Roxana - Gabriela HORINCAR Refresh Strategies and Online ... - LIP6

2.1 Divergence
We focus on the aggregator node n, that applies an aggregation query q on the source s,
where s ∈ S(q). As introduced before, query q generates a new query feed f = q(S(q)),
where the aggregation query q represents a union of filters over the source feeds in S(q).
When discussing divergence, we examine how the changes that are of interest for the
aggregation query q and that occur on the source feed s are perceived and captured by
the aggregator n.
We begin by formally defining the notion of divergence, as well as the measure proposed
to evaluate it.
Definition 2.1.1. Divergence
Let Tr represent the last time moment before time t when source s has been refreshed by
the aggregator. We define the divergence function Div(s, t, Tr) measured at time t > Tr
as the total number of new items published by the source s in the time period (Tr, t] that
were not yet fetched by the aggregator.
Definition 2.1.2. Divergence relevant to query q
We similarly define the divergence function Div(s, q, t, Tr) measured at time t as the total
number of new items relevant to query q published by the source s in the time period
(Tr, t].
We suppose that the relevant items to query q are published uniformly distributed in time,
such that the selectivity sel(q) is considered constant. If the selectivity sel(q) of the query
q and the divergence Div(s, t, Tr) of the source s are known, then we can estimate the
divergence relevant to query q as:
Div(s, q, t, Tr) = sel(q) · Div(s, t, Tr)
Observe that the values of the divergences relevant to two different queries q and q ′ defined
on the same source s and measured at the same time instant t can be different, especially if
the selectivities of the two queries are different (if sel(q) �= sel(q ′ )), then Div(s, q, t, Tr) �=
Div(s, q ′ , t, Tr).
Definition 2.1.3. Divergence relevant to a set of queries Q
Let Q be the set of disjoint queries defined on source s by an aggregator node: ∩qi = ∅,
where qi ∈ Q. We define the divergence relevant to the set of queries Q measured at time
t as the sum of the divergence functions relevant to all queries qi ∈ Q.
Div(s, Q, t, Tr) = �
Div(s, qi, t, Tr)
qi∈Q
To better understand how the divergence function behaves in time, we introduce an example
in Figure 2.1. Let Tr−1, Tr and Tr+1 represent the time moments at which source s is
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