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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 ...
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.
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 ...
If the frequency ratios of octaves are greater than a factor of 2, the tuning is stretched; if smaller than a factor of 2, it is compressed." [3] Melodic stretch refers to tunings with fundamentals stretched relative to each other, while harmonic stretch refers to tunings with harmonics stretched relative to fundamentals which are not stretched ...
C. Canberra distance; Carathéodory metric; Caristi fixed-point theorem; Cartan–Hadamard theorem; CAT(k) space; Category of metric spaces; Cauchy sequence
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with a corresponding factor graph shown on the right. Observe that the factor graph has a cycle. If we merge (,) (,) into a single factor, the resulting factor graph will be a tree. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
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.