Optimally fast incremental Manhattan plane embedding and planar tight span construction

TL;DR

This paper introduces a data structure for representing hyperconvex metric spaces and presents efficient algorithms for constructing and updating the tight span of a metric space, enabling fast testing of isometric embeddings into the Manhattan plane.

Contribution

The paper develops a rectangular complex data structure and algorithms that construct and update the tight span of a metric space in quadratic time, specifically tailored for planar embeddings.

Findings

Constructs the tight span in O(n^2) time for planar cases.

Adds a point to the tight span in O(n) time.

Tests isometric embedding into the Manhattan plane efficiently.

Abstract

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n x n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O(n^2), and to add a single point to a planar tight span in time O(n). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O(n^2), and add a single point to the space and re-test whether it has such an embedding in time O(n).