Is a cube database, simply an RDBMS—such as PostgreSQL or MySQL—configured for OLAP via a star or snowflake schema? Or is a cube database different from an RDBMS, and if so how?
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The short answer is no.
Cubes are very a different beast from a traditional database. There are several different kinds of cube storage processes depending on need (OLAP, MOLAP, ROLAP, etc.) which are all done differently depending on how real time the data needs to be.
I actually did a webcast with another Microsoft MVP a couple of weeks ago where we talked a little about the BI process and how it impacts the traditional database, which might be interesting to you. The recordings (which are free) aren't available yet, but should be soon.
Simplistic answer: A cube allows you to define dimensions (things you'd need to filter on like Dates, , regions,....), hierarchies (eg months within a year, countries for Continents,...) and measures and rules on how these measures relate with the dimensions... Then you can query values based on that using MDX
More simplistic view?
An RDBMS would be an spreadsheet and a cube a Pivotable...
Hope this helps