Week 10: test contains all material from Week 1

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.

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

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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

• Ci,j be the number of errors allowed.
• Three kinds of error mismatch (a, b), insertion(
  a, )and deletion ( , a).

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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).

s u r g e r y
0 0 0 0 0 0 0 0
s 1 0 1 1 1 1 1 1
u 2 1 0 1 2 2 2 2
r 3 2 1 0 1 2 2 3
v 4 3 2 1 1 2 3 3
e 5 4 3 2 2 1 2 3
y 6 5 4 3 3 2 2 2
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String Matching Allowing Errors

• O(n?k) algorithm, where n is the length of the
  pattern and k the number of errors allowed.
• Ci,q the number of errors for ci,iq,
• q-k, -(k1), ,0, 1, 2, , k.
• Ci,iq Ci-1, iq-1 if pitiq.
• else CI,iq1minCi-1,iq, CI, iq-1,
  Ci-1,iq-1.

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• 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.)

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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 AB,
  A.B and (A) are regular expressions.

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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 T1T2Ti is
  accepted.
• Given d(i), d(i1) 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.

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Example

• E(AAA).(BAB).
• TABBAAAB.
• 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.

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Running time

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

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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.

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GOOGLE City University

1. 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. ...
   Description Official site with information
   about courses, research, schools, and
   departments. Includes details...
2. City University 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...
3. Welcome to Dublin City University... here...
   Dublin City University, Dublin 9, Ireland. Tel.
   353 (0) 1700 5000, Fax. [phone].
   Page updated 02/09/03 legal. ... Description
   Information on facilities, services, degree
   courses, research, the campus, student life, the
   library...
4. The City University of New YorkDescription The
   University's Main Website.
5. City University 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...
6. Welcome to Oklahoma City University... At
   Oklahoma City University, 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...

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YAHOO

1. 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. ...
2. City University A 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
3. Dublin City University ... here... Dublin City
   University, Dublin 9, Ireland. Tel. 353 (0) 1
   700 5000, Fax. [phone]. Page updated
   02/09/03 legal. ...www.dcu.ie/ - 6k - Cached -
   More pages from this site
4. City University of New York The University's
   Main Website.www.cuny.edu/ - More pages from
   this site
5. 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 ...www.cityu.edu.hk/ - 26k
   - Cached - More pages from this site
6. Oklahoma City University ... At Oklahoma City
   University, our students come first. Our ...
   Oklahoma City University offers a quality,
   values-centered education. United
   ...www.okcu.edu/ - 20k - Cached - More pages
   from this site

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AllTheWeb

1. 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
2. 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
3. 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
4. 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
5. 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
6. 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. [phone], 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.

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MSN Search

1. 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
2. City University Offers prospectus, admissions,
   academic calendar, courses and degrees, and
   news.www.cityuniversity.net
3. City University Private, nonprofit institution
   located in Belleview, Washington, offers higher
   education opportunities to working professionals
   and lifelong learners.www.cityu.edu
4. 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
5. 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
6. 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...

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AOL Search

1. 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/
2. 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/
3. Welcome to Dublin City University - ... here...
   Dublin City University, Dublin 9, Ireland. Tel.
   353 (0) 1 700 5000, Fax. [phone].
   Page updated 02/09/03 legal. ...
   http//www.dcu.ie/
4. http//www.cuny.edu/ - The University's Main
   Website.http//www.cuny.edu/
5. 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
   /
6. 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/

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Ask Jeeves

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

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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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Lycos

