Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

Augmenting the connectivity of planar and geometric graphs

In this paper we study some connectivity augmentation problems. We want to make planar graphs 2-vertex (or 2-edge) connected by adding edges such that the resulting graphs remain planar. We show that it is NP-hard to find a minimum-cardinality augmentation that makes a planar graph 2-edge connected. This was known for 2-vertex connectivity. We further show that both problems are hard in a geometric setting, even when restricted to trees. For the special case of convex geometric graphs we give efficient algorithms. We also study the following related problem. Given a plane geometric graph G, two vertices s and t of G, and an integer k, how many edges have to be added to G such that G contains k edge- (or vertex-) disjoint s-t paths? For k=2 we give optimal worst-case bounds; for k=3 we characterize all cases that have a solution

Estimating national-level syringe availability to injecting drug users and injection coverage: Switzerland, 1996-2006.

BACKGROUND: Measuring syringe availability and coverage is essential in the assessment of HIV/AIDS risk reduction policies. Estimates of syringe availability and coverage were produced for the years 1996 and 2006, based on all relevant available national-level aggregated data from published sources. METHODS: We defined availability as the total monthly number of syringes provided by harm reduction system divided by the estimated number of injecting drug users (IDU), and defined coverage as the proportion of injections performed with a new syringe, at national level (total supply over total demand). Estimates of supply of syringes were derived from the national monitoring system, including needle and syringe programmes (NSP), pharmacies, and medically prescribed heroin programmes. Estimates of syringe demand were based on the number of injections performed by IDU derived from surveys of low threshold facilities for drug users (LTF) with NSP combined with the number of IDU. This number was estimated by two methods combining estimates of heroin users (multiple estimation method) and (a) the number of IDU in methadone treatment (MT) (non-injectors) or (b) the proportion of injectors amongst LTF attendees. Central estimates and ranges were obtained for availability and coverage. RESULTS: The estimated number of IDU decreased markedly according to both methods. The MT-based method (from 14,818 to 4809) showed a much greater decrease and smaller size of the IDU population compared to the LTF-based method (from 24,510 to 12,320). Availability and coverage estimates are higher with the MT-based method. For 1996, central estimates of syringe availability were 30.5 and 18.4 per IDU per month; for 2006, they were 76.5 and 29.9. There were 4 central estimates of coverage. For 1996 they ranged from 24.3% to 43.3%, and for 2006, from 50.5% to 134.3%. CONCLUSION: Although 2006 estimates overlap 1996 estimates, the results suggest a shift to improved syringe availability and coverage over time