1. Topic-Sensitive PageRank
Reference: Taher H. Haveliwala, “Topic-Sensitive PageRank: A context-sensitive ranking algorithm for websearch”, IEEE Trans. On Knowledge and Data Engineering, vol. 15, No. 4, PP
The original PageRank:
purely based on the hyperlinks of web pages.
Contents are not considered.
A vector of PageRank is used for all web pages.
Topic-Sensitive PageRank
For each topic, a vector of PageRank is created.
Each page has several PageRank values. One for each topic.

2. Creating a Page Rank vector for each topic
How to select topics?
Using a small set of topics is important for low computation cost and quick response time.
Open Directory:
16 top level topics

3. Original PageRank Rank=MRank
Rank is a vector, one element for each web page.
M is a nn matric
If there is a link from page j to page I, then
Mi,j =1/Nj, where Nj is the number of out-links of page j.

4. Another Version Let n be the total number of web pages.
P=[1/n] n1 be a vector.
d is a n1 matrix.
di=1 if page i has no out links. Otherwise, di=0.
D=ppT, and E=p[1] 1n
M’=(1-)(M+D) +E.
Rank=M’Rank=(1-)(M+D) Rank +P.

5. Topic sensitive Page Rank
Let Tj be the set of URLs in the ODP category cj.
P=vj, where
Vj,i =1/|Tj| if page j points to page i. Otherwisr vj,i=0.
The pageRank vector for topic cj is PR(, vj).
Compute the pageRAnk for all pages related to topic cj as if for the original PageRank by considering Tj.

6. The Retrieval Score
Let r j, d be the PageRank of document d given by the rank PR( , vj).
Sq,d=  j P(cj|q)•r j,d ,
P(cj|q) is the score that topic is related to q.

7. Similarity Measures for Induced Rankings Another Version
Let 1 and 2 be two rankings of documents.
OSim(1,2) indicates the degree of overlap between the top k URLs of the two rankings.
OSim (1,2) =|AB|/k.
KSimn (1,2) =
|(u, v): 1 and 2 agree on order of (u, v)|/(|U|)(|U|-1)
Let  be the true ranking given by user. To compare 1 and 2, we can use OSim (1,2) or KSimn (1,2) .

8. Weighted PageRank Assign larger rank values to more important pages.
Each outlink page gets its value proportional to it popularity.
W in (v, u) is the weight of link(v, u) calculated based on the number of inlinks of page u and the number of inlinks of all reference pages of page v.
w in (v, u) =Iu/pR(v) Ip,
Iu—number of inlinks of page u.
R(v)-the set of all pages that v points to.

9. Weighted PageRank w out (v, u) =Ou/pR(v) Op,
Ou—number of outlinks of page u.
Let B(u) be the set of pages that points to v.
PR(u)=(1-d) +d  vB(u) PR(v) w in (v, u) w out (v, u)
Reference: Wenpu Xing and Ali Ghorbani, Weighted PageRank Algorithm, Proceedings of the 2nd Annual Conference on Communication Networks and Services Research (CNSR’04), 2004.

10. 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.

11. GOOGLE: City University
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12. YAHOO
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13. 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.
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14. MSN Search
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Valley City University Admissions Site covers financial aid and tuition and offers a virtual campus tour. Site also provides access to an online application.
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20. 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.

21. 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

22. 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

23. 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.

24. 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.

25. 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].

26. 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:
if i=0 or j=0,
c[i,j]= c[i-1,j-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.

27. The algorithm to compute an LCS
1. for i=1 to m do
c[i,0]=0;
3. for j=0 to n do
c[0,j]=0;
5. for i=1 to m do
for j=1 to n do {
if x[I] ==y[j] then
c[i,j]=c[i-1,j-1]=1;
b[i,j]=1;
else if c[i-1,j]>=c[i,j-1] then
c[i,j]=c[i-1,j]
b[i,j]=2;
else c[i,j]=c[i,j-1]
b[i,j]=3; }

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

29. 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]== i=i-1
6. if b[i,j]== j=j-1
7. Goto Step 3.
The time complexity: O(nm).

30. 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.

31. Let c[i,j] be the length of an LCS of X[1...i] and X[1...j].
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] 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.