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An improved algorithm for
computing Steiner minimaltrees in Euclidean d-space
Marcia Fampa and Kurt Anstreicher
VICCOC, Vienna, December 2006
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A Feasible SolutionThe Euclidean Steiner Problem
terminal
Steiner point
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Determine:
The number of Steiner points to be used
The arcs of the tree
Geometric position of the Steiner points
Topo
logy
The Euclidean Steiner Problem
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Steiner Tree/Topology A Steiner Tree (ST) is a tree that contains the N given
terminals and k additional Steiner points, such that:
No two edges meet at a point with angle less than 120.
Each terminal point has degree between 1 and 3.
Each Steiner point has degree equal to 3.
k N-2.
A Full Steiner Tree (FST) is an ST with the maximum N-2Steiner points.
A Steiner Topology (Full Steiner Topology) is a topologythat meets the degree requirements of an ST (FST).
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Exact Algorithms
Extensive literature elucidating properties of SMTs in the
plane that do not extend to d>2.
1961 Melzak
1985 Winter GeoSteiner Algorithm 2001 Warme, Winter and Zacharisen - version 3.1of the
GeoSteiner (10000 terminals solved)
ESP in the plane
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Exact AlgorithmsGeoSteiner Algorithm
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Exact Algorithms
Find all Steiner topologies on the N given terminalsand k Steiner points, with k N-2.
For each topology optimize the coordinates of the
Steiner points.
Output the shortest tree found.
General d-space: Gilbert and Pollak (1968)
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Degenerate Steiner Topologies A topology is called a degeneracy of another if the former can be
obtained from the latter by shrinking edges.
Fact: each Steiner topology is either a full Steiner topology or adegeneracy of a full Steiner topology.
1 1
2 72
3 8
3
4 94
5105
6 6
7 8
910
degenerate Steiner points
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Exact Algorithms
Find all Steiner topologies on the N given terminals
and k Steiner points, with k=N-2.
For each topology optimize the coordinates of the
Steiner points. Output the shortest tree found.
Full
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Full Steiner Topologies The total number of full Steiner topologies for a graph
with N terminals is given by
f(2)=1, f(4)=3, f(6)=105, f(8)=10395, f(10)=2,027,025, f(12)=654,729,075
(2N-4)!
2N-2(N-2)!f(N) =
s2
a d
c
s1
b
s2
a d
cs1
b
s2
a d
c
s1
b
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Math Programming Formulation for ESTPFampa and Maculan (2004)
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General d-space: Smith (1992)
An implicit enumeration scheme to generate full
Steiner topologies and a numerical algorithm to
solve the ESP for a given topology.
Computation of SMTs on vertices of regular
polytopes led to disproof of Gilbert-Pollak
conjecture on Steiner ratio in dimensions d>2.
Exact Algorithms
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Smith (1992)Enumeration Tree
1
2
3
1 3
2
S1
T
l
a
b
T
Tree with k Steiner points,
k+2 terminals and n edges (n=2k+1)
Tree with k+1 Steiner points,
k+3 terminals and n+2 edges
T+
a
b
k+3
l n+1
n+2
Sk+1
Merging
Operation
Enumeration Tree
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Nodes at level k of tree enumerate full Steiner
topologies with k+3 terminals, k = 0,1,, N-3. Children of a given node are obtained by merging
a new terminal node with each arc in current FST.
Good: Merging operation cannot decreaseminimum length of FST - allows pruning!
Bad: No easy way to account for effect of missing
terminal nodes. Ugly: Growth of tree is super-exponential with
depth, and problems get largerat deeper levels.
Enumeration Tree
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Problem for a Given Topology
i
i2
i1
i
i2
i1
Case 1:
Case 2:
where:
Topology T with k Steiner points,
k+2 terminals and n edges (n=2k+1)
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The Merging Operation
l
a
bT
SMT(T)
Merging
Operation
T+
a
b
k+3
l n+1
n+2
Sk+1
SMT(T+)
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Fixing Variables
where:
Let:
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c
n+1
n+2
b
(c)c
n+1
n+2
b
(e)
a c
l n+1
n+2
b
(f)
ca
l n+1
n+2
b
(g)
a c
l n+1
n+2
b
(h)
a c
l n+1
n+2
b
(i)
Topologies for the Subproblems
a c
l n+1
n+2
b
(d)
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In Smith+ use conic interior-point code (MOSEK)to obtain bounds on minimum length tree for given
topology. Also use MOSEK to solve subproblems
with fixed dual variables.
Choose next terminal node to add so as to
minimize number of children created/maximize
sum of child bounds (strong branching). Note must
extend Smiths enumeration argument to allow forvarying order in which terminals are added!
Smith versus Smith+
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Computational ResultsInstance from OR-Library with 12 terminals
156661Total
285209
0.97310.850.7778
0.99650.560.85470.913790.550.8346
0.932970.630.6755
0.593850.550.7934
01050.690.7553
0150.680.4092
030031
010010
FathomNodesElimFathomNodesLevel
SmithSmith+
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Nodes on the B&B TreeOR-Library
1
10
100
1,000
10,000
100,000
1,000,000
6 8 10 12 14 16 18
Terminal nodes
Average
B&Bnodes
Smith
Smith+
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CPU Time (seconds)OR-Library
1
10
10 0
1,000
10,000
100,000
6 8 10 12 14 16 18
Terminal nodes
AverageC
PUseconds
Smith
Smith+
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Effect of dimension d
Average nodes/time for 5 randomly-generated instances with10 terminals in Rd (d=2,3,4).
4.44,680.520,805.650.89,250.0470,321.85
2.85,735.616,153.026.913,685.6368,762.84
3.1753.52,334.133.41,652.455,222.23
10.568.4717.8160.2105.016,821.62
FactorSmith+SmithFactorSmith+Smithd
Average CPU secondsAverage B&B nodesDimension
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Conic formulation provides rigorous bounds.
Fixing dual variables allows for estimate of effectof next merge via solution of smaller problem.
Novel setting for strong branching; effective in
reducing size of the tree. More to do! Key problem with use of Smithsenumeration scheme is approximating the effect ofterminals that are not present in partial Steiner
trees. May also be possible to use geometricconditions that are valid for d>2.
Conclusions