Abstract
We present some fundamental structural properties for minimum length
networks (known as Steiner minimum trees) interconnecting a given set of
points in an environment in which edge segments are restricted to uniformly oriented directions. We show that the edge segments of any full
component of such a tree contain a total of at most 4 directions if is not a
multiple of , or 6 directions if is a multiple of . This result allows us to
develop useful canonical forms for these full components.
The structural properties of these Steiner minimum trees are then used to
resolve an important open problem in the area: does there exist a polynomialtime algorithm for constructing a Steiner minimum tree, if the topology of
the tree is known? We obtain a simple linear time algorithm for constructing a Steiner minimum tree for any given set of points and a given Steiner
topology.
networks (known as Steiner minimum trees) interconnecting a given set of
points in an environment in which edge segments are restricted to uniformly oriented directions. We show that the edge segments of any full
component of such a tree contain a total of at most 4 directions if is not a
multiple of , or 6 directions if is a multiple of . This result allows us to
develop useful canonical forms for these full components.
The structural properties of these Steiner minimum trees are then used to
resolve an important open problem in the area: does there exist a polynomialtime algorithm for constructing a Steiner minimum tree, if the topology of
the tree is known? We obtain a simple linear time algorithm for constructing a Steiner minimum tree for any given set of points and a given Steiner
topology.
Original language | English |
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Place of Publication | København |
Publisher | Københavns Universitet - Datalogisk Institut |
Number of pages | 26 |
Publication status | Published - 2002 |
Publication series
Name | DIKU-rapport |
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No. | 22 |
Volume | 02 |
Keywords
- steiner tree
- steiner minimum tree
- Algorithms