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 1Qtr 2Qtr 3Qtr 4Qtr TV Product PC U.S.A VCR Canada Country Mexico

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

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

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

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

Date Product Country 1Qtr 2Qtr 3Qtr 4Qtr TV U.S.A VCR PC Canada Mexico 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 1Qtr 2Qtr 3Qtr 4Qtr Product TV U.S.A non-comp comp VCR PC Canada Country Mexico

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

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

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

Now let’s apply this technology to finding all bicliques. 1 2 3 A B C G5b1 bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC 1 2 3 A A B 2AB 2AB C 1AC 2AC B 2BC C

1 2 3 A A B C B C 2ABC 1AC 2AC 2BC 2AB 1AC 2AC 2 3 A B C G5b1 bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC 1 2 3 A 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??? A B 2AB C 1AC 2AC 1AC 2AC B 2BC C

1 2 3 A B C G5b1 bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC 1 2 3 A 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} B 2AB C 1AC 2AC A 2BC B C

1 2 3 A B C A B C 1AC 2AC 2AB 2BC 2ABC 2ABC 2 3 A B C G5b1 bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC 1 2 3 A 1AC 2AC B 2AB C 2BC A B 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} 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} C

1 2 3 A A B C B C 12 AC 2AB 1AC 2AC 2ABC 2BC 2ABC 2 3 A B C G5b1 bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC 1 2 3 A 12 AC A B 2AB C 1AC 2AC 2ABC B 2BC C 2ABC

1 2 3 A A B C B C 2AB 1AC 2AC 3AC 1BC 2BC 3BC 1AB 2AB 3AB G5b2 bCLQ3s centered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC 1 2 3 A A B 1AB 2AB 2AB 3AB C 1AC 2AC 3AC B 1BC 2BC 3BC C

1 2 3 A B C A B C 2AB 1AC 2AC 3AC 1BC 2BC 3BC 1AB 2AB 3AB G5b2 bCLQ3s centered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC 1 2 3 A B 1AB 2AB 2AB 3AB C 1AC 2AC 3AC A 1BC 2BC 3BC B C

1 2 3 A B C A B C 1ABC 2ABC 3ABC 1AC 2AC 1BC 2BC 3BC 1AB 2AB 3AB 1AC G5b2 bCLQ3s centered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC 1 2 3 A 1ABC 2ABC 3ABC B 1AB 2AB 3AB C 1AC 2AC 1AC 2AC 3AC A 1BC 2BC 3BC B C

AB 1 2 3 A B C A B C 1A BC 2A 3A 1AC 2AC 1BC 2BC 3BC 1AB 2AB 3AB 123 G5b2 bCLQ3s centered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 2AB 3AC 3BC 1 2 3 A 1A BC 2A 3A B 1AB 2AB 3AB 123 BC AB AC 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 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. C 1AC 2AC 1AC 2AC 3AC A 1BC 2BC 3BC B C

1 2 3 A B C A B C 1A BC 2A 3A 123 BC AB AC 1AC 2AC 123A BC 1BC 2BC 3BC G5b2 bCLQ3s centered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC 1 2 3 A 1A BC 2A 3A B 1AB 2AB 3AB 123 BC AB AC 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 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. C 1AC 2AC 1AC 2AC 3AC 123A BC A 1BC 2BC 3BC B C 1C AB 2C 3C 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}.

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