By Godfried T Toussaint
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Extra info for Computational geometry
Must intersect the vertical line (L through 0) at some rpoint c. below 0 . ) k k i J v ° be the vertex that yields the closest c to 0 and let c denote the corresponding role of on L c . ) and above 0. p£, and with respect to vertex Thus c* c*f c*f play the corresponding p. ) . ,ρ,, and p. ) which is precisely ^yV. v l k rj c, . Pl , will intersect J K L at V c , . Therefore K· •k c K. * c , is also an K. ) 'W I must share a side. D. Illustrating the proof of Theorem 2. K. T. Toussaint Theorem 3 : d ( p .
The maximum distance realized determines the furthest point from p. That this algorithm will not work for arbitrary sets of points is obvious from theorem 1 since points inside the CH(P) are not even accounted for. In the next section we show that the FNG(P) is not a subgraph of the FPT(P) and thus the algorithm fails even for convex polygons. 4. The Maximum Spanning Tree The maximum spanning tree (MXST) problem for a set of point sists of finding the tree connecting all points of total weight.
Proof: The procedure FIND_LTP_AND_COLLECT_LEFT_SIDE first descends on a path of T from the root to some leaf, and then ascends this path again collecting all the subtrees hanging off to the left of this path into a single tree ß " T\. At each node on the way down, the call to the function \rt? previous := Tvrightmost Tv rightmost, next : = TYleftmost return (Γχ) end CONTAINS_LTP takes a constant time. Therefore, descending from the root to a leaf of T takes time proportional to the height of T.
Computational geometry by Godfried T Toussaint