We estimate the speedup factor that can be achieved using query pack execution in two steps: first we consider one-level packs, then we extend the results towards deeper packs.
Lower and upper bounds on the speedup factor that can be achieved by
executing a one-level pack instead of separate queries can be obtained as
follows. For a pack containing queries
, let be the time needed to compute the first
answer substitution of if there are any, or to obtain failure
otherwise. Let be the part of spent within and
the part of spent in . Then
the total time needed for executing all queries separately and
the total time needed for executing the pack. Introducing
, which roughly represents the ratio of the computational
complexity in the shared part over that in the non-shared part, we have
Thus the speedup factor is bounded from above by the branching factor and by the ratio of computational complexity in the shared part over the computational complexity of the non-shared part; and a maximal speedup can be attained when (or, ), in other words when the for all queries are approximately equal.
For multi-level packs, we can estimate the efficiency gain as follows.
Given a query , let be defined as above (the total time for
finding 1 answer to or obtaining failure). Instead of and
, we now define as the time spent on level of the
pack when solving ; counting the root as level 0 and denoting the
depth of the pack with we have
Further define as the time spent on level or deeper:
with the depth of the pack.
(Thus .). We will assume a constant branching factor
in the pack. Finally, we define
with . For simplicity, in the formulae we
implicitly assume that always ranges from 1 to with the
number of queries, unless explicitly specified otherwise. We then
Note that for we have
Further, similar to in our previous analysis, define
Some algebra then gives
This inequality holds for all , hence we will find the best lower bound for
the speedup factor by maximizing the right hand side. Note that
increases and decreases monotonically with .
It is clear that if at some point becomes much larger than 1, a
speedup factor of roughly is obtained. On the other hand,
if is smaller than 1, then the behaviour of is crucial.
Our conclusion is similar to that for the one-level pack. If for some , , i.e., if in the upper part of the pack (up till level ) computations take place that are so expensive that they dominate all computations below level (even taking into account that the latter are performed times more often), then a speedup of can be expected. If , which will usually be the case for all except those near , the speedup can roughly be estimated as . The maximum of all these factors will determine the actual speedup.