Path-Aware Geometric Labels Overview

• Path-aware geometric labels are labeling schemes that adapt to significant geometric or combinatorial paths, ensuring spatial accuracy and compatibility.
• Methodologies involve greedy algorithms for convex sets, dynamic programming for simple polygons, and medial-axis extraction for optimal label placement.
• Applications span automated cartography, planar graph morphing, and hub-based network labeling, improving visual clarity and query efficiency.

Path-aware geometric labels designate labeling schemes and algorithmic approaches in computational geometry and network science where label assignments or placements adapt to geometric or combinatorial paths and ensure robust compatibility or optimality under path-based constraints. These techniques play a central role in areas such as automated geographic labeling, geometric graph drawing, hub-based distance labeling in networks, and compatibility testing for geometric embeddings. The primary focus is to encode, compute, or validate labels so that spatial or topological properties pertaining to significant paths—be they shortest paths, skeletons, or geometric Hamiltonian paths—are reflected in or preserved by the labeling, often under strict computational requirements.

1. Formal Definitions and Core Concepts

Path-aware geometric labeling can be instantiated in several mathematically precise frameworks. One axis concerns compatible paths on labeled geometric configurations: consider finite point sets $P, Q \subset \mathbb{R}^2$, each with bijective labelings. Two noncrossing geometric graphs $G_P$ and $G_Q$, realized as straight-line embeddings over $P$ and $Q$ respectively, are deemed compatible if every face $f \in \mathcal{F}(G_P)$ is matched by a face in $G_Q$ with the same cyclic label order. For the case of path graphs (Hamiltonian paths), such compatibility underpins path-aware labelings: a labeling is path-aware if, for all considered embeddings, the induced orderings of labels along every path/facial cycle agree (Arseneva et al., 2020).

In the domain of area labeling for cartography, path-aware refers to selecting and placing area labels so that their support curves or boxes are aligned to significant geometric paths within polygonal regions, such as medial axis branches or skeleton paths, ensuring both maximum visibility and adherence to region geometry (Krumpe et al., 2020).

Similarly, in network labeling, hub-based distance labels can be constructed with path-aware schemes by selecting hubs located on central subpaths of shortest-path trees, as parameterized by the skeleton dimension of the network graph (Kosowski et al., 2016).

2. Algorithmic Foundations

2.1 Compatible Paths on Labeled Point Sets

Determining if two labeled point sets admit compatible noncrossing path embeddings is NP-complete in general. This intractability motivates efficient algorithms for specific cases:

• Convex Position: If all points lie in convex position, a greedy algorithm leverages the property that path prefixes form contiguous convex hull intervals, producing a compatible label ordering (if one exists) in $O(n)$ time (Arseneva et al., 2020).
• Simple Polygons: Dynamic programming over intervals with precomputed visibility graphs allows $O(n^2)$-time detection of compatible paths inside simple polygons.
• Monotone Embeddings: For embeddings restricted to paths monotone to a direction, Goodman-Pollack enumeration yields an $O(n^2 \log n)$ approach for matching monotone orderings.

2.2 Medial-Skeleton-Based Label Placement

In area labeling, path-aware geometric labels are computed by:

1. Constructing a conforming Delaunay triangulation over the polygon's vertex set.
2. Forming a pruned medial-axis skeleton (Voronoi-based), removing edges near the boundary.
3. Extracting $k$ longest distinct skeleton paths with sufficient clearance determined by the label's aspect ratio.
4. Fitting circular arcs to these skeleton paths.
5. Optimizing the placement of bent label boxes along these arcs to maximize angular extent while avoiding boundary intersections.

This pipeline yields arc-shaped labels that adhere closely to the region’s intrinsic geometry and can be executed in $O(k n \log n)$ time (Krumpe et al., 2020).

