Optimal distortion embedding of complete binary trees into lines

We study the problem of embedding an unweighted complete binary tree into a line with low distortion. Very recently, Mathieu and Papamanthou (2008) [9] showed that the inorder traversal of the complete binary tree of height h gives a line embedding of distortion O(2h), and conjectured that the current lower bound of Ω(2h/h ) increases to Ω(2h), i.e., the upper bound of O(2ℎ) is best possible. In this paper, we disprove their conjecture by providing a line embedding of the complete binary tree of height h with optimal distortion Θ(2h/h)