🚀 go-pugleaf

I met a problem recently. 

given an integer vector (x1, x2, ..., xn), could you find 
an efficient ALGORITHM to get minimum M and  an integer vector
(y1, y2, ..., yn) so that the following holds: 
(1). (y1, ..., yn) is a PERMUTATION VECTOR from [0, M-1]: 
     yi does not equal to yj, if i does not equal to j; 
     all yi belongs to [0,M-1];
(2)  ((x1+y1) mod M, (x2+y2) mod M, ..., (xn+yn) mod M) is also a 
     PERMUTATION VECTOR from [0, M-1] as defined above. 

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because (x1, x2, ..., xn) is equivalent to (x1+1, x2+2, ..., xn+n) 
in term of finding M and solution, we can assume xi>=0 for 1<=i<=n.

I can get some kind of upper-bound about M, such as 
n+max{xi}-min{xi}. 

A follow-up question is, for multiple integer vectors, find out
common M and vector Y which has the same property. This question 
is most interesting to me. 

could anyone point some references about it? it looks like so 
plain! any results in PERMUTATION GROUP can help? 

I guess it is NP-complete. Any NP-complete problem in the similiar
manner? 




-- 
Duanyang Guo	Computer Science Department, Columbia University