Better bounds for the worst case of the stretch-length tradeoff in geometric networks
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The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study
Consider a network linking the points of a rate-1 Poisson point process on the plane. Write ! (s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function ! (s), and on the analogous “worst-case” function !(s) where t...
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xxx Working notes. Three main things remain to do. (1) Either include Tamar’s argument [now in separate document] for Theorem 5, or replace by some better construction for worst-case upper bound. (2) Put in the average-case lower bound calculus calculations needed in section 6. (3) General version of the worst-case lower bound method in section 7.3. A fourth thing one might contemplate: (4) Is ...
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