1. The geometric GMST problem with grid clustering Presented by 楊劭文, 游岳齊, 吳郁君, 林信仲, 萬高維 Department of Computer Science and Information Engineering, National Taiwan University

2. Outlines Geometric GMST with grid clustering Proof of NP-hardness –R–Reduction –O–Optimal structure –O–Optimal cost Dynamic programming algorithm Polynomial time approximation scheme 2Special Topics on Graph Algorithms

3. Minimum Spanning Tree a tree formed from a subset of the edges in a given undirected graph, with two properties: – (1) it spans the graph, i.e., it includes every vertex in the graph, and – (2) it is a minimum, i.e., the total weight of all the edges is as low as possible. 3Special Topics on Graph Algorithms

4. Generalized Minimum Spanning Tree A partition of the vertex set V into clusters Find a tree of minimum cost containing at least one vertex in each cluster 4Special Topics on Graph Algorithms

5. Applications Applications are encountered in telecoms. 5Special Topics on Graph Algorithms

6. Geometric GMST w/grid clustering The graph is complete All vertices are the points situated inside the k × l planar integer grid Edge cost: Euclidean distance between the points in the plane All points in the same cell form a cluster k × l grid is the smallest integer grid containing all points 6Special Topics on Graph Algorithms

7. Geometric GMST w/grid clustering 7Special Topics on Graph Algorithms

8. Outlines Geometric GMST with grid clustering Proof of NP-hardness –R–Reduction –O–Optimal structure –O–Optimal cost Dynamic programming algorithm Polynomial time approximation scheme 8Special Topics on Graph Algorithms

9. Theorem 1 The geometric GMST is strongly NP-hard, even if we restrict to instances in which all nonempty grid cells are connected and each grid cell contains at most two points Proof by reducing from the problem exact cover by 3-sets (X3C) 9Special Topics on Graph Algorithms

10. Exact Cover by 3-Sets A ground set X = {1, 2, …, n}, n = 3q S1S1 S2S2 S3S3 S4S4 x1x1 x3x3 x4x4 x2x2 x5x5 x6x6 C = {S 1, S 2, …, S m } – For 1 ≤ i ≤ m, S i is a subset of X – |S i | = 3 10Special Topics on Graph Algorithms

11. Exact Cover by 3-Sets Is there a set C’ such that – C’ ⊆ C – The elements of C’ are disjoint and – For each x i C’, Ux i = X x1x1 x3x3 x4x4 x2x2 x5x5 x6x6 S1S1 S2S2 S3S3 S4S4 11Special Topics on Graph Algorithms

12. x2x2 x1x1 S1S1 S2S2 S3S3 12Special Topics on Graph Algorithms

13. x 1 S 3 x 2 S 2 13Special Topics on Graph Algorithms

14. Outlines Geometric GMST with grid clustering Proof of NP-hardness –R–Reduction –O–Optimal structure –O–Optimal cost Dynamic programming algorithm Polynomial time approximation scheme 14Special Topics on Graph Algorithms

15. Connecting Edge Connecting Edge (dotted edge) Its length d is slightly larger than √2. Assume d is arbitrary close to √2. 15Special Topics on Graph Algorithms

16. Lemma1 No edge in T opt is larger than d, where T opt is some optimal solution. 16Special Topics on Graph Algorithms

17. Optimal subgraph 17Special Topics on Graph Algorithms

18. Lemma2 The subgraph induced by an arbitrary optimal solution and nonempty cells of an arbitrary block is connected. 18Special Topics on Graph Algorithms

19. Optimal Subgraph 19Special Topics on Graph Algorithms

20. Two possible structures Two possible structure in a column. – By lemma1 and lemma2 Trunk: the structure in a column. 20Special Topics on Graph Algorithms

21. Outlines Geometric GMST with grid clustering Proof of NP-hardness –R–Reduction –O–Optimal structure –O–Optimal cost Dynamic programming algorithm Polynomial time approximation scheme 21Special Topics on Graph Algorithms

22. Calculate the Total Cost For any n ≥ 1 let be the total cost of the edges in a trunk Let > 0 be a small enough number. 22Special Topics on Graph Algorithms

