Proximity Drawings: Three Dimensions Are Better than Two (Extended Abstract)Penna, Paolo and Vocca, Paola (1998) Proximity Drawings: Three Dimensions Are Better than Two (Extended Abstract). In: Graph Drawing 6th International Symposium, GD’ 98, August 1315, 1998 , pp. 275287(Official URL: http://dx.doi.org/10.1007/3540376232_21). Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/3540376232_21
AbstractWe consider weak Gabriel drawings of unbounded degree trees in the threedimensional space. We assume a minimum distance between any two vertices. Under the same assumption, there exists an exponential area lower bound for general graphs. Moreover, all previously known algorithms to construct (weak) proximity drawings of trees, generally produce exponential area layouts, even when we restrict ourselves to binary trees. In this paper we describe a lineartime polynomialvolume algorithm that constructs a strictlyupward weak Gabriel drawing of any rooted tree with O(logn)bit requirement. As a special case we describe a Gabriel drawing algorithm for binary trees which produces integer coordinates and n^3area representations . Finally, we show that an infinite class of graphs requiring exponential area, admits linearvolume Gabriel drawings.The latter result can also be extended to \beta drawings, for any 1< \beta <2, and relative neighborhood drawings.
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