Approximating the Average Stretch Factor of Geometric Graphs
Let G be a geometric graph whose vertex set S is a set of n points in ℝd. The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the sum of all \(n \choose 2\) stretch factors determined by all pairs of points in S. We show that for paths, cycles, and trees, this sum can be approximated, within a factor of 1 + ε, in O(npolylog(n)) time. For plane graphs, we present a (2 + ε)-approximation algorithm with running time O(n5/3polylog(n)), and a (4 + ε)-approximation algorithm with running time O(n3/2polylog(n)).
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