Seedlings in the Theory of Shortest Paths

This article explores three developments that arise from the fundamental theorem of Beardwood, Halton, and Hammersley on the asymptotic behavior of the shortest path through n random points. The first development concerns the role of martingales in the theory of shortest paths, especially their role in large deviation inequalities. The second development concerns the use of Lipschitz spacefilling curves to obtain analytical bounds in the theory of the TSP, and it provides some bounds that refine those of Bartholdi and Platzman on the worst case performance of the spacefilling heuristic for the TSP. The final topic addresses the relationship between Karp’s partitioning heuristic and the BHH theorem.