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The stretch factor is important in the theory of geometric spanners, weighted graphs that approximate the Euclidean distances between a set of points in the Euclidean plane. In this case, the embedded metric S is a finite metric space, whose distances are shortest path lengths in a graph, and the metric T into which S is embedded is the ...
If an embedding maps all pairs of vertices with distance to pairs of vectors with distance in the range [,] then its stretch factor or distortion is the ratio /; an isometry has stretch factor one, and all other embeddings have greater stretch factor. [1] The graphs that have an embedding with at most a given distortion are closed under graph ...
A t-path is defined as a path through the graph with weight at most t times the spatial distance between its endpoints. The parameter t is called the stretch factor or dilation factor of the spanner. [1] In computational geometry, the concept was first discussed by L.P. Chew in 1986, [2] although the term "spanner" was not used in the original ...
Greedy geometric spanner of 100 random points with stretch factor t = 2 Greedy geometric spanner of the same points with stretch factor t = 1.1. In computational geometry, a greedy geometric spanner is an undirected graph whose distances approximate the Euclidean distances among a finite set of points in a Euclidean space.
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number < such that for all x and y in M,
The stretch factor of the entire spanner is the maximum stretch factor over all pairs of points within it. Recall from above that θ = 2 π / k {\displaystyle \theta =2\pi /k} , then when k ≥ 9 {\displaystyle k\geq 9} , the Θ {\displaystyle \Theta } -graph has a stretch factor of at most 1 / ( cos θ − sin θ ) {\displaystyle 1 ...
With a stretching exponent β between 0 and 1, the graph of log f versus t is characteristically stretched, hence the name of the function. The compressed exponential function (with β > 1) has less practical importance, with the notable exception of β = 2, which gives the normal distribution.
The Gromov–Hausdorff space is path-connected, complete, and separable. [5] It is also geodesic, i.e., any two of its points are the endpoints of a minimizing geodesic. [6] [7] In the global sense, the Gromov–Hausdorff space is totally heterogeneous, i.e., its isometry group is trivial, [8] but locally there are many nontrivial isometries.