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

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

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

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

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

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

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

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

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

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

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

x2x2 x1x1 S1S1 S2S2 S3S3 12Special Topics on Graph Algorithms

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

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

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

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

Optimal subgraph 17Special Topics on Graph Algorithms

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

Optimal Subgraph 19Special Topics on Graph Algorithms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Special Topics on Graph Algorithms

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

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

THE END Thanks 45Special Topics on Graph Algorithms