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.

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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

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Applications

• Applications are encountered in telecoms.

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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

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Geometric GMST w/grid clustering

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Outlines

• Geometric GMST with grid clustering• Proof of NP-hardness

– Reduction– Optimal structure– 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)

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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

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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

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x2

x1

S1S2 S3

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x1 S3

x2 S2

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Outlines

• Geometric GMST with grid clustering• Proof of NP-hardness

– Reduction– Optimal structure– Optimal cost

• Dynamic programming algorithm• Polynomial time approximation scheme

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Connecting Edge

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

than √2.• Assume d is arbitrary

close to √2.

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Lemma1

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

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Optimal subgraph

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Lemma2

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

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Optimal Subgraph

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Two possible structures

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

• Trunk: the structure in a column.

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Outlines

• Geometric GMST with grid clustering• Proof of NP-hardness

– Reduction– Optimal structure– Optimal cost

• Dynamic programming algorithm• Polynomial time approximation scheme

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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.

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• 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

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Definition

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

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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

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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

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433

3

31 332

dt

dtdt

ntt

tddAAec nn

X3CGMST

• If an exact cover exists

• if no cover exists

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Outlines

• Geometric GMST with grid clustering• Proof of NP-hardness

– Reduction– Optimal structure– Optimal cost

• Dynamic programming algorithm• Polynomial time approximation scheme

28Special Topics on Graph Algorithms

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’)

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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.

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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.

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Outlines

• Geometric GMST with grid clustering• Proof of NP-hardness

– Reduction– Optimal structure– Optimal cost

• Dynamic programming algorithm• Polynomial time approximation scheme

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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

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Partitioning into Slices

• Define . 2811 kf

Slice 1

Slice 2

Slice 3

Slice △

Row

k 1k

k2

k

1#Rows

1O

1O

1O

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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

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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

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TAPPX: (1+ ε)-approximation

1.

2.

OPT

OPTAPPX

TcTcTc

:Claim

1

22i

iAPPX TcTc

1i

iOPT FcTc TOPT

1F2F

F

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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

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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

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

4.

228

2211

i

ii

iOPTAPPX FcTcTcTc

?

228

OPT

OPTAPPX

TcTcTc

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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

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Combining (4) and (5)

6.

1281

9 assuming 299

92928

kf

kf

kfTcTcTc

OPT

OPTAPPX

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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.

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THE ENDThanks

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