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Charles M. - BWT Notes BWA datastructure lecture # Suffix array # Burrows Wheeler Transform - variation on suffix array, useful for data compression and indexing # FM index - uses both suffix arrays and BWT for fast lookup suffix array definition: X[0:n] with a '$' character spliced in the range of [0,n]. e.g. string S: z a b c d e index: 0,1,2,3,4,5 # BWT definition: B[i] = X[S[i]-1] 1) look at suffixes of a string a de cde bcde abcde zabcde 2) sort order of suffixes: Pos Suffix 6 $ 1 abcde$ 2 bcde$ 3 cde$ 4 de$ 5 e$ 0 zabcde$ 3) rotate the string: zabcde[$] abcde$[z] bcde$z[a] . . . $zabcd[e] ^ (last column is the burrows wheeler transform) left column (since it is in alphabet order) just record where a character starts since there is repetition of a character. -> $ $ $ -> A A A A ... -> C C ... 4) Record Occupancy: Definitions: C(a) # Where char's start. Occ(a,i) # Array of number of char's before i in B. (e.g. # of a's in B[0,i]) - Can store part of the Occupancy array to obtain greater compression at the cost of slightly greater cpu time. B # BWT array Use last_to_first(i) function which allows you to take the BWT string and find the 1st column row index of a BWT character. Use that row index to find the previous character in the BWT string (It's the last column character of the same row index). Occupancy function: C(B[i]) + Occ(B[i],i) # Search algorithm Given Pattern P R(P): where pattern starts Rbar(P): where pattern ends Rbar(aP) = C(a) + Occ(a, Rbar(P)) R(aP) = C(a) + Occ(a,R(P)-1)+1
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