Worst-Case Bounds for Subadditive Geometric Graphs

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 Ω(n). We look more closely at some specific graphs: the worst-case sum of dth powers is O(1) for minimum matching, but Ω(log n) for traveling salesman tour, which answers a question of Snyder and Steele.

• Type: Conference
• Year: 1993
• Where: COMPGEOM
• Authors: Marshall W. Bern, David Eppstein
• Keywords: COMPGEOM 1993, Discrete Geometry, Dth Powers, Edge Lengths, Worst-case Sum