2.3 Skeletal Hub Labeling in Networks

In distance labeling for graphs, skeleton-based hub labeling utilizes:

• Construction of shortest-path trees for each root, pruning by reach to obtain skeletons $T_r^*$ of limited "width," as defined by skeleton dimension $\text{sk}(G)$.
• For each node pair, finding a hub on the central subpath of the unique $uv$-path (by symmetric interval selection), ensuring that any label intersection contains an actual path hub.
• Randomization for selecting hub edges uniformly and efficiently (Kosowski et al., 2016).

3. Structural Properties and Complexity

A key structural concept is skeleton dimension $\text{sk}(G)$, which measures the maximum width (number of persistent branches at any level) in pruned shortest-path trees. Small skeleton dimension enables compact path-aware hub labeling in road networks and similar graphs, with expected label size $O(\text{sk} \log D)$, $D$ being the network’s diameter.

For compatible paths, the complexity is summarized as:

Scenario	Time Complexity	Key Property
Convex Position	$O(n)$	Hull interval structure
Simple Polygons	$O(n^2)$	Visibility intervals
Monotone Paths	$O(n^2\log n)$	Goodman–Pollack orderings

For area labeling, the dominant cost arises from skeleton construction and repeated path extraction along with arc fitting and placement optimization, typically feasible in near-real-time for large polygons (Krumpe et al., 2020).

4. Applications and Practical Relevance

Path-aware geometric labeling schemes are foundational in several disciplines:

• Automated Cartography: Placement of curved labels respecting medial-axis paths enhances readability and conformality for complex regions such as countries or lakes (Krumpe et al., 2020).
• Planar Graph Morphing and Drawing: Path-aware labelings ensure that morphs between geometric graphs preserve facial orderings, crucial for animation and simultaneous representations (Arseneva et al., 2020).
• Efficient Shortest-Path Indexing: Compact path-aware hub labels based on skeleton dimension enable sublinear distance queries in large-scale networks with provable bounds (Kosowski et al., 2016).
• Shape Morphing and Geometric Interpolation: Path-aware schemes allow for deformation or interpolation between labeled configurations while maintaining combinatorial consistency.

A plausible implication is that path-aware geometric labeling serves as an underpinning for robust interoperability and interpretability across geometric, combinatorial, and spatial data representations.

5. Connections to Related Frameworks

Path-aware geometric labeling intersects with several established frameworks:

• Highway Dimension: While highway dimension underlies classical hub labeling schemes, skeleton dimension offers a more tractable and sometimes sharper measurement for path-aware purposes, particularly in low-dimensional geometric graphs (Kosowski et al., 2016).
• Medial Axis and Voronoi Geometry: The use of medial skeletons for label path extraction leverages long-standing geometric primitives for capturing the "centrality" and thickness of a shape, critical for area-conformal labeling (Krumpe et al., 2020).
• Simultaneous Embedding and Morph Compatibility: Compatible path concepts directly bridge to work on simultaneous planar embedding and morphing, where preserving face or path orderings across embeddings is required (Arseneva et al., 2020).

6. Limitations and Open Problems

Compatibility testing for arbitrary embeddings remains computationally hard (NP-complete), restricting efficient label determinations to special configurations—convex, polygonal, or monotone cases. In network settings, hub label sizes are fundamentally tied to the skeleton (or highway) dimension, and while practical sizes are small for many graphs, tight worst-case bounds or efficient computation in more complex domains remain challenging.

This suggests avenues for further research on extending efficient path-aware labeling frameworks to broader families of geometric or topological contexts, as well as refining structural parameters that govern label size and compatibility.

7. Summary Table: Approaches and Their Domains

Approach	Application Domain	Structural Principle
Compatible Paths (Arseneva et al., 2020)	Geometric embeddings	Cyclic label order preservation
Medial-Axis Labeling (Krumpe et al., 2020)	Area labeling	Skeleton paths and arc fitting
Skeleton Hub Labeling (Kosowski et al., 2016)	Network labeling	Central paths in shortest-path trees

These methodologies collectively form the computational basis of path-aware geometric labeling, enabling robust, structure-conforming labeling in cartography, networks, and geometric algorithmics.