Tree Drawings Revisited

03/19/2018

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by   Timothy M. Chan, et al.

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We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that 1. every tree of size n (with arbitrarily large degree) has a straight-line drawing with area n2^O(√( n n)), improving the longstanding O(n n) bound; 2. every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area n√( n)( n)^O(1), improving the longstanding O(n n) bound; 3. every binary tree of size n has a straight-line orthogonal drawing with area n2^O(^*n), improving the previous O(n n) bound by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996); 4. every binary tree of size n has a straight-line order-preserving drawing with area n2^O(^*n), improving the previous O(n n) bound by Garg and Rusu (2003); 5. every binary tree of size n has a straight-line orthogonal order-preserving drawing with area n2^O(√( n)), improving the O(n^3/2) previous bound by Frati (2007).