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Simultaneous Embedding of Two Paths on the Grid

Stephen Kobourov, William Lenhart, Giuseppe Liotta, Daniel Perz, Pavel Valtr, Johannes Zink

Abstract

We study the problem of simultaneous geometric embedding of two paths without self-intersections on an integer grid. We show that minimizing the length of the longest edge of such an embedding is NP-hard. We also show that we can minimize in $O(n^{3/2})$ time the perimeter of an integer grid containing such an embedding if one path is $x$-monotone and the other is $y$-monotone.

Simultaneous Embedding of Two Paths on the Grid

Abstract

We study the problem of simultaneous geometric embedding of two paths without self-intersections on an integer grid. We show that minimizing the length of the longest edge of such an embedding is NP-hard. We also show that we can minimize in time the perimeter of an integer grid containing such an embedding if one path is -monotone and the other is -monotone.
Paper Structure (4 sections, 3 theorems, 2 figures)

This paper contains 4 sections, 3 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. NP-Completeness
  3. Minimizing the parameter for two monotone paths
  4. Omitted Proof of \ref{['sec:nph']}

Key Result

Theorem 1

Simultaneous embedding of two paths on a grid is NP-complete for the objectives of minimizing the maximum edge length and minimizing the sum of edge lengths.

Figures (2)

  • Figure 1: Simultaneous embedding of two paths (all edges have length 1) obtained in our NP-hardness reduction from the NotAllEqual3SAT instance given on top. Edges of $P_1$ are colored blue, edges of $P_2$ are colored light red, and shared edges are colored purple.
  • Figure 2: The paths $P_1$ and $P_2$ of \ref{['fig:nph']}.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3