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Closed and Iceberg Cubes Reduction necessity • Data cube produces large outputs – 1,015,367 tuples (39MB) – 210,343,580 tuples (8GB)(200 times) • Two methods to reduce outputs – Iceberg cube – Closed cube Closed Iceberg cube Cells and Measures • Cell – In an n-dimension data cube, a cell c = (a1,a2,…,an: m) (where m is a measure) is called a k-dimensional group-by cell, if and only if there are exactly k (k<=n) values among {a1,a2,…,an} which are not * (i.e., all). – Further denote M(c) = m and V(c) = (a1,a2,…,an). Notion of Cover • Cover – Given two cells c = (a1,a2,…,an:m) and c’ = (a1’,a2’,…,an’:m’), we denote V(c)<= V(c’) if for each ai (i = 1,…,n) which is not *, ai’ = ai. – A cell c is said to be covered by another cell c’ if c’’ such that V(c)<=V (c’’)<=V (c’), M(c’’) = M(c’). Closed and iceberg cells • Closed cell – A cell is called a closed cell if it is not covered by any other cells. • Closed Iceberg cell – Closed cell which satisfies the iceberg constraints Closed and iceberg cell contd • Let the measure be count, and the iceberg constraint be count>=2. • Cell1 = (a1,b1,c1,*: 2), and cell2 = (a1,*, *, * : 3) are closed iceberg cells; • Cell3 = (a1,*, c1,* : 2) and cell4 = (a1, b2, c2, d2 : 1)are not, because the former is covered by cell1, where as the latter does not satisfy the iceberg constraint. Methods of computation • Top-down method • Bottom-up method • These methods of computing the cubes such as BUC, Multi array aggregation and Star Cubing shall be explained in detail in the next resource in this module.