Approximation Algorithms for MAX-MIN tiling

1. Approximation Algorithms for MAX-MIN tiling Authors Piotr Berman, Bhaskar DasGupta, S. Muthukrishman Published on Journal of Algorithms, 2003 Presented by Sera Kahruman

2. Outline • Definition • Approximation algorithms • Greedy Slice and Dice • Recursive slice and dice-Binary space partitionings • Conclusion

3. Outline • Definition • Approximation algorithms • Greedy Slice and Dice • Recursive slice and dice-Binary space partitionings • Conclusion

4. Definition Given a 2-dimensional array A[1,…,n][1,…,n] s.t. each A[i][j] stores a non-negative number • A tile is a subarray A[l,…,r][t,…,b] of A • Weight of a tile T, w(T) = sum of all array entries in that tile • A tiling of A is a collection of tiles of A s.t. each entry of A is contained in exactly one tile

5. Max-Min Tiling Problem Given a weight bound W, find a tiling of A such that: • Each tile is of weight at least W • The number of tiles is maximized Max-Min tiling is NP-hard! Data mining…..an application area

6. More definitions • (r, s) approximation of max-min tiling is a solution that produces a tiling of A with at least rp* tiles each of which has weight at least s(W-c) where; • p* = max number of tiles used in an optimal solution • c≥0 is a constant

7. Slice and Dice Technique • Partition the input array into a number of slices (rectangles) using some criteria (depending on the problem)…..slicing step • Search for a better solution by looking at the neighbor slices and repartitioning the entries…..dicing step

8. Outline • Definition • Approximation algorithms • Greedy Slice and Dice • Recursive slice and dice-Binary space partitionings • Conclusion

9. Approximation algorithms for Max-Min tiling • Greedy slice and dice • Linear in number of non-zero entries (m) • Greedily slice the array into strips and dice each slice • Recursive slice and dice: binary spacepartitionings • Use BSP of isothetic rectangles • Special case : Every cut of BSP is either a vertical or horizontal line Results

10. Greedy Slice and Dice Algorithm Observation 1: Given an instance array A of the MAX-MIN tiling problem, let A’ be the array obtained from A in which every element larger than 1 is replaced by 1. Then, any feasible solution for A’ is also a feasible solution for A and vice versa.

11. Greedy Slice & Dice Algorithm • Input array : array of row lists where each row i contains an entry (j, x) for each A[i][j]=x > 0 O( m+n) space • m = number of non-zero entries • Scale the array such that W=1 if necessary • After scaling w(A) becomes an upper bound • A good tile is a tile whose weight is at least 1 • The algorithm finds a solution with t good tiles Such that w(A)< r*t + 2

12. Slicing Step • Slices are composed of complete rows • Start with the bottom row and proceed upwards • Find minimal slices • Last slice is the remainder slice, others regular

13. Example : Slicing step R4 S3 R3 S2 R2 S1 R1 • Compute L and the array t where: • L= number of regular slices • t[i] = Row index such that Si consists of the rows in between t[i-1] and t[i] Can compute in O(m +n) time!

14. Dicing Step • Partition each slice Si using the slicing algorithm considering columns instead of rows • V-slices (vertical slices) are produced • α(i) = # of V-slices obtained from Si • Vi,1 , …. , Vi, α(i) are regular, Vi, α(i)+1 remainder • Combine Vi, α(i)+1 with Vi, α(i) for preliminary partition

15. Example: Dicing step Done in O(m+ n) time

16. Dicing step : Cont’ • Estimate weight of each part • w(VLi,j)< 1 • Summation of all w(VRi,j)of Si < 1 • w(VCi,j)≤ 1 • Split each V-slice Vi,j into 3 parts • VLi,j :all columns except last • VRi,j : last column except its topmost element • VCi,j :topmost element of last column C L R

17. Cont’ • Guarantee an approx. ratio of 4 (use at least floor(w(A))/4 tiles) Proof : Let t be the # of tiles obtained using the algorithm • w(Si) < 2+ 2α(i) ≤ 4α(i) • w(A) < 1 + w(S1)+……..+w(SL) < 1 + 4*(α(1)+……+ α(L) ) < 1 + 4*t 4*t ≥ floor(w(A)) • Do additional dicing to improve the ratio

18. Additional Dicing Step • Each additional dicing on Si-1USi is done in O(β+2 ) where β is total number non-zeros in Si-1USi • Total running time still O(m+n) • Define σ(i) =1 if # of tiles increased when Si-1U Si processed σ(i) =0 ow

19. Cont’ • With the additional dicing step, approximation ratio becomes 3. Proof: need to show that t ≥ floor((w(A)+1)/3) w(A) < 3*t + 2 w(S1)+……..+w(SL) < 3*t + 1 Since need to show : Proved by induction!!!

20. Outline • Definition • Approximation algorithms • Greedy Slice and Dice • Recursive slice and dice-Binary space partitionings • Conclusion

21. Recursive Slice & Dice • Based on binary space partitioning of a set of isothetic rectangles • isothetic rectangle = rectangle with edges parallel to principal axes • Given a rectangular region R containing a set of n disjoint isothetic rectangles, a BSP of R consists of recursively partitioning R by a horizontal or vertical line into 2 sub regions and continuing in this manner for all the sub regions until each obtained region has at most 1 rectangle. If a rectangle is cut by a line, it is split into disjoint rectangles • Size(BSP)= # of regions produced • A set of rectangles form a tiling if they partition some rectangular region R. • Use dynamic programming – min size BSP in O(n5) time.

22. Theorems Proof of Theorem 3: Consider an optimal solution OPT using p* tiles. So there exists a BSP of OPT that has at most 2p* regions in which each rectangle of OPT is cut into at most 4 pieces( from thm 2) If a rectangle is cut into i pieces, at least one piece has weight at least W/i. fraction of rectangles which has a weight of at least W/4 = 1/2 fraction of rectangles which has a weight of at least W/3 = 4/9

23. Recursive Slice & Dice How to compute such a BSP of A? • Definitions • HBP : Hierarchical binary partition An HBP of an array is either the entire array or the unions of HBPs of the two sub arrays of the array. • Γ(I,j,k,l,p) = HBP of the sub array A[ i,.,j ][ k,…,l ] in which there exists p tiles of weight at least s*W. • A BSP of OPT is also a HBP of A • Use dynamic programming to find {Γ(1,n,1,n,p) | 0 ≤ p ≤ n2}

24. Time analysis • There are at most O(n4) sub arrays of A • If the weight of the sub array A[ i,.,j ][ k,..,l ] is at least s*W, then • # of ways to construct a HBP of it is (j - i) + (l – k ) + 3 ≤ 2n-1…………O( n ) • 0 ≤ p ≤ n2 Since we are looking for the answer for each p, Total running time is O(n4 * n * n2)=O(n7 )

25. Outline • Definition • Approximation algorithms • Greedy Slice and Dice • Recursive slice and dice-Binary space partitionings • Conclusion

26. Conclusion • Authors introduce two approximation algorithms for MAX-MIN tiling problem using slice and dice technique • They state that efficient approximation algorithms for this problem in higher dimensions can be improved using variations of this approach

27. QUESTIONS ???

28. Homework • Formulate MAX-MIN tiling problem as an optimization problem. • In the paper, authors also mention MIN-MAX tiling problem. Define MIN-MAX tiling problem and state the main differences between the two problems.