Week 10: test contains all material from Week 1– Week 8, not including this week

1. . • Week 10: test contains all material from Week 1– Week 8, not including this week

2. Week 9: String Matching Allowing Errors • Problem: Given a short pattern P of length m, a long text T of length n, and a maximum allowed number of errors k, find all the text positions where the pattern occurs with at most k errors.

3. Dynamic Programming • C[i,j] be the number of errors allowed. • Three kinds of error: mismatch (a, b), insertion( a, )and deletion ( , a).

4. The matrix The dynamic programming algorithm search ‘survey’ in the text ‘surgery’ with two errors. Bold entries indicate matching positions. Running time O(nm).

5. String Matching Allowing Errors • O(nk) algorithm, where n is the length of the pattern and k the number of errors allowed. • C[i,q] the number of errors for c[i,i+q], • q=-k, -(k+1), …,0, 1, 2, …, k. • C[i,i+q]= C[i-1, i+q-1] if pi=ti+q. • else C[I,i+q]=1+min{C[i-1,i+q], C[I, i+q-1], C[i-1,i+q-1]}.

6. An NFA for approximate string matching of the pattern ‘survey’ with two errors. The shaded states are those active after reading the text ‘surgery’. Unlabelled transitions match any character.(Also O(nk) time.)

7. Regular expressions Matching • Regular expression: • Any letter x in {},is a regular expression, where  is the set of all letters. • if A and B are regular expression, then A|B, A.B and (A)* are regular expressions.

8. Regular expressions Matching • Given an regular expression E and a string T, find all the substrings in T that match E. • Let d(i) be the set of all states in the automaton that can be reached after T1T2…Ti is accepted. • Given d(i), d(i+1) can be computed easily. • There is a starting and final state in the automaton. • Whenever the final state is reach, we find a substring in T that match the expression.

13. Example: • E=(A|AA).(B|AB). • T=ABBAAAB. • D(1)={a,b,d,c, e, f, g, i} • D(2)={a,b,d, h,l }, • D(3)={a,b,d}. • D(4)={a,b,d,c,e,f,g,i} • D(5)={a,b,d,c,e,f,g, i,j} • D(6)={a,b,d,c,e,f,g,i,j} • D(7)={a, h, k,l}.

14. Running time • O(n2), where n is the size of the automaton since d(s, i) could contain O(n) states.

15. Choices of Search Engines • Many search engines exist to compete for users • The results are not necessarily the same • Different users prefer different search engines • Search results may, in the future, be biased towards paid advertisements.

16. GOOGLE: City University • CityUniversity London - the University for business and the ...... The University for business and the professions. Contact Us | About CityUniversity| Maps & Directions, AZ Index | Site Map | Help. Prospective Students. ...Description: Official site with information about courses, research, schools, and departments. Includes details... • CityUniversity HomeA University which believes in forward-thinking business and leadership skills, exposing students to the latest technology in all courses of study. Description: A private, nonprofit institution founded to serve working adults wanting to pursue educational opportunit... • Welcome to Dublin CityUniversity... here... Dublin CityUniversity, Dublin 9, Ireland. Tel. +353 (0) 1700 5000, Fax. +353 (0) 1 836 0830. Page updated: 02/09/03 legal. ...Description: Information on facilities, services, degree courses, research, the campus, student life, the library... • The CityUniversity of New YorkDescription: The University's Main Website. • CityUniversity of Hong KongAD Working Group lends staff, students an ear. Several issues stillevoked strong emotions as staff members and students voiced their ...Description: Formerly the City Polytechnic of Hong Kong. Includes information on university, links to learning... • Welcome to Oklahoma CityUniversity... At Oklahoma CityUniversity, our students come first. Our ... Oklahoma CityUniversity offers a quality, values-centered education. United ...Description: Admissions, academic programs and alumni relations, sports, services, news, calendar of events and...

17. YAHOO • CityUniversity London - the University for business and the ...... The University for business and the professions. Contact Us | About CityUniversity | Maps & Directions, AZ Index | Site Map | Help. Prospective Students. ... • City UniversityA University which believes in forward-thinking business and leadership skills, exposing students to the latest technology in all courses of study.www.cityu.edu/ - 3k - Cached - More pages from this site • Dublin City University... here... Dublin CityUniversity, Dublin 9, Ireland. Tel. +353 (0) 1 700 5000, Fax. +353 (0) 1 836 0830. Page updated: 02/09/03 legal. ...www.dcu.ie/ - 6k - Cached - More pages from this site • City University of New YorkThe University's Main Website.www.cuny.edu/ - More pages from this site • City University of Hong KongAD Working Group lends staff, students an ear. Several issues still evoked strong emotions as staff members and students voiced their ...www.cityu.edu.hk/ - 26k - Cached - More pages from this site • Oklahoma City University... At Oklahoma CityUniversity, our students come first. Our ... Oklahoma CityUniversity offers a quality, values-centered education. United ...www.okcu.edu/ - 20k - Cached - More pages from this site

