Question

Problem 1. (Calculating Edit Distance Using Dynamic Programming) A direct implementation of the above recursive scheme will work, but it is spectacularly inefficient. If both input strings have N characters, then the number of recursive calls will exceed 2N. To overcome this performance bug, we use dynamic programming. Dynamic programming is a powerful algorithmic paradigm, first introduced by Bellman in the context of operations research, and then applied to the alignment of biological sequences by Needleman and Wunsch. Dynamic programming now plays the leading role in many computational problems, including control theory, financial engineering, and bioinformatics, including BLAST (the sequence alignment program almost universally used by molecular biologists in their experimental work). The key idea of dynamic programming is to break up a large computational problem into smaller subproblems, store the answers to those smaller subproblems, and, eventually, use the stored answers to solve the original problem. This avoids recomputing the same quantity over and over again. Instead of using recursion, use a nested loop that calculates opt [1] [j] in the right order so that opt [1 + 1] [j + 1], opt (i + 1] [j], and opt[i] [j + 1) are all computed before we try to compute opt(1) [j]. Write a program edit_distance.py that reads strings x and y from standard input and computes the edit-distance matrix opt as described above. The program should output x, y, the dimensions (number of rows and columns) of opt, and opt itself, using the following format: The first and second lines should contain the strings x and y. • The third line should contain the dimensions of the opt matrix, separated by a space. The following lines should contain the rows of the opt matrix, each ending in a newline character. Use stdio.writof with the format string '%3d to write out the elements of the matrix.

Answer 1

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num Ros, numals, Ope) def piatoptValues ( Stdfo coteln () Stdfo orteln (y) Sidfo. coftteln ( Str CnumRoos) "4 Str (numcols)) for Yoo in ange CnumRos): for col fn range Cnumcols): Staocortel d,int (Opt Erouco Std to.corttef Cn) TO read the tert fle from the Consele flesnopen CSYS. agv , #Read X Xfiein reodLine . StP Cin') #Read Y YAlean Tcad Line C) Strtp(in') #coe can fird the lergth of and y ad Store # the values fn M ard N M-len Ca) N= lenty) Opt L

Create CM+1x(Nt) Mati ept oth the elements al2ed to conere M and N are lergths of xandy OPELLO for n rarge(N41 for k in arge Cti J fnitiali2e Opt CMC (3<n) and opt iCNI CR to appyoprffote values for 3in ranae Co, N): OPE CMIC33 CN-) for fn rarge Co,M) OPt CiIN2 (Mt) Compute the Yest 6f opt TO for fn range CM-1,-1.-): for Ph range (N-1,-4, -1): XC13= y 33 OPt 33 31min (opt Cit3+d, opt C9t th+a, SpECtut+) the too stirgs atnae ore #not al then get the minimum value #as shoron.

else: OPE Ci C3 min Copt t L+ +oPt Cft314 2, OPE I 3+) Call the functen to Rsplay the regred output PrintOpt Values Ciy,Mt, Nt,OPt) output $ Pthens eait distance py data/ eampleto.tyt AACAGTTACC TAAG.GTCA 11 12 13 1s 16 81 20 ic ti (3 16 te It 12 14 t2 o 3 3 It 13 5 1 3 2 2.