Worst-Case Bounds for Subadditive Geometric Graphs (1993) [4 citations — 1 self]
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author = {Marshall Bern and David Eppstein},
title = {Worst-Case Bounds for Subadditive Geometric Graphs},
booktitle = {Proc. 9th ACM Symp. Comp. Geom},
year = {1993},
pages = {183--188},
publisher = {Press}
}
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Abstract:
We consider graphs such as the minimum spanning tree, minimum Steiner tree, minimum matching, and traveling salesman tour for n points in the d-dimensional unit cube. For each of these graphs, we show that the worst-case sum of the dth powers of edge lengths is O(log n). This is a consequence of a general "gap theorem": for any subadditive geometric graph, either the worst-case sum of edge lengths is O(n (d-1)/d ) and the sum of dth powers is O(log n), or the sum of edge lengths is #(n). We look more closely at some specific graphs: the worst-case sum of dth powers is O(1) for minimum matching, but #(log n) for traveling salesman tour, which answers a question of Snyder and Steele. 1. Introduction A worst-case, or a priori , bound on a geometric graph is a bound that depends only on the assumption that all vertices lie within a given container. Such a bound does not depend on the specific locations of vertices, nor on any probabilistic assumptions. Early papers especially ...
Citations
| 106 | Steiner Minimal Trees – Gilbert, Pollak - 1968 |
| 17 | Subadditive Euclidean functionals and non-linear growth in geometric probability – STEELE - 1981 |
| 9 | Probabilistic and worst case analyses of classical problems of combinatorial optimization in Euclidean space – Steele - 1990 |
| 5 | On Optimal Matchings – Ajtai, Komlós, et al. - 1984 |
| 5 | How long can a Euclidean Traveling Salesman Tour be – Karloff - 1989 |
| 5 | Worst-case growth rates of some classical problems of combinatorial optimization – Steele, Snyder - 1989 |
| 3 | A priori inequalities for the euclidean traveling salesman – Snyder, Steele - 1992 |
| 2 | Worst-case minimal rectilinear Steiner trees in all dimensions – Snyder - 1990 |
| 2 | Worst-case greedy matchings in the unit d-cube – Snyder, Steele - 1990 |
| 1 | Random planar matching and bin packing – Shor - 1985 |
| 1 | Lower bounds for rectilinear Steiner trees in bounded space – Snyder - 1991 |
| 1 | Computational Aspects of VSLI – Ullman - 1984 |
