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Mining for Empty Rectangles in Large Data Sets

Mining for Empty Rectangles in Large Data Sets. Jeff Edmonds Jarek Gryz Dongming Liang Renee Miller. A. B. 3. 6. 1. 7. 3. 8. 1 2 3. 0. 0. 1. 6. 1. 0. 0. 7. 0. 0. 1. 8. Matrix representation.  A,B (R. S). al. um. 0. A. B. 3. 6. 0. 1. 7. 0. 3. 8. 1 2 3. 0.

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Mining for Empty Rectangles in Large Data Sets

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

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

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

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

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

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

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

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

  9. Designing an Algorithm Exit Exit Exit 0 km Exit 79 km 75 km 79 km to school Exit

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

  11. 1 ( x ,y ) r r 1 Stack of steps step 0 0 y* 1 1 0 0 0 1 0 ( x ,y ) ( x ,y ) ( x ,y ) ( x ,y ) ( x ,y ) 3 2 1 4 5 4 1 2 5 3 1 0 1 0 0 0 0 x* 1 Y 1 <x,y> * X

  12. Constructing Maximal Rectangles <x,y> *

  13. Constructing Maximal Rectangles • Too Narrow • Maximal • Too short <x,y> *

  14. 0 <x,y> * Constructing staircase(x,y)from staircase(x-1,y) 1 1 0 Case 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 <x-1,y> * 1 0 1 0 0 0 0

  15. Constructing staircase(x,y)from staircase(x-1,y) 1 Case 2 ( x ,y ) r r 1 1 Y 1 1 0 1 0 0 0 0 0 1 0 ( x ,y ) 1 1 <x-1,y> * 1 0 ( x, y ) 1 0 0 0 0 X

  16. Constructing staircase(x,y)from staircase(x-1,y) Delete 1 • Too Narrow • Maximal • Too short ( x ,y ) r r 1 1 Keep Y 1 1 0 0 0 1 0 0 0 0 0 1 0 ( x ,y ) 1 1 <x,y> * <x-1,y> * 1 0 ( x, y ) 1 0 0 0 0 X

  17. y* Constructing x*& y* 1 ( x ,y ) r r 1 1 1 0 1 0 0 1 0 0 0 0 0 1 0 ( x ,y ) 1 1 <x,y> * 1 0 ( x, y ) x* 1 0 0 0 0

  18. Location of last 1 seen in each column 1 1 1 Y 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0 0 0 0 0 0 0 0 0 <x,y> X

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

  20. Timing Only work that is not constant Time Delete 1 • Too Narrow • Maximal • Too short ( x ,y ) r r 1 1 Y 1 1 0 0 0 1 0 0 0 0 0 1 0 ( x ,y ) 1 1 <x,y> * 1 0 ( x, y ) 1 0 0 0 0 X

  21. 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 <x-1,y> * 1 0 1 0 0 0 0 Amortized # of steps deleted (per <x,y>) = # of steps created (per <x,y>) £ 1 Timing

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

  23. Designing an Algorithm Exit Exit Exit 0 km Exit 79 km 75 km 79 km to school Exit

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