Geometry-Driven Path Augmentation

• Geometry-driven path augmentation encompasses techniques that exploit spatial structures to enrich path sets and improve network metrics like diameter and radius.
• It employs efficient algorithms, such as O(n) sweeps and dynamic programming, to identify optimal shortcuts and enhance path diversity in planning applications.
• The approach also extends to inferring nonlinear system dynamics by interpolating sparse observations using data-driven Riemannian metrics.

Geometry-driven path augmentation comprises a collection of methodologies in geometric network optimization, motion planning, data-driven inference, and topological path enumeration, wherein the geometric structure of the ambient space, network, or configuration directly modulates the augmentation or enrichment of path sets. The principal goal is to improve network performance metrics (diameter, radius, parity, etc.), enable richer path diversity (topologically or geometrically distinct trajectories), or recover dynamical system features from limited observations by exploiting geometric properties and augmenting the underlying path space. This domain intersects computational geometry, algorithmic network theory, robotics, image analysis, and stochastic inference.

1. Geometric Augmentation to Optimize Network Metrics

One foundational strand is minimizing the "continuous diameter" of a geometric network by augmenting its path structure with shortcuts. In the Euclidean plane, a geometric network $G=(V,E)$ consists of straight-line segments; a path $P$ is an open chain, and a cycle $C$ is a closed chain. The continuous diameter $\operatorname{diam}(G)$ is the maximal distance along network paths between any two points on $G$, not restricted to vertices.

For a path $P$, augmenting with a single shortcut $S=(a,b)$—where $a$, $b$ are arbitrary points on $P$—permits reduction of the continuous diameter with an $O(n)$ algorithm leveraging monotonicity: the maximal network distance as a function of one endpoint's position is unimodal, so prefix/suffix distance arrays and two endpoint sweeps yield the optimum efficiently (Carufel et al., 2015).

For cycles $C$, a single shortcut cannot decrease diameter, as the longest induced cycle segment post-augmentation remains at least half the original perimeter. However, two shortcuts suffices; optimal pairs either segment a convex cycle into three equal-length arcs or utilize diametral chords. The combinatorics for non-convex cycles involve reflex vertex analysis, and for convex cycles, an $O(n)$ sweep over arc-length separators computes optimum pairs (Carufel et al., 2015).

Similarly, in geometric trees, the existence of a diameter-reducing shortcut is determined by the intersection set $C$ of all diametral paths in the tree. An $O(n \log n)$ algorithm preprocesses the tree for diametral intersection queries and restricts candidate shortcuts to extremal points in $C$, guaranteeing strictly diameter-reducing augmentation iff $C$ is not a singleton or a segment (Carufel et al., 2016).

2. Augmentation for Radius, Parity, and Related Metrics

Beyond diameter, geometry-driven path augmentation has been developed for minimization of radius or satisfaction of vertex parity constraints.

For paths in a metric space, the optimal augmentation to minimize the discrete graph radius (smallest maximal distance from center to any vertex) can be found in $O(n)$ using a monotonic pointer sweep and careful decomposition of candidate configurations (prefix, suffix, etc.) for centers and shortcuts (Wang et al., 2020, Johnson et al., 2019). The continuous (any point as center) formulation allows a similar sweep, but supporting more center types.

Parity-constraint augmentation addresses the construction of plane geometric supergraphs where a specified vertex subset $R$ attains odd degree in the augmentation, and all others have even degree. For paths, the key structural dichotomy is "pseudo-convexity": if the path is not pseudo-convex, a T-join via plane-spanning-tree suffices; otherwise, a convex hull-based dynamic programming on the weak dual tree solves the problem in $O(n \log n)$ time, leveraging planarity and local combinatorial properties (Christiansen et al., 14 Feb 2025).

3. Geometry-Driven Path Diversification for Planning

Geometry-driven augmentation is crucial in kinodynamic and multi-modal planning, especially for robotic systems requiring the enumeration of multiple topologically or geometrically distinct paths.

A prominent approach decomposes a workspace into obstacle-adjacency cells and constructs a medial-axis-like skeleton, enabling the refinement of homotopy classes into finer cell-sequence-based classes (Wang et al., 2022). This cell-based labeling allows distinguishing and augmenting paths in narrow passages that standard homotopy methods miss, using the workspace's spine to repair or enrich homotopy-equivalent trajectories.

Efficient tangent graph-based path enumeration encodes geometrically distinctive paths as locally-shortest chains in grid maps. Here, geometric constraints (line-of-sight, locally-collide, edge transfer) ensure each path encoded in the tangent graph corresponds to a unique homotopy class—enumerated via breadth-first search with priority constraints to mitigate combinatorial explosion. Empirically, this achieves enumeration of hundreds of distinct trajectories in ~100 ms in city-scale maps, vastly outperforming winding-number or signature-based methods (Yao et al., 2023).