18. AllTheWeb • Apartment Listings in University City  (sponsored)St. Louis, Missouri area apartment listings at Apartments.com. Free nationwide apartment search with visual rental listings online.http://www.apartments.com • City University - Washington  (sponsored)Contact information and resources such as yellow page information, phone number, address, maps and directions as provided by QwestDex.http://service.bfast.com • City University Apartments - Rent.com  (sponsored)Rent.com has millions of free apartment listings nationwide. Get $100 when you sign a lease near your school - it's easy.http://www.rent.com • City University London - the University for business and the professions... Contact Us | About City University | Maps & Directions A-Z Index | Site Map | Help ... Description: Official site with information about courses, research, schools, and departments. Includes details of news and events.more hits from:  http://www.city.ac.uk/  -  13 KB • Flash UpgradeDescription: A University which believes in forward-thinking business and leadership skills, exposing students to the latest technology in all courses of study.http://www.cityu.edu/  -  27 KB • Welcome to Dublin City University... KnowledgeWorks, wins DCU Mallin-invent award Full text you can go anywhere in the world from here... Dublin City University, Dublin 9, Ireland. Tel. +353 (0) 1 700 5000, Fax. +353 (0) 1 836 0830. Page updated: 02/09/03 legal search ... Description: Information on facilities, services, degree courses, research, the campus, student life, the library and DCU news.

19. MSN Search • City University of New York Details the CUNY system's campuses and admissions requirements. Eye photos of notable graduates such as actor Judd Hirsch.www.cuny.edu • City University Offers prospectus, admissions, academic calendar, courses and degrees, and news.www.cityuniversity.net • City University Private, nonprofit institution located in Belleview, Washington, offers higher education opportunities to working professionals and lifelong learners.www.cityu.edu • Valley City University Admissions Site covers financial aid and tuition and offers a virtual campus tour. Site also provides access to an online application.www.vcsu.nodak.edu/admissions • Oklahoma City University The Oklahoma City University site contains information about admissions, academic departments and student life. The site also provides library resources, faculty information and news.frodo.okcu.edu • Dublin City University, Dublin - TripAdvisor Browse reviews of the Dublin City University in Dublin, County Dublin, Ireland. Check consumer reviews, articles and opinions about the Dublin City University.www.tripadvisor.com/Hotel_Review-g186605-d209093-Reviews-Dublin_City_U...

20. AOL Search • City University London - the University for business and the ... - ... The University for business and the professions. Contact Us | About City University | Maps & Directions, AZ Index | Site Map | Help. Prospective Students. ... http://www.city.ac.uk/ • City University Home - A University which believes in forward-thinking business and leadership skills, exposing students to the latest technology in all courses of study. http://www.cityu.edu/ • Welcome to Dublin City University - ... here... Dublin City University, Dublin 9, Ireland. Tel. +353 (0) 1 700 5000, Fax. +353 (0) 1 836 0830. Page updated: 02/09/03 legal. ... http://www.dcu.ie/ • http://www.cuny.edu/ - The University's Main Website.http://www.cuny.edu/ • City University of Hong Kong - AD Working Group lends staff, students an ear. Several issues still evoked strong emotions as staff members and students voiced their ... http://www.cityu.edu.hk/ • Welcome to Oklahoma City University - ... At Oklahoma City University, our students come first. Our ... Oklahoma City University offers a quality, values-centered education. United ... http://www.okcu.edu/

21. Ask Jeeves • CityUniversity of Hong Kong Formerly the City Polytechnic of Hong Kong. Includes information on university, links to learning resources, research and departmental information.....From:www.cityu.edu.hk/ • CityUniversity Text Version. The University for business and the professions ... Contact Us | About CityUniversity | Maps & Directions A-Z Index | Site Map | Help...From:www.city.ac.uk/ • New Jersey CityUniversity Official site includes graduate and undergraduate information, news, events, online learning, calendars, and athletics.From:www.njcu.edu/ • Elizabeth City State University Admission Application Introduction 2004-2005 APPLICATION FOR ADMISSION. Welcome to the Elizabeth City State University Online Application. New Freshmen...From:www.ncmentor.org/applications/unc/apply/elizabeth_city_state_univ... • CityUniversity A University which believes in forward-thinking business and leadership skills, exposing students to the latest technology in all courses of study.From:www.cityu.edu/ • The City College of New York City College of New York (CUNY) A senior college of The CityUniversity of New York. Located in Manhattan. 138th Street at Convent Avenue. A Harlem masterpiece in the neo-Gothic...From:www.ccny.cuny.edu/

