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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 ...
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,
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.
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 ...
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 function composition of two metric maps is another metric map, and the identity map: on a metric space is a metric map, which is also the identity element for function composition.