the geometric gmst problem with grid clustering presented by 楊劭文, 游岳齊, 吳郁君,...

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

– Reduction– Optimal structure– 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.


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


Applications

• Applications are encountered in telecoms.


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


Geometric GMST w/grid clustering


Outlines





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)


Exact Cover by 3-Sets

• A ground set X = {1, 2, … , n}, n = 3q

S1 S2 S3 S4

x1 x3 x4x2 x5 x6

• C = {S1, S2, …, Sm}– For 1 ≤ i ≤ m, Si is a subset of X

– |Si| = 3


Exact Cover by 3-Sets

• Is there a set C’ such that– C’ ⊆ C– The elements of C’ are disjoint and– For each xi C’, Uxi = X

x1 x3 x4x2 x5 x6

S1 S2 S3 S4


x2

x1

S1S2 S3


x1 S3

x2 S2


Outlines





Connecting Edge

• Connecting Edge (dotted edge)• Its length d is slightly larger

than √2.• Assume d is arbitrary

close to √2.


Lemma1

• No edge in Topt is larger than d, where Topt is some optimal solution.


Optimal subgraph


Lemma2

• The subgraph induced by an arbitrary optimal solution and nonempty cells of an arbitrary block is connected.


Optimal Subgraph


Two possible structures

• Two possible structure in a column.– By lemma1 and lemma2

• Trunk: the structure in a column.


Outlines





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.


• 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


Definition

• let Z = c( ) be its cost.• = Z−3(m−1)(n+1)• let be the contribution of column j•


Connecting edge

• For a connecting edge e in a column j wedefine its averaged connecting cost as

where is the number of connecting edges in column j.

• We have


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


433

3

31 332

dt

dtdt

ntt

tddAAec nn

X3CGMST

• If an exact cover exists

• if no cover exists


Outlines





Definitionst {1, 2, ∈ . . . , − 3}

Ct: The tth columnSt: 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 the connectivity

f (St,M): a generalized minimum spanning forest

Ct Ct+2 Ct+3Ct+1

St-1

St

… …

M

M’

f (St,M)

f (St-1,M’)


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.


Dynamic programming algorithm

The recursive relation:

Consistency

Enumerate St and M

Enumerate St-1 and M’

Enumerate Tt

kO 3 292 kO

2162 kO

Adding 2kO

kO 3 292 kO

kS 3

kkM 33

4k points

Number of St O 31Special Topics on Graph Algorithms

Theorem 2

The dynamic programming algorithm solves the geometric GMST with connected nonempty grid cells in time

2346 2

2 kO kk

The computation time is polynomial if k is fixed.


Outlines





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 .

kkfkf 1,

1 0


Partitioning into Slices

• Define . 2811 kf

Slice 1

Slice 2

Slice 3

Slice △

Row

k 1k

k2

k

1#Rows

1O

1O

1O


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

11 ki

ki

iT


Obtaining the GST for the Graph

• Picking edges greedily yields GST .

Slice 1

Slice 2

Slice 3

Slice △

Row

k 1k

k2

k

1

APPXT


TAPPX: (1+ ε)-approximation

1.

2.

OPT

OPTAPPX

TcTcTc

:Claim

1

22i

iAPPX TcTc

1i

iOPT FcTc TOPT

1F2F

F


Lower Bound of c(Fi)

3. 2226 ii TcFc

Slice iiF

32.8284271222,:3 Lemma Recall ecGEe

connected makes

2 cell ain distancelongest with 6most at Adding

iF


Lower Bound of c(Fi)

3. 2226 ii TcFc

Slice iiT

rows bottom andupper in the edges addingby obtained is that Recall iT

edges additional theseRemove


Combining (1), (2) and (3)

4.

228

2211

i

ii

iOPTAPPX FcTcTcTc

?

228

OPT

OPTAPPX

TcTcTc


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.

5.

9kf

19

kfTc OPT


Combining (4) and (5)

6.

1281

9 assuming 299

92928

kf

kf

kfTcTcTc

OPT

OPTAPPX


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.


THE ENDThanks


the geometric gmst problem with grid clustering presented by 楊劭文, 游岳齊, 吳郁君,...

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