Applying OLAP Pre-Aggregation Techniques to ... - Jacobs University

5.4 Answering Scaling Operations Using Pre-Aggregated Data 83
Algorithm 3 PRE-AGGREGATESSELECTION
Require: A workload Q, and a storage space constraint c
1: P = {top scaling operation}
2: while (c > 0 and |P | != |Q| ) do
3: p = highestBenefit(Q, P )
4: if (c - |p| > 0) then
5: c = c - |p|
6: P = P ∪ p
7: end if
8: else c = 0
9: return P
of selected pre-aggregates and n is the number of vertices in the lattice), which arises
from the cost of sorting the pre-aggregates by benefit per unit size.
5.3.1 Complexity Analysis
Let m be the number of queries in the lattice. Suppose we have no queries selected
except for the top query, which is mandatory. The time to answer a given query in the
workload is the time taken to compute the query using the top query and calculating
it according to our cost model. We denote this time by T o . Suppose that in addition
to the top query, we choose a set of queries P . Denote the average time to answer a
query by T p . The benefit of the set of queries P is the reduction in average time to
answer a query, that is, T o − T p . Thus, minimizing the average time to answer a query
is equivalent to maximizing the benefit of a set of queries.
Let p 1 , p 2 , ..., p k be the k queries selected by the PRE-AGGREGATESSELECTION
algorithm. Let b i be the benefit achieved by the selection of p i , for i = 1, 2, ..., k.
That is, b i is the benefit of p i , with respect to the set consisting of the top query and
p 1 , p 2 , ..., p i−1 . Let P = p 1 , p 2 , ..., p k .
Let O = o 1 , o 2 , ..., o k be an optimal set of k queries, i.e., those queries giving
the maximum benefit. Let m i be the benefit achieved by the selection of o i , for i =
1, 2, ..., k. That is, m i is the benefit of o i , with respect to the set consisting of the top
query and o 1 , o 2 , ..., o i−1 .
Harinarayan et al [92] proved that the benefit of the greedy algorithm can never
be less than (e-1)/e = 0.63 times the benefit of the optimum choice of pre-aggregated
queries.
5.4 Answering Scaling Operations Using Pre-Aggregated Data
We say that a pre-aggregate p answers query q if there exists some other query q ′
which when executed on the result of p, provides the result of q. The result can be
either exact with respect to q (q ′ ◦ p ≡ q), or only an approximation (q ′ ◦ p ≈ q).
In practice, the result is often an approximation because of the effect of resampling
the original dataset. The same effect is observed in the traditional image pyramids