Minimum Steiner trees, roots of a polynomial, and other magic

Soukup, Jiří

doi:10.1145/1217082.1217084

Cited by 2 publications

(1 citation statement)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, one must begin the algorithm with some upper bound "STUB" on the length of the Steiner minimal tree. The simplest approach is to use oo; One could also use the length of the minimum line-segment spanning tree, which may be found in O(dN 2) time using Prim's algorithm, or indeed the length of any heuristic tree [3], [17], [46], [47], [53]. However, it is my experience that the backtracking algorithm itself soon generates a very good upper bound, so that this idea will not improve the performance substantially.…”

Section: Theorem 1 the Coordinates Of The Steiner Points In The Smt mentioning

confidence: 99%

How to find Steiner minimal trees in euclideand-space

1992

View full text Add to dashboard Cite

Abstract. This paper has two purposes. The first is tO present a new way to find a Steiner minimum tree (SMT) connecting N sites in d-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 in d-space for d > 2, and also the first one which fits naturally into the framework of "backtracking" and "branch-and-bound." Finding SMTs of up to N = 12 general sites in d-space (for any d) now appears feasible.We tabulate Steiner minimal trees for many point sets, including the vertices of most of the regular and Archimedean d-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 dimensions d with d > 3, but it is probably true when d = 2.)The second purpose is to present some new theoretical results regarding the asymptotic computational complexity of finding SMTs to precision e.We show that in two-dimensions, Steiner minimum trees may be found exactly in exponential 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 the exact optimum SMT, or trees with lengths < (1 + e) x optimum, where e is arbitrarily small, are considered here. Key Words. Steiner trees, Gilbert-Pollak conjecture, Subexponential algorithms, Regular polytopes, Sensitivity diagrams. Basic Facts and Definitions.This section presents basic facts and definitions which we will need. Readers familiar with Steiner tree terminology may skip to Section 1.A set of real numbers is said to be in "general position" if they satisfy no nontrivial algebraic equation with integer coefficients. A positive real-valued function f(N), which may be bounded for all sufficiently large N by Received November, 1988; revised September, 1989, and February, 1990. Communicated by F. K. Hwang. 138 W.D. Smith where 1 < C a < C 2, is said to be "exponential." A function which grows faster than any upper bound of this type is "superexponential," and one which cannot be bounded by any lower bound of this type is "subexponential." Z and R represent the sets of integers and real numbers, respectively.The Steiner minimal tree (SMT) on N particles 21..-s in d-space, d > 2, is the shortest tree containing the sites. A "Steiner tree" is a tree of line segments with the following properties:1. It contains N "regular sites" 21'")~u, and possibly K additional "Steiner nodes" ~N+ 1 "'" 2N+K" 2. Each Steiner node has valence 3, and the edges emanating&ore it lie in a p...

show abstract