Intersection algorithm, and skip pointers

When searching a two$-$word query, the following two postings lists are returned from

the inverted index:

$[2,13,18,23,45,78,88,90,123,178,245,269,290,313,389,390]$

$[2,313]$

a$.$ List all the comparison pairs when using the standard postings lists

b$.$ To speed up the intersection between the postings lists, skip pointers are

considered to add to the postings listings. Suppose the skip length is $($P is the

length of the postings list$)$

a$.$ Add the skip pointers to the first postings list

b$.$ After the skip pointers are added to the first postings list, list all the

comparison pairs

$2.$ Positional Indexes

Consider the following documents:

Doc $1$: new home sales top forecast

Doc $2$: home sales rise in july

a$.$ Build positional indexes on top of these documents using the format DocID:

$<($position$1,$ position$2),...>.$ For example, the positional index for the word

$$sales$$ is as follows:

sales: $1$:$<3>$; $2$: $<2>$

b$.$ A phrase query $$word$1$ word$2$ retrieves the occurrences where word$1$ is

immediately followed by word$2.$ A $/$k query $$word$1/$k word$2($k is a posi@ve integer$)$

retrieves the occurrences of word$1$ within k words of word$2$ on either side. For

example, k $=1$ demands that word$1$ be adjacent to word$2,$ but word$1$ may come

either before or aHer word$2.$

For the following queries, return all the docs and corresponding posi@ons for which

the query condi@ons are met. If none of the documents meet the criteria, return

None.

i$.$new sales$$

ii$.$new home$$

iii. $$new $/2$ sales$$