A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. International Journal of Computational Geometry and Applications, 6(3):333--356, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1.3 Can quadratic area be beaten for classes of planar graphs The obvious candidate to consider is the class of trees. There are many papers proving that linear or almost linear area bounds can be achieved for particular classes of trees, including the result by Garg, Goodrich and Tamassia [17] and the result by Chan [3] Summarizing tables and more references can be found in the book by Di Battista, Eades, Tamassia, and Tollis [10] O(nlog log n) area is the best result for trees of unbounded degree. Problem 2. Open) Are there other classes of planar graphs that can be drawn in ....

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333-356, 1996.

.... Chrobak, Goodrich, and Tamassia [7] who studied convex grid drawings of triconnected planar graphs in an integer grid of quadratic area; and the many papers proving that linear or almost linear area bounds can be achieved for classes of trees, including the result by Garg, Goodrich and Tamassia [23] and the result by Chan [5] Summarizing tables and more references can be found in the book by Di Battista, Eades, Tamassia, and Tollis [14] While the problem of computing small sized crossing free straight line drawings in the plane has a long tradition, its 3D counterpart has become the ....

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333-356, 1996.

.... Chrobak, Goodrich, and Tamassia [4] who studied convex grid drawings of triconnected planar graphs in an integer grid of quadratic area; and the many papers proving that linear or almost linear area bounds can be achieved for classes of trees, including the result by Garg, Goodrich and Tamassia [12] and the result by Chan [2] Summarizing tables and more references can be found in the book by Di Battista, Eades, Tamassia, and Tollis [9] While the problem of computing small sized crossing free straight line drawings in the plane has a long tradition, the 3d counterpart has received less ....

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333--356, 1996.

.... Chrobak, Goodrich, and Tamassia [4] who studied convex grid drawings of triconnected planar graphs in an integer grid of quadratic area; and the many papers proving that linear or almost linear area bounds can be achieved for classes of trees, including the result by Garg, Goodrich and Tamassia [12] and the result by Chan [2] Summarizing tables and more references can be found in the book by Di Battista, Eades, Tamassia, and Tollis [9] Research supported in part by the CNR Project Geometria Computazionale Robusta con Applicazioni alla Gra ca ed al CAD , the project Algorithms for ....

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333-356, 1996.

....parent to the left child is to the left of the curve from the parent to the right child. Table 1 summarizes the known results for various combinations of the four criteria. The two O(n log n) area bounds in the table, due to Crescenzi, Di Battista, and Piperno [2] and Garg, Goodrich, and Tamassia [6], were obtained by simple recursive algorithms and proved to be tight in the worst case for their corresponding types of drawings. The type of drawings considered by Crescenzi et al. is not order preserving, and thus, one cannot reconstruct the binary tree uniquely from the drawing (without the ....

.... trees) ideal drawings satisfying all criteria 1 4 can be constructed using only O(n) area; see the references [3, 4, 7, 11, 13] Also not in the table are results regarding orthogonal drawings, i.e. drawings in which all line segments are either horizontal or vertical; see the references [1, 2, 6, 8, 11, 14]. 2 planar upward order preserv. straight line area references yes yes no no O(n) 6] yes yes no yes O(n log log n) 11] yes yes (strictly) no yes O(n log n) 2, 10] yes yes (strictly) yes no O(n log n) 6] yes yes (strictly) yes (strongly) yes O(n 1 ) this paper Table 1: Area bounds for ....

[Article contains additional citation context not shown here]

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Int. J. Comput. Geom. Appl., 6:333-356, 1996.

.... Chrobak, Goodrich, and Tamassia [4] who studied convex grid drawings of triconnected planar graphs in an integer grid of quadratic area; and the many papers proving that linear or almost linear area bounds can be achieved for classes of trees, including the result by Garg, Goodrich and Tamassia [12] and the result by Chan [2] Summarizing tables and more references can be found in the book by Di Battista, Eades, Tamassia, and Tollis [9] While the problem of computing small sized crossing free straight line drawings in the plane has a long tradition, its 3d counterpart has become the ....

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333-356, 1996.

....parent to the left child is to the left of the curve from the parent to the right child. Table 1 summarizes the known results for various combinations of the four criteria. The two O(n log n) area bounds in the table, due to Crescenzi, Di Battista, and Piperno [2] and Garg, Goodrich, and Tamassia [6], were obtained by simple recursive algorithms and proved to be tight in the worst case for their corresponding types of drawings. The type of drawings considered by Crescenzi et al. is not order preserving, and thus, one cannot reconstruct the binary tree uniquely from the drawing (without the ....

.... trees) ideal drawings satisfying all criteria 1 4 can be constructed using only O(n) area; see the references [3, 4, 7, 10, 12] Also not in the table are results regarding orthogonal drawings, i.e. drawings in which all line segments are either horizontal or vertical; see the references [1, 2, 6, 8, 10, 13]. Note that despite its naturalness, our strong definition of order preserving drawings (criterion 3) seems to be unstudied before. One may insist on an even stronger condition where the curves are not only monotone increasing decreasing in the x direction, but strictly increasing decreasing. ....

[Article contains additional citation context not shown here]

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Int. J. Comput. Geom. Appl., 6:333--356, 1996.

....does not settle the question whether binary trees can be drawn in this fashion in O(n) area. Thus, our result is signi cant from a theoretical view point. In fact, we already know of one category of drawings, namely, planar upward orthogonal grid drawings, for which n log log n is a tight bound [5], i.e. any binary tree can be drawn in this fashion in area O(n log log n) and there exists a family of binary trees that require area n log log n) in any such drawing. So, a natural question arises, if n log log n is a tight bound for planar straight line grid drawings also. Of course, our ....

....straight line grid drawings also. Of course, our result imply that this is not the case. Besides, our drawing technique and proofs are signi cantly di erent from those of [1] and [9] We now summarize some other known results on planar grid drawings of binary trees. Let T be a binary tree. [5] presents an algorithm for constructing an upward polyline drawing of T in O(n) area. 6] and[11] present algorithms for constructing an orthogonal drawing of T in O(n) area. 7] gives an algorithm for constructing upward layered drawing in O(n ) area. 2] gives an algorithm for constructing ....

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333-356, 1996.

No context found.

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. International Journal of Computational Geometry and Applications, 6(3):333--356, 1996.

No context found.

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333--356, 1996.

No context found.

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. International Journal of Computational Geometry and Applications, 6(3):333--356, 1996.

No context found.

A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl., 6:333-356, 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC