Before defining IP-tables in section 7.2 we want to make three observations which serve as a motivation:
).
Imagine now a temporal relation R and a partition P with the
breakpoints . Then the number |Rk| of
tuples in a fragment Rk can be determined by (i) the number of
tuples that overlap from preceding fragments Rj (j<k) plus (ii)
the tuples that start in the partition range
(pk-1,pk] = [pk-1+1,pk] that corresponds to Rk.
Obviously (i) corresponds to the number of intervals which contain the
first point in the partition range but start before, i.e. the number
, whereas (ii) can be determined by summing up the
values . Thus
| (23) |
,
, 
, 
for a temporal relation R. The values of the
unknown functions can be derived by using equations (5.1),
(5.2) and (5.3) or their derivatives in
figure 5.1. For the following, we choose and to be stored explicitly whereas and are derived when
necessary.
and 
for those t at which at least one interval starts or ends: for all
other timepoints, t, it is and with t'
being the next start- or endpoint (of some interval) before t. This
corresponds to the observation made in theorem 1 in
section 5.4 and allows us to concentrate on the
start- and endpoints![[*]](foot_motif.gif)
of the intervals rather than the
entire time span.