Angle-Restricted Steiner Arborescences for Flow Map Layout

TL;DR

This paper introduces angle-restricted Steiner arborescences, called flux trees, for flow map layouts, analyzing their properties, computational complexity, and proposing approximation algorithms for practical use.

Contribution

It defines flux trees with angle restrictions, studies their properties, proves NP-hardness of optimal solutions, and provides a 2-approximation algorithm for spiral trees.

Findings

Optimal flux trees are planar and composed of spirals and straight lines.

Computing optimal flux trees is NP-hard.

A 2-approximation algorithm for spiral trees is developed.

Abstract

We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling lines smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. We study the properties of optimal flux trees and show that they are planar and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. We show that computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid "positive monotone" obstacles.