Recent developments in neighbor-augmented graph search generalize the concept to "topo-geometrically distinct" path enumeration. Vertices are augmented with local path neighborhoods, and equivalence of search branches is determined by overlap of these neighborhoods. This universal construction produces multiple locally-optimal, geometrically distinct paths even when topologically indistinguishable (e.g., around genus-zero obstacles or complex high-curvature regions). Applications include tether-constrained robot planning, where both simulation and physical experiments validate the approach on multi-room environments with restrictive path constraints (Sahin et al., 2023).

4. Geometry-Aware Augmentation in Stochastic System Inference

A less traditional but increasingly important branch applies geometry-driven path augmentation to infer properties of sparsely observed nonlinear stochastic systems. Here, the method interpolates unobserved system paths using bridges that are constrained not only by observation times but by geodesics of a data-driven Riemannian metric on the inferred state space. The augmentation is defined by soft constraints favoring proximity to these geodesics during the EM inference of drift functions (using Gaussian process priors).

The algorithm alternates between augmentation (sampling path ensembles between observations with geodesic penalties) and drift field update, resulting in recovery of nonlinear dynamics (including non-conservative vector fields) at sampling rates far coarser than standard temporal bridge methods allow. Quantitative benchmarks (weighted RMSE) show a 40–60% error reduction over conventional approaches at typical noise levels and sparse sampling (Maoutsa, 2023).

5. Analytical Methods and Algorithmic Complexity

Geometry-driven path augmentation typically relies on algorithmic primitives exploiting geometric monotonicity, unimodality, and combinatorial decomposition:

• Prefix/suffix distance arrays and two-pointer sweeps allow $O(n)$ or $O(n \log n)$ algorithms for diameter, radius, and center minimization.
• Range-minimum, sorted-matrix, and parametric search yield efficient evaluation of candidate path augmentations in higher dimensions or for complex metrics (Große et al., 2016, Wang, 2016).
• Cell-decomposition (Delaunay triangulation, medial axis) and product graphs enable finer-than-homotopy classification and planning (Wang et al., 2022).
• Tangent-graph paradigms and priority-limited BFS enable real-time construction of abundant topologically distinctive paths at scale (Yao et al., 2023).
• Secondary neighborhood search, cut-point insertion, and local equivalence checking underpin multi-path enumeration via neighbor-augmented graphs in general configuration spaces (Sahin et al., 2023).

Table: Selected algorithmic complexities in key geometry-driven augmentation problems

Problem Type	Complexity	Reference
Path diameter (cont.)	O(n)	(Carufel et al., 2015)
Cycle diameter (2-shortcut)	O(n)	(Carufel et al., 2015)
Tree diameter (cont.)	O(n log n)	(Carufel et al., 2016)
Path radius (disc.)	O(n)	(Wang et al., 2020)
Plane path parity	O(n log n)	(Christiansen et al., 14 Feb 2025)
Tangent paths (BFS K-topo)	O(K·d·logK)	(Yao et al., 2023)
Neighbor-aug. graph search	O(k_n·	V_exp

6. Extensions to Curves, High Dimensions, and Data-Driven Spaces

Geometry-driven augmentation generalizes from polygonal networks to arbitrary rectifiable curves, leveraging arc-length parameterization and extending optimality proofs and algorithmic monotonicity (Carufel et al., 2015, Carufel et al., 2016). Well-separated pair decompositions enable $(1+\epsilon)$-approximation of minimal-diameter augmentations in Euclidean spaces of constant dimension in linear time (Große et al., 2016).

In image analysis, region-based homogeneity features and curvature regularization are embedded in local Finsler metrics, facilitating geodesic-based segmentation via Eikonal PDE and dual-cut optimal contour concatenation, with efficiencies dependent on Fast Marching complexity (Chen et al., 2020).

7. Limitations, Open Problems, and Interpretive Notes

Several geometry-driven augmentation problems in general graphs (adding $k>1$ shortcuts, $k$-center variants) rapidly become NP-hard, and only single-edge or special-case algorithms maintain tractability. While homotopy-based augmentation is insufficient to classify paths in high dimensions or under robot configuration constraints, cell- and neighborhood-driven methods address these gaps. Data-driven metric learning relies on sufficient concentration of invariant densities in the underlying dynamical system, with high noise and poor initializations potentially degrading inference accuracy (Maoutsa, 2023).

Geometry-driven path augmentation thus encapsulates a suite of algorithmic and theoretical advances in network optimization, motion planning, inference, and computational geometry, unified by the explicit exploitation of ambient geometric structure to generate, select, and optimize path spaces.