Introduction

On Monday, we introduced the vector space model.
In it, we imagine each document in the corpus is represented with a vector where the coordinates of the vector correspond to terms in the vocabulary of the whole corpus: i.e., there is one column for each term that appears anywhere in the corpus.
For a given document, we said we would you a TF-IDF score for the value a coordinate where TF, term frequency, measures how frequently the terms appears within a document, and IDF, inverse document frequency measures the number of bits of information you would learn if you knew a given document has that term (rarer words give more information).
Given two such documents how similar they are can be computed using cosine similarity.
We gave an algorithm to rank closeness of documents to a fixed query using these vectors.
We begin today by looking at an algorithm to rank documents according to their proximity score -- how close the search terms occur within the same document.

Proximity Ranking

Results and ranking in the VSM depend only on term frequency, TF, and Inverse Document Frequency, IDF.
The proximity ranking method in contrast depends explicitly on how close the terms in the term vector occur, but only implicitly depends of TF and IDF plays no role at all.
Given a term vector `langle t_1, ... t_n rangle`, we define a cover for this vector to be an interval `[u,v]` in the collection that contains all the terms in the vector, such that no smaller interval `[u', v']` contained in `[u,v]` also has a match to all the terms in the vector.
For example, if the document 1 in our collection began as "To be or not to be", and the term vector was "to be" then the intervals `[1:1, 1:2]` and `[1:5, 1:6]` would be covers, but `[1:1, 1:3]` would not be as it contains `[1:1, 1:2]`.
Covers are allowed to overlap. If "meow meow" was the term vector and document 2 began "meow meow meow meow meow meow ..." then the intervals `[1:1, 1:2]` and `[1:2, 1:3]` would both be covers.
A given token though can appear in at most `n cdot l` covers, where `l` is the length of the shortest posting list.

Algorithm for Finding Covers

nextCover(t[1],.., t[n], position) 
{
    v:= max_(1≤ i ≤ n)(next(t[i], position));
    if(v == infty)
        return [ infty, infty];
    u := min_(1≤ i ≤ n)(prev(t[i], v+1))
    if(docid(u) == docid(v) ) then 
        return [u,v]; 
       // covers need to be in the same document
    else
        return nextCover(t[1],.., t[n], u);
}

Ranking Covers

We now want to come up with a formula that scores documents according to the covers it contains for the term query.
We would like that smaller covers are worth more than larger covers.
We would also like more covers in a document to count more than fewer covers in a document.
Keeping this in mind, suppose document `d` has covers `[u_1, v_1]`, `[u_2, v_2]`, ...
Then we define the score for `d` to be:
`mbox(score)(d) = sum(frac(1)(v_i - u_i + 1))`.

Ranking Algorithm with Proximity Scores

rankProximity(t[1],.., t[n], k)
// t[] term vector
// k number of results to return 
{
    u := - infty;
    [u,v] := nextCover(t[1],.., t[n], u);
    d := docid(u);
    score := 0;
    j := 0;
    while( u < infty) do
        if(d < docid(u) ) then
        // if docid changes record info about last docid
            j := j + 1;
            Result[j].docid := d;
            Result[j].score := score;
            d := docid(u);
            score := 0;
        score := score + 1/(v - u + 1);
        [u, v] := nextCover(t[1],.., t[n], u);
    if(d < infty) then
        // record last score if not recorded
        j := j + 1;
        Result[j].docid := d;
        Result[j].score := score;
    sort Result[1..j] by score;
    return Result[1..k];   
}

Using an analysis similar to that used for galloping search in the book, you can prove this algorithm has running time:
`O(n^2 l cdot log(L/l))`.

In-Class Exercise

Suppose we have two documents `d_1` = "Which is correct: all the people, all of the people, or all people?" and `d_2`="All the king's horses and all the king's men".
Work out step by step how the call rankProximity("all", "the", 2) would process this corpus.
Please post your solution to the Sep 20 In-Class Exercise Thread.

Boolean Retrieval

Besides being used as implicit filters in some web search engines, explicit support for Boolean queries is important for some specific applications such as digital libraries and in the legal domain.
For example, when we do a patent search, we want an exhaustive list of patents returned which match the term query, not just the top 10.
In the Boolean retrieval model we thus return sets of results rather than ranked lists.
The standard boolean operators AND, OR, and NOT are used to construct queries.
Here "A AND B" means the intersection of sets A and B; "A OR B" means the union of the sets A and B; and NOT A means those documents in the collection not contained in A.
A term `t` corresponds in a query to the collection of documents containing `t`.
So ("quarrel" OR "sir") AND "you" represents the set of documents that contain "you" and also contain at least one of the two terms "quarrel" and "sir".
An algorithm to solve a Boolean query locates candidate solutions to the query, where each candidate solution represents a range of documents that together satisfy the query.

Extending Our ADT for Boolean Retrieval

To simplify our Boolean search algorithm, we define two functions that operate over Boolean queries:

docRight(Q, u)
end point of the first candidate solution to `Q` starting after document `u`

docLeft(Q, v)
start point of the last candidate solution to `Q` ending before document `v`
For terms we define docRight(t, u) := nextDoc(t, u) and docLeft(t,v) := prevDoc(t,v).

For AND and OR operators we define:

docRight(A AND B, u) := max(docRight(A, u), docRight(B,u))
docLeft(A AND B, v) := min(docLeft(A,v), docLeft(B,v))
docRight(A OR B, u) := min(docRight(A, u), docRight(B,u))
docLeft(A OR B, v) := max(docLeft(A,v), docLeft(B,v))

Notice the above rules give us a recursive algorithm given a Boolean query `Q` to compute docRight, docLeft.

Algorithm to Return the Next Solution to a Positive Boolean Query (No NOT's).

nextSolution(Q, position)
{
    v := docRight(Q, position);
    if v = infty then
        return infty;
    u := docLeft(Q, v+1);
    if(u == v) then
        return u;
    else
        return nextSolution(Q, v);
}

Algorithm to Return All Solutions to a Positive Boolean Query

u :=  -infty
while u < infty do
    u := nextSolution(Q, u);
    if(u < infty) then
        report docid(u);

If we implement nextDoc, prevDoc with galloping search, the complexity of this algorithm is `O(n cdot l cdot log(L/l))`

Handling Queries with NOT

To handle negations we first use De Morgan's rules to push negations so that they are only on terms.
i.e., NOT (A AND B ) = NOT A OR NOT B and NOT (A OR B) = NOT A AND NOT B
Then we enhance our methods nextDoc and prevDoc, so that they can directly handle negations. i.e., So nextDoc(NOT t, u) and prevDoc(NOT t, u) make sense.
Finally, we set docRight(NOT t, u) := nextDoc(NOT t, u) and docLeft(NOT t,v) := prevDoc(NOT t,v).

Proximity Ranking, Boolean Retrieval

Outline