TopicSensitive Page Rank Reference Taher H Haveliwala TopicSensitive

Topic-Sensitive Page. Rank Reference: Taher H. Haveliwala, “Topic-Sensitive Page. Rank: A context-sensitive ranking algorithm for websearch”, IEEE Trans. On Knowledge and Data Engineering, vol. 15, No. 4, PP. 784 -796. The original Page. Rank: • purely based on the hyperlinks of web pages. • Contents are not considered. • A vector of Page. Rank is used for all web pages. Topic-Sensitive Page. Rank • For each topic, a vector of Page. Rank is created. • Each page has several Page. Rank values. One for each topic.

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: http: //www. dmoz. org • 16 top level topics

Original Page. Rank • 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.

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 p. T, and E=p [1] 1 n • M’=(1 - )(M+D) + E. • Rank=M’ Rank=(1 - )(M+D) Rank + P.

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 page. Rank vector for topic cj is PR( , vj). • Compute the page. RAnk for all pages related to topic cj as if for the original Page. Rank by considering Tj.

The Retrieval Score • Let r j, d be the Page. Rank 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.

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).

Weighted Page. Rank • 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.

Weighted Page. Rank • 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 Page. Rank Algorithm, Proceedings of the 2 nd Annual Conference on Communication Networks and Services Research (CNSR’ 04), 2004.

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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Meta. Search 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 Allthe. Web, Google, MSN. • Then the results from the individual search engine are combined into a single result set to present to users.

Longest common subsequence • Definition 1: Given a sequence X=x 1 x 2. . . xm, another sequence Z=z 1 z 2. . . zk is a subsequence of X if there exists a strictly increasing sequence i 1 i 2. . . 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

• 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

• 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.

Longest common subsequence problem • Input: Two sequences X=x 1 x 2. . . xm, and Y=y 1 y 2. . . 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 2 m subsequence of X), test if such a subsequence is also a subsequence of Y, and select the one with the longest length.

Charactering a longest common subsequence • Theorem (Optimal substructure of an LCS) • Let X=x 1 x 2. . . xm, and Y=y 1 y 2. . . yn be two sequences, and • Z=z 1 z 2. . . 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].

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 x i=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.

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. else if 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 }

Example 3: X=BDCABA and Y=ABCBDAB.

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. 2. 3. 4. i=m j=n; if i==0 or j==0 then exit; 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).

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.

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: c[i, j]= j if i=0 i if j=0, c[i-1, j-1]+1 if i, j>0 and x i=yj, min{c[i, j-1]+1, c[i-1, j]+1} if i, j>0 and xi yj.