Euclidean minimum spanning tree

<p>A Euclidean minimum spanning tree of a <a href="/facts/Finite_set/za1newlP">finite set</a> of points in the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean plane</a> or higher-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> connects the points by a system of <a href="/facts/Line_segment/V8E0M4qk">line segments</a> with the points as endpoints, minimizing the total length of the segments. In it, any two points can reach each other along a path through the line segments. It can be found as the <a href="/facts/Minimum_spanning_tree/kgkGmY9s">minimum spanning tree</a> of a <a href="/facts/Complete_graph/AYD6hUqI">complete graph</a> with the points as vertices and the <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distances</a> between points as edge weights.
</p><p>The edges of the minimum spanning tree meet at angles of at least 60°, at most six to a vertex. In higher dimensions, the number of edges per vertex is bounded by the <a href="/facts/Kissing_number/UE3mtV0d">kissing number</a> of tangent <a href="/facts/Unit_sphere/5UeoTcug">unit spheres</a>. The total length of the edges, for points in a <a href="/facts/Unit_square/6aJUASbv">unit square</a>, is at most proportional to the square root of the number of points. Each edge lies in an empty region of the plane, and these regions can be used to prove that the Euclidean minimum spanning tree is a subgraph of other <a href="/facts/Geometric_graph/4vIAySX3">geometric graphs</a> including the <a href="/facts/Relative_neighborhood_graph/8PUNYtWy">relative neighborhood graph</a> and <a href="/facts/Delaunay_triangulation/6VRk7cS9">Delaunay triangulation</a>. By constructing the Delaunay triangulation and then applying a graph minimum spanning tree algorithm, the minimum spanning tree of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 given planar points may be found in time 
  
    
      
        O
        (
        n
        log
        ⁡
        n
        )
      
    
    {\displaystyle O(n\log n)}
  
, as expressed in <a href="/facts/Big_O_notation/weFFjSWg">big O notation</a>. This is optimal in some models of computation, although faster <a href="/facts/Randomized_algorithms/Fngl1zgk">randomized algorithms</a> exist for points with integer coordinates. For points in higher dimensions, finding an optimal algorithm remains an <a href="/facts/Open_problem/VuR6yLsW">open problem</a>.
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Euclidean minimum spanning tree open-in-new

Euclidean minimum spanning tree