How to Fit a Tree in a Box

TL;DR

This paper explores optimal embedding of trees in grids, demonstrating that perfect binary trees can be efficiently embedded in square grids and establishing NP-hardness for certain upward embedding tests.

Contribution

It proves that perfect binary trees can be optimally embedded in square grids and shows NP-hardness of upward embedding testing for binary trees.

Findings

Perfect binary trees can be embedded on a √n by √n grid.

Testing upward embedding with a fixed combinatorial embedding is NP-hard.

Provides bounds and complexity results for tree embeddings.

Abstract

We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with $n$ nodes can be drawn on a $\sqrt n$ by $\sqrt n$ grid. We also show that testing whether a given binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.