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  1. Remember the Sales Data Cube? Each cell contains a sales measurement, e.g., the number of sales (may contain many other measurements of product-date-country instances) We will attempt to apply this technology to the task of finding bicliques later, after reviewing the technology. Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  2. Total of all product sales by country and quarter Total sales by country and dateRollup (aggregate under +) along product (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  3. Total annual sales by country and product Rollup along date (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  4. Total of all product sales by product and date Total of all product sales by product and date Rollup along country (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  5. sales by product, country sales by country sales by country sales by country, date sales by product sales by product sales by product, country Total sales Total sales Total sales sales by date sales by date All rollups (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC sales by product, country and quarter VCR Canada Country Mexico

  6. TV VCR PC Partial Rollup: climbing up a concept hierarchy(instead of eliminating Product altogether by summing over all products, rollup partially on Product, from (VCR, PC, TV) to computer (includes PC only) and non-computer (includes VCR + TV) Date 2Qtr 1Qtr 3Qtr 4Qtr Product U.S.A non-comp comp Canada Country Mexico

  7. TV VCR PC SLICE e.g., slice off PC Date 2Qtr 1Qtr 3Qtr 4Qtr Product U.S.A Canada Country Mexico

  8. 3Qtr 4Qtr PC Mexico DICE (e.g. dice off PC, the last two quarters, the country Mexico) Date 2Qtr 1Qtr Product TV U.S.A VCR Canada Country

  9. secondary Country Mexico Canada U.S.A Date 2Qtr 1Qtr 3Qtr 4Qtr 4Qtr TV Product U.S.A PC VCR 3Qtr Date Canada Country 2Qtr Mexico 1Qtr TV PC VCR Product tertiary Pivot/Rotate primary

  10. bCLQ3scentered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 Now let’s apply this technology to finding all bicliques. 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A A B 2AB 2AB C 1AC 2AC 1 2 B 3 2BC C A B C

  11. 2ABC RollUp along the front-to-back dimension using the hub intersection and spoke union gives the expanded hub-and-spoke biclique, hub={2}, spokes={A,B,C} or hub={2A}, spokes={B,C} or the hub-union (of hubs {B},{C}), spoke-intersection (of spokes {2,A}). Rather than view it as an intersection-union of hubs and spokes, I think it suffices to just take the union??? bCLQ3scentered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A A B 2AB C 1AC 2AC 1AC 2AC 1 2 B 3 2BC C A B C

  12. 12AC RollUp along the left-right dimension using the hub intersection and the spoke union gives the one expanded biclique, (hub={AC}, spokes={1,2} bCLQ3scentered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A B 2AB C 1AC 2AC 1 2 A 3 2BC C A B B C

  13. 1AC 2AC 2BC 2ABC 2ABC RollUp along the top-bottom dim using hub intersection and spoke union gives the expanded hub-and-spoke biclique, (hub={2}, spokes={A,B,C} RollUp along the top-bottom dim using hub intersection and spoke union gives the expanded hub-and-spoke biclique, (hub={2}, spokes={A,B,C} bCLQ3scentered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A B 2AB C 1 2 A 3 C A B B C

  14. 12 AC 2ABC 2ABC bCLQ3scentered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A A B 2AB C 1AC 2AC 1 2 B 3 2BC C A B C

  15. bCLQ3scentered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC A B C 3 3 3 1 1 1 1 3 2 1 1 1 3 3 1 1 1 3 1 2 3 G5b2 A A B 1AB 2AB 3AB 2AB C 1AC 2AC 3AC 1 2 B 3 1BC 2BC 3BC C A B C

  16. bCLQ3scentered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC A B C 3 3 3 1 1 1 1 3 2 1 1 1 3 3 1 1 1 3 1 2 3 G5b2 A B 1AB 2AB 3AB 2AB C 1AC 2AC 3AC 1 2 A 3 1BC 2BC 3BC C A B B C

  17. 1ABC 2ABC 3ABC bCLQ3scentered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC A B C 3 3 3 1 1 1 1 3 2 1 1 1 3 3 1 1 1 3 1 2 3 G5b2 A B 1AB 2AB 3AB C 1AC 2AC 3AC 1AC 2AC 1 2 A 3 1BC 2BC 3BC C A B B C

  18. 123 AB 123 AC 1A BC 2A BC 3A BC 123 BC UnionRollUp along front-back dim gives expanded bicliques, hub={1,A} spoke={B,C}. hub={2,A} spoke={B,C}, hub={3,A} spoke={B,C}. UnionRollUp along left-right dim gives expanded bicliques, hub={A,B} spokes={1,2,3}, hub={A,C} spokes={1,2,3}, hub={B,C} spokes={1,2,3}. Note: hub is always the combo of fixed values. bCLQ3scentered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 2AB 3AC 3BC A B C 3 3 3 1 1 1 1 3 2 1 1 1 3 3 1 1 1 3 1 2 3 G5b2 A B 1AB 2AB 3AB C 1AC 2AC 3AC 1AC 2AC 1 2 A 3 1BC 2BC 3BC C A B B C

  19. 123 AB 123 AC 123A BC 1A BC 2A BC 3A BC 123 BC UnionRollUp along front-back dim gives expanded bicliques, hub={1,A} spoke={B,C}. hub={2,A} spoke={B,C}, hub={3,A} spoke={B,C}. UnionRollUp along left-right dim gives expanded bicliques, hub={A,B} spokes={1,2,3}, hub={A,C} spokes={1,2,3}, hub={B,C} spokes={1,2,3}. Note: hub is always the combo of fixed values. 1C AB 3C AB 2C AB UnionRollUp along the top-bottom dim gives the expanded biclique, hub={1,C} spokes={A,B}; hub={2,C} spokes={A,B}; hub={3,C} spokes={A,B}. bCLQ3scentered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC A B C 3 3 3 1 1 1 1 3 2 1 1 1 3 3 1 1 1 3 1 2 3 G5b2 A B 1AB 2AB 3AB C 1AC 2AC 3AC 1AC 2AC 1 2 A 3 1BC 2BC 3BC C A B B C

  20. 3 C bCLQ3scentered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC bCLQ3s 1AB 2AB A B C A B 2 2 3 3 3 DICE (e.g. dice off 3 AND C.) 1 1 1 1 3 1 1 1 2 2 1 1 1 3 2 1 1 2 3 1 1 1 3 2 1 G5b2 G5b3 A B 1AB 2AB 1 1 2 2 A 3 C A A B B B C