Políticas de operación

We introduce a data structure to store the intermediate query results (i.e., the results to the query before applying the aggregation function). It is a simple tree data structure which has 2 node types; RVars representing random variables, and Nodes which represent an XML element containing a value. An RVar has 1 or more Node children and has exactly 1 Node parent. A Node has 0 or more RVar children and 0 or 1 RVar parent. When a Node has no RVar parent, it is the root of the tree. Additionally, a Node stores the probability of itself given its ancestors exist, and it contains the values associated with it. The root of the tree will have probability 1, since it does not belong to any RVar. The tree hierarchically stores intermediate query results based on their random variable string. The two types of nodes are required since the semantics are different between them. As defined earlier, the different valuations of one random variable are mutually exclusive. This translates to only 1 of the Node children of an RVar being true at the same time. On the other hand, all RVar children of any Node do exist at the same time. Therefore, it was necessary to keep track of the random variables associated with the values to know which values are mutually exclusive and which can exist simultaneously.

Consider an example of a simple probabilistic document with nested random variables in Figure 4.2 and a query which calculates the sum of all the <leaf> values: sum(//leaf). In order to construct the tree, the non-aggregate part of the query is executed first to yield all<leaf> nodes and their random variable strings. The random variable string of a<leaf> element is its random variable attribute combined with the random variable attribute of all its ancestors. The tree is then built from these <random variable string, value> tuples. The random variable string (e.g., “X=0 Y=0”) is processed from left to right to determine a value’s destination node. The resulting tree is displayed in Figure 4.3, where represent RVars and represent Nodes.

Such a tree is constructed for each aggregation function, with which the various summary values are calculated. Each aggregation function uses a slightly different implementation of the tree, which was inevitable since different operations are applied to a sequence of values at each node depending on the aggregate function (addition for Count/Sum and obvious operations for Min, Max, and Avg). In addition, the summary values are not the same for all functions. In the Min/Max case we can go beyond just summary values and are able to yield the top-k result values as well. That is, we return the actual result values to the aggregation function, ordered by descending probability and limited to an arbitrary number k. For Count/Sum/Avg yielding the top-k result values was not possible due to the characteristics of those aggregate functions, specifically the fact the number of distinct result values for those functions can

<root> <branch rv="X=0"> <leaf>2</leaf> <leaf>12</leaf> <leaf rv="Y=0">18</leaf> <leaf rv="Y=1">22</leaf> <leaf rv="Z=0">16</leaf> <leaf rv="Z=1">25</leaf> </branch> <branch rv="X=1"> <leaf>5</leaf> <leaf>17</leaf> <leaf rv="A=0">23</leaf> <leaf rv="B=0">1</leaf> <leaf rv="B=0">7</leaf> <leaf rv="B=0">20</leaf> </branch> </root>

Figure 4.2: Document with nested random variables

X 1 B A 0 0 0 Y Z 1 0 (5, 17) 1 0 (1, 7, 20) 23 25 16 22 18 (2, 12)

Figure 4.3: Tree structure of values before aggregation

grow as large as the number of possible worlds and there is no way to efficiently produce each possible value, or just the kmost likely ones.

4.2.1 Tree Confidence

A property of the tree that is shared among all implementations is the computation of the confidence of the tree. The confidence corresponds to the probability of the union of the random variable strings associated with the leaf nodes in the tree. The computation of this probability is defined recursively in our tree structure. In order to efficiently compute it, we use the property A∪B = (Ac_∩Bc₎c_{, i.e. the} union of two independent sets is equal to the complement of the intersection of their complements. We have discussed this property in Section 3.2.1.

We use this principle to compute the confidence value of any {Node, RVar} tree by defining a recursive

confidence function on either type of node. The confidence of an RVar node is defined as the sum of

confidences of its Node children. Taking the sum is allowed since the child nodes are mutually exclusive. For a Node we apply the union property described above. That is, the confidence of a Node is computed

by taking the complement of the multiplication of the confidence complements of all its RVar children, multiplied with the probability of the Node itself – the probability associated with the single random variable assignment that the Node represents, for example “X=0”. If a Node has no RVar children, the confidence is equal to this probability. Formally, the confidence is thus calculated as below. In this equation,ndenotes a Node,rdenotes an RVar,prob(n) denotes the probability of a Noden,n.Rdenotes the set of RVars belonging to Node n, and r.N denotes the set of Nodes belonging to RVar r. The formulas are based on the work of Koch and Olteanu [16], who have defined a similar tree structure – defined as a ws-tree – in their research on conditioning probabilistic databases. When defining our tree

structure we were not aware of their work yet.

conf(n) =    prob(n) if n.R=_∅ prob(n)·(1− Q r∈n.R 1−conf(r)) otherwise conf(r) = X n∈r.N conf(n)

Figure 4.4: The recursive computation of the confidence in a {Node, RVar} tree

The confidence is used in the summary computation of the Min and Max aggregation functions, as well in non-aggregate queries to calculate the probability of each unique value. In such a non-aggregate case, a unique value can have multiple associated random variable strings which each yield the specific value. In order to calculate the overall probability of the value – i.e. the confidence of the set of possible worlds that contain that value – we build a tree structure filled with all the random variable assignments and compute its confidence. The resulting probability is then equal to the sum of probabilities of possible worlds the value exists in. The remainder of this section will describe the procedures we use to yield the summary values for each of the aggregation functions.