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Transcript of The geometric GMST problem with grid clustering Presented by 楊劭文, 游岳齊, 吳郁君,...
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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
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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
2Special Topics on Graph Algorithms
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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
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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
4Special Topics on Graph Algorithms
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Applications
• Applications are encountered in telecoms.
5Special Topics on Graph Algorithms
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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
6Special Topics on Graph Algorithms
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Geometric GMST w/grid clustering
7Special Topics on Graph Algorithms
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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
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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
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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
10Special Topics on Graph Algorithms
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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
14Special Topics on Graph Algorithms
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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.
15Special Topics on Graph Algorithms
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Lemma1
• No edge in Topt is larger than d, where Topt is some optimal solution.
16Special Topics on Graph Algorithms
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Optimal subgraph
17Special Topics on Graph Algorithms
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Lemma2
• The subgraph induced by an arbitrary optimal solution and nonempty cells of an arbitrary block is connected.
18Special Topics on Graph Algorithms
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Optimal Subgraph
19Special Topics on Graph Algorithms
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Two possible structures
• Two possible structure in a column.– By lemma1 and lemma2
• Trunk: the structure in a column.
20Special Topics on Graph Algorithms
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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
21Special Topics on Graph Algorithms
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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.
22Special Topics on Graph Algorithms
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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
23Special Topics on Graph Algorithms
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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
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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
25Special Topics on Graph Algorithms
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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
26Special Topics on Graph Algorithms
433
3
31 332
dt
dtdt
ntt
tddAAec nn
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X3CGMST
• If an exact cover exists
• if no cover exists
27Special Topics on Graph Algorithms
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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
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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’)
29Special Topics on Graph Algorithms
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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.
30Special Topics on Graph Algorithms
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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
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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.
32Special Topics on Graph Algorithms
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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
33Special Topics on Graph Algorithms
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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
34Special Topics on Graph Algorithms
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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
35Special Topics on Graph Algorithms
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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
36Special Topics on Graph Algorithms
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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
37Special Topics on Graph Algorithms
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TAPPX: (1+ ε)-approximation
1.
2.
OPT
OPTAPPX
TcTcTc
:Claim
1
22i
iAPPX TcTc
1i
iOPT FcTc TOPT
1F2F
F
38Special Topics on Graph Algorithms
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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
39Special Topics on Graph Algorithms
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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
40Special Topics on Graph Algorithms
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Combining (1), (2) and (3)
4.
228
2211
i
ii
iOPTAPPX FcTcTcTc
?
228
OPT
OPTAPPX
TcTcTc
41Special Topics on Graph Algorithms
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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
42Special Topics on Graph Algorithms
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Combining (4) and (5)
6.
1281
9 assuming 299
92928
kf
kf
kfTcTcTc
OPT
OPTAPPX
43Special Topics on Graph Algorithms
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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.
44Special Topics on Graph Algorithms
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THE ENDThanks
45Special Topics on Graph Algorithms