... 1, 2, 3, ... K

Note that if N\%(2*K) == 0 then we can perform shuffling as depicted above for each (N/(2*K)) set of 2*K size each otherwise we have to shuffle the remaining elements with the elements present before them to make a block of size 2*K and need to shuffle them as we did above.

We can also claim that no answer exist if K \gt \lfloor{N/2}\rfloor because the number that we can put in the position 1 to \lfloor{N/2}\rfloor are greater than \lceil{N/2}\rceil and the number that we can put in the position \lceil{N/2}\rceil+1 to N are less than \lfloor{N/2}\rfloor+1 . So we can’t put any number in position \lceil{N/2}\rceil.