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April 2nd, 2014, 04:16 AM  #1 
Newbie Joined: Mar 2014 Posts: 2 Thanks: 0  Find cardinality of set
I have faced the following problem recently: We have a sequence A of M consecutive integers, beginning at A[1] = 1: 1,2,...M (example: M = 8 , A = 1,2,3,4,5,6,7,8 ) We have the set T consisting of all possible subsequences made from L_T consecutive terms of A, which do not overlap. (example L_T = 3 , subsequences are {1,2,3},{4,5,6},{7,8,9},...). Let's call the elements of T "tiles". We have the set S consisting of all possible subsequences of A that have length L_S. ( example L_S = 4, subsequences like {1,2,3,4} , {1,3,7,8} ,...{4,5,7,8} ). We say that an element s of S can be "covered" by K "tiles" of T if there exist K tiles in T such that the union of their sets of terms contains the terms of s as a subset. For example, subsequence {1,2,3} is possible to cover with 2 tiles of length 2 ({1,2} and {3,4}), while subsequnce {1,3,5} is not possible to "cover" with 2 "tiles" of length 2, but is possible to cover with 2 "tiles" of length 3 ({1,2,3} and {4,5,6}). Let C be the subset of elements of S that can be covered by K tiles of T. Find the cardinality of C given M, L_T, L_S, K. Any ideas would be appreciated how to tackle this problem. 

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algorithm, cardinality, find, set 
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