Multi-Sided Boundary Labeling

TL;DR

This paper introduces polynomial-time algorithms for multi-sided boundary labeling, enabling crossing-free leader layouts on multiple sides, including cases with adjacent sides, and optimizing for label maximization and leader length.

Contribution

It extends boundary labeling algorithms to multiple sides with efficient solutions for crossing-free layouts, label maximization, and leader length minimization.

Findings

Polynomial-time algorithm for multi-sided boundary labeling.

Efficient methods for testing crossing-free layout existence.

Algorithms for maximizing labeled sites and minimizing leader length.

Abstract

In the Boundary Labeling problem, we are given a set of $n$ points, referred to as sites, inside an axis-parallel rectangle $R$, and a set of $n$ pairwise disjoint rectangular labels that are attached to $R$ from the outside. The task is to connect the sites to the labels by non-intersecting rectilinear paths, so-called leaders, with at most one bend. In this paper, we study the Multi-Sided Boundary Labeling problem, with labels lying on at least two sides of the enclosing rectangle. We present a polynomial-time algorithm that computes a crossing-free leader layout if one exists. So far, such an algorithm has only been known for the cases in which labels lie on one side or on two opposite sides of $R$ (here a crossing-free solution always exists). The case where labels may lie on adjacent sides is more difficult. We present efficient algorithms for testing the existence of a crossing-free leader layout that labels all sites and also for maximizing the number of labeled sites in a crossing-free leader layout. For two-sided boundary labeling with adjacent sides, we further show how to minimize the total leader length in a crossing-free layout.