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

eaches (and possibly exceeds) the size of its publication window:
Div(s, t, Tr) ≥ Ws.
In consequence, we make a distinction between the total number of items published by a
source since the last refresh (stream divergence) and the number of only those available
ones (window divergence).
Definition 2.1.5. Stream divergence relevant to query q
Let F (s, t) be the publication stream of source feed s. Let function new(F (s, t)) be the
sequence of new items generated by s since its last refresh moment Tr. We define the
stream divergence as:
DivF (s, q, t, Tr) = |q(new(F (s, t)))|
Note that the divergence function introduced before (Definition 2.1.2) corresponds to the
stream divergence.
Definition 2.1.6. Window divergence relevant to query q
Let A(s, t) be the publication window of source feed s. Let function new(A(s, t)) return
the sequence of new items published since time moment Tr and still available at source s
at time moment t. We define the window divergence as:
DivA(s, q, t, Tr) = |q(new(A(s, t)))|
Observe that stream and window divergence functions relevant to query q are equal if the
source s is not yet saturated at time moment t:
if |new(A(s, t))| = |new(F (s, t))| < Ws, then DivA(s, q, t, Tr) = DivF (s, q, t, Tr)
Otherwise, as source s becomes saturated, stream divergence exceeds the window divergence
value:
if |new(A(s, t))| = Ws, |new(F (s, t))| > Ws, then DivA(s, q, t, Tr) < DivF (s, q, t, Tr)
We introduce an example in Figure 2.2 in order to better understand the evolution in time
of the stream and window divergence functions when different types of aggregation queries
are used: a simple union q1, with sel(q1) = 1, and a filtering query q2, with sel(q2) < 1.
We consider that source s becomes saturated at time instant Tsat. Both the stream and
window divergence functions relevant to query q1 reach the capacity of the publication
window Ws and those relevant to query q2 reach a value estimated near sel(q2) · Ws. In
Figure 2.2 the saturation point is marked by red lines.
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