

Popular aggregate operators like MIN and MAX cannot be handled by the techniques discussed in the previous section due to the lack of an inverse operation. This section summarizes approaches that work for these two operators as well. Some of them take advantage of specific monotonicity properties, while others apply whenever the domain of the measure attribute forms a commutative semigroup under the aggregate operator.
In Ho et al. (1997), a technique for the computation of range max queries is proposed. The data cube is recursively partitioned into smaller hyperrectangular chunks (similar to a multidimensional quad tree). For each chunk the position of the maximum value in the chunk is stored. The corresponding tree is traversed topdown, using pruning techniques to reduce the number of accesses. Lee, Ling, & Li (2000) use a similar data structure but a different pruning strategy. They obtain constant average query cost and logarithmic update cost in the size of the data cube (for fixed dimensionality).
Poon (2001) proposes a general approach for commutative semigroups. Similar to iterative data cubes, Poon's approach combines onedimensional schemes to obtain a multidimensional preaggregation technique. Query, update, and storage costs are and respectively. Here L_{i} ∊ {l,…, log N_{i}} denotes a usercontrolled parameter and γ(N_{i}) a slowgrowing function. Applied to invertible operators the query, update, and and respectively. Note, however, that techniques like IDC—which are optimized to take advantage of the inverse operator—achieve better cost tradeoffs for invertible operators. For instance, for L_{i}=1, an IDC that uses PS in each dimension achieves the same query but lower update and storage costs (cf., Table 1). Similarly IDC using RPS (DDC) in each dimension offers a better solution than the above technique for L_{i} = 2 (L_{i} = log N_{i}).