• 1. City University London - the University for
  business and the... Contact Us About City
  University Maps Directions A-Z Index Site
  Map Help More results from www.city.ac.uk  
  September 16, 2003 - 14 KB
• 2. Flash UpgradeA University which believes in
  forward-thinking business and leadership skills,
  exposing students to the latest technology in all
  courses of study.www.cityu.edu   November 1,
  2003 - 27 KB
• 3. The City University of New YorkThe
  University's Main Website.www.cuny.edu  
  December 31, 1969 - 121 B
• 4. 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. [phone], Fax. 353 (0)
  1 836...More results from www.dcu.ie   October
  29, 2003 - 5 KB
• 5. City University of Hong KongFormerly the City
  Polytechnic of Hong Kong. Includes information on
  university, links to learning resources, research
  and departmental information and student
  information.More results from www.cityu.edu.hk
    October 29, 2003 - 26 KB
• 6. HCU HomePage -English- contact us For any
  suggestion and requests to this web site,
  www-admin_at_hiroshima-cu.ac.jp Feel free to link
  this web site. Hiroshima City University More
  results from www.hiroshima-cu.ac.jp   May 11,
  2003 - 26 KB

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Teoma

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

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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.

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Longest common subsequence

• Definition 1 Given a sequence Xx1x2...xm,
  another sequence Zz1z2...zk is a subsequence of
  X if there exists a strictly increasing sequence
  i1i2...ik of indices of X such that for all
  j1,2,...k, we have xijzj.
• Example 1 If Xabcdefg, Zabdg is a subsequence
  of X. Xabcdefg,Zab d g

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• 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 Xabcdefg and Yaaadgfd. Zadf is a
  common subsequence of X and Y.
• Xabc defg
• Yaaaadgfd
• Za d f

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• 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.

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Longest common subsequence problem

• Input Two sequences Xx1x2...xm, and
• Yy1y2...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.

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Charactering a longest common subsequence

• Theorem (Optimal substructure of an LCS)
• Let Xx1x2...xm, and Yy1y2...yn be two
  sequences, and
• Zz1z2...zk be any LCS of X and Y.
• 1. If xmyn, then zkxmyn and Z1..k-1 is an
  LCS of X1..m-1 and Y1..n-1.
• 2. If xm ?yn, then zk?xm implies that Z is an LCS
  of X1..m-1 and Y.
• 2. If xm ?yn, then zk?yn implies that Z is an LCS
  of X and Y1..n-1.

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The recursive equation

• Let ci,j be the length of an LCS of X1...i
  and X1...j.
• ci,j can be computed as follows
• 0
  if i0 or j0,
• ci,j ci-1,j-11 if
  i,jgt0 and xiyj,
• maxci,j-1,ci-1,j if i,jgt0
  and xi?yj.
• Computing the length of an LCS
• There are n?m ci,js. So we can compute them in
  a specific order.

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The algorithm to compute an LCS

• 1. for i1 to m do
• 2. ci,00
• 3. for j0 to n do
• 4. c0,j0
• 5. for i1 to m do
• 6. for j1 to n do
• 7.
• 8. if xI yj then
• 9. ci,jci-1,j-11
• 10 bi,j1
• 11. else if ci-1,jgtci,j-1 then
• 12. ci,jci-1,j
• 13. bi,j2
• 14. else ci,jci,j-1
• 15. bi,j3
• 14

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• Example 3 XBDCABA and YABCBDAB.

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Constructing an LCS (back-tracking)

• We can find an LCS using bi,js.
• We start with bn,m and track back to some cell
  b0,i or bi,0.
• The algorithm to construct an LCS
• 1. im
• 2. jn
• 3. if i0 or j0 then exit
• 4. if bi,j1 then
• ii-1
• jj-1
• print xi
• 5. if bi,j2 ii-1
• 6. if bi,j3 jj-1
• 7. Goto Step 3.
• The time complexity O(nm).

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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 Xabc and Yabb. Both abbc and abcb are
  the shortest common supersequences for X and Y.

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• Recursive Equation
• Let ci,j be the length of an LCS of X1...i
  and X1...j.
• ci,j can be computed as follows
• j
  if i0
• i
  if j0,
• ci,j ci-1,j-11 if
  i,jgt0 and xiyj,
• minci,j-11,ci-1,j1 if
  i,jgt0 and xi?yj.

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