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Processing Boolean queries

How do we process a query using an inverted index and the basic Boolean retrieval model? Consider processing the simple conjunctive query :
\term{Brutus} \oper{AND} \term{Calpurnia}
over the inverted index partially shown in Figure 1.3 (page [*]). We:

  1. Locate Brutus in the Dictionary
  2. Retrieve its postings
  3. Locate Calpurnia in the Dictionary
  4. Retrieve its postings
  5. Intersect the two postings lists, as shown in Figure 1.5 .
The intersection is the crucial one: we need to efficiently intersect postings lists so as to be able to quickly find documents that contain both terms. (This operation is sometimes referred to as merging postings lists: this slightly counterintuitive name reflects using the term merge algorithm for a general family of algorithms that combine multiple sorted lists by interleaved advancing of pointers through each; here we are merging the lists with a logical AND operation.)

Figure: Intersecting the postings lists for Brutus and Calpurnia from Figure 1.3 .
Brutus & $\longrightarrow$\ &
...rightarrow$\ & \framebox{2}$\rightarrow$\framebox{31}

Figure 1.6: Algorithm for the intersection of two postings lists $p_1$ and $p_2$.
\begin{figure}\begin{algorithm}{Intersect}{p_1, p_2}
answer \= \langle\;\rangle ...
\end{IF}\end{IF}\end{WHILE} \\

There is a simple and effective method of intersecting postings lists using the merge algorithm (see Figure 1.6 ): we maintain pointers into both lists and walk through the two postings lists simultaneously, in time linear in the total number of postings entries. At each step, we compare the docID pointed to by both pointers. If they are the same, we put that docID in the results list, and advance both pointers. Otherwise we advance the pointer pointing to the smaller docID. If the lengths of the postings lists are $x$ and $y$, the intersection takes $O(x+y)$ operations. Formally, the complexity of querying is $\Theta(N)$,where $N$ is the number of documents in the collection.[*]Our indexing methods gain us just a constant, not a difference in $\Theta$ time complexity compared to a linear scan, but in practice the constant is huge. To use this algorithm, it is crucial that postings be sorted by a single global ordering. Using a numeric sort by docID is one simple way to achieve this.

We can extend the intersection operation to process more complicated queries like:
{(\term{Brutus} \oper{OR} \term{Caesar})} \oper{AND} \oper{NOT} \term{Calpurnia}
Query optimization is the process of selecting how to organize the work of answering a query so that the least total amount of work needs to be done by the system. A major element of this for Boolean queries is the order in which postings lists are accessed. What is the best order for query processing? Consider a query that is an AND of $t$ terms, for instance:
\term{Brutus} \oper{AND} \term{Caesar} \oper{AND} \term{Calpurnia}
For each of the $t$ terms, we need to get its postings, then AND them together. The standard heuristic is to process terms in order of increasing document frequency: if we start by intersecting the two smallest postings lists, then all intermediate results must be no bigger than the smallest postings list, and we are therefore likely to do the least amount of total work. So, for the postings lists in Figure 1.3 (page [*]), we execute the above query as:
{(\term{Calpurnia} \oper{AND} \term{Brutus})} \oper{AND} \term{Caesar}
This is a first justification for keeping the frequency of terms in the dictionary: it allows us to make this ordering decision based on in-memory data before accessing any postings list.

Consider now the optimization of more general queries, such as:
{(\term{madding} \oper{OR} \term{crowd})} \oper{AND} (\term{igno...
...} \term{strife}) \oper{AND} (\term{killed} \oper{OR} \term{slain})
As before, we will get the frequencies for all terms, and we can then (conservatively) estimate the size of each OR by the sum of the frequencies of its disjuncts. We can then process the query in increasing order of the size of each disjunctive term.

Figure 1.7: Algorithm for conjunctive queries that returns the set of documents containing each term in the input list of terms.
\begin{figure}\begin{algorithm}{Intersect}{\langle t_1, \ldots, t_n \rangle}
...erms \= rest(terms)
\end{WHILE} \\

For arbitrary Boolean queries, we have to evaluate and temporarily store the answers for intermediate expressions in a complex expression. However, in many circumstances, either because of the nature of the query language, or just because this is the most common type of query that users submit, a query is purely conjunctive. In this case, rather than viewing merging postings lists as a function with two inputs and a distinct output, it is more efficient to intersect each retrieved postings list with the current intermediate result in memory, where we initialize the intermediate result by loading the postings list of the least frequent term. This algorithm is shown in Figure 1.7 . The intersection operation is then asymmetric: the intermediate results list is in memory while the list it is being intersected with is being read from disk. Moreover the intermediate results list is always at least as short as the other list, and in many cases it is orders of magnitude shorter. The postings intersection can still be done by the algorithm in Figure 1.6 , but when the difference between the list lengths is very large, opportunities to use alternative techniques open up. The intersection can be calculated in place by destructively modifying or marking invalid items in the intermediate results list. Or the intersection can be done as a sequence of binary searches in the long postings lists for each posting in the intermediate results list. Another possibility is to store the long postings list as a hashtable, so that membership of an intermediate result item can be calculated in constant rather than linear or log time. However, such alternative techniques are difficult to combine with postings list compression of the sort discussed in Chapter 5 . Moreover, standard postings list intersection operations remain necessary when both terms of a query are very common.


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