23. we can move some points by a very small distance – The cost of a red trunk remains – The cost of a blue trunk is – Connecting blocks in a red trunk costs d – The connection cost for a blue trunk is as follows. Connecting block i with block i + 1 in column j costs d − if i ∈ and d otherwise Differences between Red Trunk & Blue Trunk 23Special Topics on Graph Algorithms

24. Definition let Z = c( ) be its cost. = Z−3(m−1)(n+1) let be the contribution of column j 24Special Topics on Graph Algorithms

25. Connecting edge For a connecting edge e in a column j we define its averaged connecting cost as where is the number of connecting edges in column j. We have 25Special Topics on Graph Algorithms

26. Use Blue Trunk the averaged connecting cost c(e) for each of the three connecting edges e in this column is if a column j contains at least one connecting edge e that connects block i with block i+1 while, then the averaged connecting cost c(e) is at least 26Special Topics on Graph Algorithms

27. X3C  GMST If an exact cover exists if no cover exists 27Special Topics on Graph Algorithms

28. Outlines Geometric GMST with grid clustering Proof of NP-hardness –R–Reduction –O–Optimal structure –O–Optimal cost Dynamic programming algorithm Polynomial time approximation scheme 28Special Topics on Graph Algorithms

29. Definitions t ∈ {1, 2,..., − 3} C t : The t th column S t : subset of V containing exactly one point from each nonempty cell in C t+1,C t+2, and C t+3. T t : edge set on S t-1 U S t M: zero-one transitive matrix represents the connectivity f (S t,M): a generalized minimum spanning forest CtCt C t+2 C t+3 C t+1 S t-1 StSt …… M M’ f (S t,M) f (S t-1,M’) 29Special Topics on Graph Algorithms

30. Lemma 3 Assume that all nonempty grid cells are connected, then an optimal solution of a geometric GMST with grid clustering does not contain edges of length greater than 2√2. By Lemma 3, any forest f(S t, M) can be obtained as a forest f(S t-1, M’) extended by a subset T t of edges on the point set S t-1 ∪ S t. 30Special Topics on Graph Algorithms

31. Dynamic programming algorithm The recursive relation: Consistency Enumerate S t and M Enumerate S t-1 and M’ Enumerate T t Adding 4k points Number of S t 31Special Topics on Graph Algorithms

32. Theorem 2 The dynamic programming algorithm solves the geometric GMST with connected nonempty grid cells in time The computation time is polynomial if k is fixed. 32Special Topics on Graph Algorithms

33. Outlines Geometric GMST with grid clustering Proof of NP-hardness –R–Reduction –O–Optimal structure –O–Optimal cost Dynamic programming algorithm Polynomial time approximation scheme 33Special Topics on Graph Algorithms

34. Polynomial Time Approximation Scheme (PTAS) Assume all nonempty grid cells are connected. The number is at least. The PTAS is based on the DP. It is a - approximation where. 34Special Topics on Graph Algorithms

35. Partitioning into Slices Define. Slice 1 Slice 2 Slice 3 Slice △ Row#Rows 35Special Topics on Graph Algorithms

36. Finding GST for each Slice GMSTs are obtained by applying DP. Obtain a GST by adding edges only in the upper/bottom rows of the slice. Slice i 36Special Topics on Graph Algorithms

37. Obtaining the GST for the Graph Picking edges greedily yields GST. Slice 1 Slice 2 Slice 3 Slice △ Row 37Special Topics on Graph Algorithms

38. T APPX : (1+ ε)-approximation 1. 2. T OPT 38Special Topics on Graph Algorithms

39. Lower Bound of c(F i ) 3. Slice i 39Special Topics on Graph Algorithms

40. Lower Bound of c(F i ) 3. Slice i 40Special Topics on Graph Algorithms

41. Combining (1), (2) and (3) 4. 41Special Topics on Graph Algorithms

42. Upper Bound of c(T OPT ) Consider 3×3 subgrid with nonempty center. There are at least such subgrids. It takes at least length 1 for the center to connect to its boundary. 5. 42Special Topics on Graph Algorithms

43. Combining (4) and (5) 6. 43Special Topics on Graph Algorithms

44. Open Questions, Further Research PTAS for geometric GMST with non- intersecting square clusters of variable sizes. Fast constant approximation algorithms for geometric GMST with grid clustering. – DP as a subroutine of PTAS is impractical. 44Special Topics on Graph Algorithms

45. THE END Thanks 45Special Topics on Graph Algorithms