Short-length routes in low-cost networks via Poisson line patterns
In designing a network to link n cities in a square of area n, one might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all cities. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two differences? For typical configurations the shortest network length is order n and the average straight-line distance is order n^1/2, so it seems implausible that one can construct a network in which the first difference is o(n) and the second difference is o(n^1/2). But in fact one can do better: for an arbitrary configuration one can construct a network where the first difference is o(n) and the second difference is almost as small as O(log n). The construction is conceptually simple: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. The key ingredient is a new result about the Poisson line process. Consider two points at distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length 2r + C log r. Turning to lower bounds we show that, under a weak equidistribution assumption, if the first difference is O(n) then the second difference cannot be o(sqrt(log n)).