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Concept
Geometric spanner
Summary
A geometric spanner or a t-spanner graph or a t-spanner was initially introduced as a weighted graph over a set of points as its vertices for which there is a t-path between any pair of vertices for a fixed parameter t. 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.
In computational geometry, the concept was first discussed by L.P. Chew in 1986, although the term "spanner" was not used in the original paper.
The notion of graph spanners has been known in graph theory: t-spanners are spanning subgraphs of graphs with similar dilation property, where distances between graph vertices are defined in graph-theoretical terms. Therefore geometric spanners are graph spanners of complete graphs embedded in the plane with edge weights equal to the distances between the embedded vertices in the corresponding metric.
Spanners may be used in computational geometry for solving some proximity problems. They have also found applications in other areas, such as in motion planning, in telecommunication networks: network reliability, optimization of roaming in mobile networks, etc.
There are different measures which can be used to analyze the quality of a spanner. The most common measures are edge count, total weight and maximum vertex degree. Asymptotically optimal values for these measures are edges, weight and maximum degree (here MST denotes the weight of the minimum spanning tree).
Finding a spanner in the Euclidean plane with minimal dilation over n points with at most m edges is known to be NP-hard.
Many spanner algorithms exist which excel in different quality measures. Fast algorithms include the WSPD spanner and the Theta graph which both construct spanners with a linear number of edges in time. If better weight and vertex degree is required the Greedy spanner can be computed in near quadratic time.
Theta graph
The Theta graph or -graph belongs to the family of cone-based spanners.
Official source
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