An Streicher 2

7/30/2019 An Streicher 2

1/24

An improved algorithm for

computing Steiner minimaltrees in Euclidean d-space

Marcia Fampa and Kurt Anstreicher

VICCOC, Vienna, December 2006


2/24

A Feasible SolutionThe Euclidean Steiner Problem

terminal

Steiner point


3/24

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


4/24

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).


5/24

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


6/24

Exact AlgorithmsGeoSteiner Algorithm


7/24

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)


8/24

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


9/24

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


10/24

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


11/24

Math Programming Formulation for ESTPFampa and Maculan (2004)


12/24

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


13/24

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


14/24

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


15/24

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)


16/24

The Merging Operation

l

a

bT

SMT(T)

Merging

Operation

T+

a

b

k+3

l n+1

n+2

Sk+1

SMT(T+)


17/24

Fixing Variables

where:

Let:


18/24

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)


19/24

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+


20/24

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+


21/24

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+


22/24

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+


23/24

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


24/24

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

An Streicher 2

Documents

Transcript of An Streicher 2