How to find Steiner minimal trees in euclideand-space
This paper has two purposes. The first is to present a new way to find a Steiner minimum tree (SMT) connectingN sites ind-space,d >- 2. We present (in Appendix 1) a computer code for this purpose. This is the only procedure known to the author for finding Steiner minimal trees ind-space ford > 2, and also the first one which fits naturally into the framework of “backtracking” and “branch-and-bound.” Finding SMTs of up toN = 12 general sites ind-space (for anyd) now appears feasible.
We tabulate Steiner minimal trees for many point sets, including the vertices of most of the regular and Archimedeand-polytopes with <- 16 vertices. As a consequence of these tables, the Gilbert-Pollak conjecture is shown to be false in dimensions 3–9. (The conjecture remains open in other dimensions; it is probably false in all dimensionsd withd ≥ 3, but it is probably true whend = 2.)
The second purpose is to present some new theoretical results regarding the asymptotic computational complexity of finding SMTs to precision ɛ.
We show that in two-dimensions, Steiner minimum trees may be found exactly in exponential time O(CN) on a real RAM. (All previous provable time bounds were superexponential.) If the tree is only wanted to precision ɛ, then there is an (N/ɛ)O(√N)-time algorithm, which is subexponential if 1/ɛ grows only polynomially withN. Also, therectilinear Steiner minimal tree ofN points in the plane may be found inNO(√N) time.
J. S. Provan devised an O(N6/ɛ4)-time algorithm for finding the SMT of a convexN-point set in the plane. (Also the rectilinear SMT of such a set may be found in O(N6) time.) One therefore suspects that this problem may be solved exactly in polynomial time. We show that this suspicion is in fact true—if a certain conjecture about the size of “Steiner sensitivity diagrams” is correct.
All of these algorithms are for a “real RAM” model of computation allowing infinite precision arithmetic. They make no probabilistic or other assumptions about the input; the time bounds are valid in the worst case; and all our algorithms may be implemented with a polynomial amount of space. Only algorithms yielding theexact optimum SMT, or trees with lengths ≤ (1 + ɛ) × optimum, where ɛ is arbitrarily small, are considered here.
Key wordsSteiner trees Gilbert-Pollak conjecture Subexponential algorithms Regular polytopes Sensitivity diagrams
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