22. HotBot 1.City University of New YorkDetails the CUNY system's campuses and admissions requirements. Eye photos of notable graduates such as actor Judd Hirsch.www.cuny.edu/ - August 10, 2003 - 25 KB 2.City UniversityPrivate, nonprofit institution located in Belleview, Washington, offers higher education opportunities to working professionals and lifelong learners.www.cityu.edu/ - August 23, 2003 - 27 KB 3.New Jersey City UniversityExplore the course requirements for this liberal arts institution. Link to admissions and financial aid information. ... New Jersey City University. 2039 Kennedy Boulevard Jersey City, New Jersey 07305-1597 ... www.njcu.edu/ - November 1, 2003 - 20 KB 4.City University London - the University for business and the...... Contact Us | About City University | Maps & Directions. A-Z Index | Site Map | Help ... www.city.ac.uk/ - September 28, 2003 - 14 KB 5.Oklahoma City UniversityTake a tour of the campus, and explore a roster of academic programs. ... At Oklahoma City University, our students come first. ... www.okcu.edu/ - October 26, 2003 - 20 KB 6.University of Missouri, Kansas CityUniversity based in Kansas City, Missouri, presents and overview of its academic programs, and offers campus news. ... UNIVERSITY OF MISSOURI-KANSAS CITY. Bulletin. Winter 2004 Fee Update, more... ... www.umkc.edu/ - October 23, 2003 - 16 KB

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25. MetaSearch Engine • Metasearch Engines are designed to increase the coverage of web by forwarding users’ queries to multiple search engines • Users’ requests are sent to multiple search engines such as AlltheWeb, Google, MSN. • Then the results from the individual search engine are combined into a single result set to present to users.

26. Longest common subsequence • Definition 1: Given a sequence X=x1x2...xm, another sequence Z=z1z2...zk is a subsequence of X if there exists a strictly increasing sequence i1i2...ik of indices of X such that for all j=1,2,...k, we have xij=zj. • Example 1: If X=abcdefg, Z=abdg is a subsequence of X. X=abcdefg,Z=ab d g

27. Definition 2: Given two sequences X and Y, a sequence Z is a common subsequence of X and Y if Z is a subsequence of both X and Y. • Example 2: X=abcdefg and Y=aaadgfd. Z=adf is a common subsequence of X and Y. X=abc defg Y=aaaadgfd Z=a d f

28. Definition 3: A longest common subsequence of X and Y is a common subsequence of X and Y with the longest length. (The length of a sequence is the number of letters in the seuqence.) • Longest common subsequence may not be unique. • Example: abcd acbd Both acd and abd are LCS.

29. Longest common subsequence problem • Input: Two sequences X=x1x2...xm, and Y=y1y2...yn. • Output:a longest common subsequence of X and Y. • A brute-force approach Suppose that mn. Try all subsequence of X (There are 2m subsequence of X), test if such a subsequence is also a subsequence of Y, and select the one with the longest length.

30. Charactering a longest common subsequence • Theorem (Optimal substructure of an LCS) • Let X=x1x2...xm, and Y=y1y2...yn be two sequences, and • Z=z1z2...zk be any LCS of X and Y. • 1. If xm=yn, then zk=xm=yn and Z[1..k-1] is an LCS of X[1..m-1] and Y[1..n-1]. • 2. If xmyn, then zkxm implies that Z is an LCS of X[1..m-1] and Y. • 2. If xmyn, then zkyn implies that Z is an LCS of X and Y[1..n-1].

31. The recursive equation • Let c[i,j] be the length of an LCS of X[1...i] and X[1...j]. • c[i,j] can be computed as follows: 0 if i=0 or j=0, c[i,j]= c[i-1,j-1]+1 if i,j>0 and xi=yj, max{c[i,j-1],c[i-1,j]} if i,j>0 and xiyj. Computing the length of an LCS • There are nm c[i,j]’s. So we can compute them in a specific order.

32. The algorithm to compute an LCS • 1. for i=1 to m do • 2. c[i,0]=0; • 3. for j=0 to n do • 4. c[0,j]=0; • 5. for i=1 to m do • 6. for j=1 to n do • 7. { • 8. if x[I] ==y[j] then • 9. c[i,j]=c[i-1,j-1]=1; • 10 b[i,j]=1; • 11. elseif c[i-1,j]>=c[i,j-1] then • 12. c[i,j]=c[i-1,j] • 13. b[i,j]=2; • 14. else c[i,j]=c[i,j-1] • 15. b[i,j]=3; • 14 }

33. Example 3: X=BDCABA and Y=ABCBDAB.

34. Constructing an LCS (back-tracking) • We can find an LCS using b[i,j]’s. • We start with b[n,m] and track back to some cell b[0,i] or b[i,0]. • The algorithm to construct an LCS 1. i=m 2. j=n; 3. if i==0 or j==0 then exit; 4. if b[i,j]==1 then { i=i-1; j=j-1; print “xi”; } 5. if b[i,j]==2 i=i-1 6. if b[i,j]==3 j=j-1 7. Goto Step 3. • The time complexity: O(nm).

35. Shortest common supersequence • Definition:Let X and Y be two sequences. A sequence Z is a supersequence of X and Y if both X and Y are subsequence of Z. • Shortest common supersequence problem: Input: Two sequences X and Y. Output: a shortest common supersequence of X and Y. • Example: X=abc and Y=abb. Both abbc and abcb are the shortest common supersequences for X and Y.

36. Recursive Equation: • Let c[i,j] be the length of an LCS of X[1...i] and X[1...j]. • c[i,j] can be computed as follows: j if i=0 i if j=0, c[i,j]= c[i-1,j-1]+1 if i,j>0 and xi=yj, min{c[i,j-1]+1,c[i-1,j]+1} if i,j>0 and xiyj.