The geometric GMST problem with grid clustering

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 • Reduction • Optimal structure • Optimal cost • Dynamic programming algorithm • Polynomial time approximation scheme Special 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. Special 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 Special Topics on Graph Algorithms

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

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

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

10. S1 S2 S3 S4 x1 x2 x3 x4 x5 x6 Exact Cover by 3-Sets • A ground set X = {1, 2, … , n}, n = 3q • C = {S1, S2, …, Sm} • For 1 ≤ i ≤ m, Si is a subset of X • |Si| = 3 Special 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 xiC’, Uxi = X S1 S2 S3 S4 x1 x2 x3 x4 x5 x6 Special Topics on Graph Algorithms

12. S2 S1 S3 x1 x2 Special Topics on Graph Algorithms

13. x1 S3 x2 S2 Special Topics on Graph Algorithms

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

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

17. Optimal subgraph Special Topics on Graph Algorithms

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

19. Optimal Subgraph Special Topics on Graph Algorithms

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

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

23. Differences between Red Trunk & Blue Trunk • 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 Special 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 Special 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 Special 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 Special Topics on Graph Algorithms

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

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

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

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(St, M) can be obtained as a forest f(St-1, M’) extended by a subset Tt of edges on the point set St-1∪St. Special Topics on Graph Algorithms

31. Dynamic programming algorithm The recursive relation: Consistency Enumerate St and M Enumerate St-1 and M’ Enumerate Tt 4k points Adding Number of St Special 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. Special Topics on Graph Algorithms

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

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

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

38. TAPPX: (1+ ε)-approximation TOPT Special Topics on Graph Algorithms

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

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

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

42. Upper Bound of c(TOPT) • 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. Special Topics on Graph Algorithms

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

45. Thanks THE END Special Topics on Graph Algorithms