2. Mining for Empty Rectangles in Large Data Sets Jeff Edmonds Jarek Gryz Dongming Liang Renee Miller

3. 2 0 0 1 1 0 0 0 0 1 1 2 3 6 7 8 Matrix representation A B 3 1 3 6 7 8  A,B (R S)

4. 3 0 0 1 1 0 0 0 0 1 1 2 3 6 7 8 Find All Maximal 0-Rectangles  A,B (R S) 0 0 0 00 0 al 0 0 0 um A B 3 1 3 6 7 8

5. 4 0 0 1 1 0 0 0 0 1 95 96 97 BMW Z3 Honda L2 Toyota 6A Example  A,B (R S) 00 Car Year … First BMW Z3 series cars were made in 1997.

6. 5 Relation to Previous Work [Lui, Ku, Hsu] & [Orlowski] Our Work Problem: Purpose: Machine Learning Computational Geometry Query Optimization between points in real plane within a 0-1 matrix Find all maximal empty rectangles # of maximal 0-rectangles: O( (# 1’s) 2 ) O( #0’s ) [Namaad, Hsu, Lee]

7. 6 Relation to Previous Work Our Work Time: Space: O(|X||Y|) O(min (|X|, |Y|)) only two rows of matrix kept in memory O( # 1’s log(#1’s) + # rectangles ) = O(|X||Y|) O( #0’s ) = O(|X||Y|) [Lui, Ku, Hsu] & [Orlowski] [Namaad, Hsu, Lee]

8. 7 Relation to Previous Work Our Work Practical Implementation: Scalable: Scales Badly Scales well wrt # of tuples in join # of maximal rectangles # of values |X| & |Y| Intensive random memory access  Requires a single scan of the sorted data Practical? IBM paid us $25,000 to patent it! [Lui, Ku, Hsu] & [Orlowski] [Namaad, Hsu, Lee]

9. 8 Structure of Algorithm loop y = 1..|Y| loop x = 1..|X| Output all maximal 0-rectangles with as bottom-right corner Maintain the loop invariant 1 1 1 1 1 X 0 Y 0 1 Timing O(1) amortized time per *

10. 9 Designing an Algorithm Define ProblemDefine Loop Invariants Define Measure of Progress Define StepDefine Exit ConditionMaintain Loop Inv Make ProgressInitial ConditionsEnding 79 km to school Exit 79 km75 km Exit 0 kmExit

11. 10 1 1 1 1 1 0 0 1 X Y * Define the Loop Invariant We have read the matrix up to and cannot reread the matrix. We must output all maximal 0-rectangles with as bottom-right corner What must we remember?

12. 11 0 step 1 1 1 1 1 0 ( x,y ) rr 11 22 33 44 55 Stack of steps 1 1 X Y * 10000 1 0 0 0 0 0 x*x* y*y*

13. 12 * Constructing Maximal Rectangles

14. 13 Too Narrow Maximal Too short * Constructing Maximal Rectangles

15. 14 * Constructing staircase(x,y) from staircase(x - 1,y) 1 1 1 1 1 1 1 10000 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 Case 1 * 0

16. 15 1 1 1 1 1 1 1 X Y 0 10000 1 0 0 0 0 0 ( x,y ) rr 11 0 * Constructing staircase(x,y) from staircase(x - 1,y) 0 Case 2

17. 16 1 1 1 1 1 1 1 X Y 0 10000 1 0 0 0 0 0 ( x,y ) rr 11 0 Too Narrow Maximal Too short * Constructing staircase(x,y) from staircase(x - 1,y) 0 0 Delete Keep * 0

18. 17 Constructing x * & y * 1 1 1 1 1 1 1 0 10000 ( x,y ) rr 11 0 * 0 0 0 0 0 0 0 1 0 x*x* y*y*

19. 18 X Y 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 Location of last 1 seen in each column *

20. 19 Structure of Algorithm loop y = 1..|Y| loop x = 1..|X| Construct staircase(x,y) Output all maximal 0-rectangles with as bottom-right corner 1 1 1 1 1 X 0 Y 0 1 Timing O(1) amortized time per Third *

21. 20 1 1 1 1 1 1 1 X Y 0 10000 1 0 0 0 0 0 ( x,y ) rr 11 0 Too Narrow Maximal Too short * Timing 0 0 Delete 0 Only work that is not constant Time

22. 21 Timing Amortized # of steps deleted (per ) = # of steps created (per )  * 1 1 1 1 1 1 1 10000 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

23. 22 Number of Maximal Rectangles # of maximal 0-rectangles: O( (# 1’s) 2 ) [Namaad, Hsu, Lee] Running time of alg = O( #0’s )  

24. 23 Designing an Algorithm Define ProblemDefine Loop Invariants Define Measure of Progress Define StepDefine Exit ConditionMaintain Loop Inv Make ProgressInitial ConditionsEnding 79 km to school Exit 79 km75 km Exit 0 kmExit