Revisión del concepto: de Género a Estudios de género

Before we introduce our new holistic twig join algorithm, we first discuss the properties of different streaming techniques. Figure 5.8 shows that it is possible that all current elements are blocked for a given query and document. Now we prove that for a certain class of twig pattern query, a particular streaming scheme can prevent the situation whereby all current elements are blocked. In other words, based on such particular streaming scheme, we can design a holistic algorithm to guarantee its optimality for this certain class of query. We first introduce an auxiliary lemma.

Lemma 5.1. Given two streaming schemes α and β, suppose α is a refinement of β. Suppose we have a set of streams in β being further partitioned under α, we have

(1) A current-matching element in β is also a current-matching element in α. (2) A current-useless element in β is also a current-useless element in α.

It is easy to find example to prove that the opposite direction in each of the two conclusions in the lemma is not true.

Lemma 5.2. Under Tag Streaming, for an A-D only query Q, there always exists a stream Tq such that the current element in Tq is either a current-match or current-

useless element for Q.

Proof. We use induction on the sub-queries of Q.

(Base case) Suppose a node q in Q which is the parent of leaf nodes q1, q2, . . . , qn

and Tq has not ended. Note that if any stream Tqi has ended, current(Tq) is a useless

element for Qq and we are done. Otherwise all streams Tq1, . . . , Tqn do not end and

it is obvious that each sub-twig Qqi for i from 1 to n has a real match. There are the

following cases:

(1) If current(Tq).end < current(Tqi).start for some qi, then current(Tq) is a

useless element for Qq.

(2) Else if current(Tq).start > current(Tqi).start for some qi, then current(Tqi)

is a matching element for Qq.

(3) Otherwise current(Tq) is an ancestor of current(Tqi) for each qi, and current(Tq)

is a matching element for Qq.

Note that in the base case we can find either a current-useless or current-matching element with respect to (w.r.t) Qq.

(Induction) Suppose a node q in Q has child nodes q1, . . . , qn, by the induction

hypothesis, if there is a node qi for which current(Tqi) is not a matching element for

Qqi, we must find either a useless element for Qqi (which is also a useless element

for Qq) under the above case 1. or a matching element for a sub-twig of Qqi but

2. and thus done. Otherwise each current(Tqi) is a matching element for Qqi, we

proceed with the same argument (with the three cases) in the base case. Note that the induction step ends when q is the root of Q and in such case current(Tq) is a

matching element.

Lemma 5.3. Under Tag+Level streaming, for an A-D only or P-C only query Q, there always exists a stream T such that the current element e in T is either a current- useless or current-matching element.

Proof. According to Lemma 5.1, for an A-D only twig pattern, the above statement is true.

Given a P-C only query Q with n nodes, we can partition the streams into a few groups each of which has n streams and contains possible matches to Q. For example, streams T2

A, TD3, TB3 and TC4 form a group for the query A[./D]/B/C in Figure 5.4.

Notice that it is impossible that two elements from streams of different groups can be in the same match. For each group of n streams, we can perform the same analysis as in the proof of Lemma 5.2 to find out either a current-useless or current-matching element for Q.

Lemma 5.4. Under Prefix-Path streaming, for an A-D only or P-C only and one- branching-node only queries, there always exists a stream T such that the current element e in T is either a current-useless or current-match element.

Proof. According to Lemma 5.1, for an A-D or P-C only twig pattern, the above statement is true. For only one-branching-node queries, we first prove the special case where the root node is also the branching node. Suppose the PPS stream Tmin

1. Tmin is of class q where q is not the root of Q and L is the path of Tmin. Suppose

qlis a leaf node of Q and a descendant of q. Suppose Tqminl is a stream of class ql

having path L′ _{with the following properties: (1) L}′ _{= L + L}′′ _{and the string L}′′

matches the path query Qq (2) Tqminl has the current element with the minimum

start value among all streams of class ql. Now if current(Tmin) is not an ancestor

of current(Tmin

ql ), then current(Tmin) is a useless element and we have done.

Otherwise take any stream Tm of class qm where qm is a node between q and ql

and having a path which is prefix of L′

and for which L is a prefix. If for any such Tm, current(Tm).end < current(Tqminl ).start then current(Tm) is a useless

element; otherwise current(Tmin) is a matching element.

2. Tmin is of class q where q is the root of Q. We can consider the leaf nodes for

each branching-node of Q and repeat the same argument as (1).

Finally, in the case where the branch node is not the root node, we can reduce it to the first case.

Interested readers may wonder if there is a streaming scheme whereby the all blocked situation never occurs, we point out that at least the costly F B−BiSimulation scheme [41] is able to do that because the scheme is so refined that paper [41] shows that from the label (i.e. index node) of each stream, F B − BiSimulation can tell if all elements in the stream are in a match or not.

As the final point of this section, astute reader may wonder how to explain the optimality for the algorithm TwigStackList proposed in Chapter 3, which is based on Tag Streaming scheme but has the larger optimality class than that defined in Lemma 5.2. In fact, TwigStackList guarantees the optimality for some (not all) cases where all current elements are current-blocked by using the look -ahead technique.

(b) Twig Query

c

a

c

b

c

d

b

d

(a) Data tree

a

Figure 5.11: Illustration to the optimality for TwigStackList in all-current-blocked cases

Example 5.6. See the query and data in Figure 5.11. Let us consider the TwigStack- Listalgorithm based on Tag Streaming scheme. At the beginning, all current elements are a1, b1, c1 and d1. According to the case (i) in Figure 5.10, d1 and c1 are current-

blocked elements and consequently a1, b1 are also current-blocked elements. In such

all-current-blocked case, TwigStackList buffers c1, c2 in the main memory and finds a

real match (a1, b1, c2, d1). This all-current-blocked case is “conquered” in TwigStack-

List by using the look-ahead technique.

The above example demonstrates that the look-ahead technique can solve the problem for all-current-blocked cases when P-C relationships occur in only non- branching edges. Unfortunately, this technique is not effective to conquer other sub- optimality cases caused by all-current-blocked. The main reason is the limited main memory size so that we cannot buffer too many elements in the main memory ( See Example 3.7 in Chapter 3 for more explanation ). Next, we will demonstrate a new algorithm, called GeneralTwigStackList, which not only can be applied on different streaming schemes, but also use the look-ahead technique to enlarge the optimality

query class identified by the above lemmas in this section.

5.5

Twig Join Algorithm

In this section, we describe a twig pattern matching method GeneralTwigStackList applicable to any streaming schemes discussed so far. Our method can work correctly for all twig patterns and meanwhile they are also optimal for certain classes of twig patterns depending on the streaming scheme used.