Piecewise Linear Morse–Smale Segmentations

• PLMSS is a segmentation technique that decomposes piecewise-linear domains into monotone regions based on gradient flows from unique minima to maxima.
• It leverages efficient algorithms such as pointer-doubling and parallel strategies to achieve scalable performance on high-performance computing platforms.
• The method underpins robust scientific visualization and analysis, with correction frameworks ensuring topological consistency even under lossy compression.

Piecewise Linear Morse–Smale Segmentations (PLMSS) are a foundational construct in computational topology for the analysis and decomposition of scalar fields. A PLMSS partitions a piecewise-linear domain—such as a triangulated or tetrahedral mesh—into monotone regions determined by the gradient flow from unique local minima to unique local maxima. This segmentation underlies feature-aware visualization, topological simplification, and robust scientific postprocessing, particularly in high-performance computing and large-scale simulations (Li et al., 5 Jan 2026, Will et al., 2024, Li et al., 2024, Maack et al., 2023, Subhash et al., 2020, Ludmány et al., 2021, Thanh et al., 2024). PLMSS serves as both a theoretical and computational bridge between continuous Morse–Smale theory and practical data-driven analysis.

1. Mathematical Framework and Definitions

PLMSS is constructed from a piecewise-linear scalar field $f:|K|\rightarrow\mathbb{R}$ defined on a simplicial complex $K$. The function $f$ is assumed Morse in the PL sense—critical points correspond to topologically nondegenerate vertices—by enforcing injectivity on vertex values via strategies such as Simulation of Simplicity (Li et al., 5 Jan 2026, Maack et al., 2023, Will et al., 2024).

• Critical points are identified as follows: a vertex $v$ is a minimum if its lower link $Lk^-(v)=\emptyset$, a maximum if $Lk^+(v)=\emptyset$, and a $p$-saddle if $Lk^-(v)$ has $p$ connected components; the index is defined accordingly (Li et al., 5 Jan 2026, Maack et al., 2023, Ludmány et al., 2021).
• Discrete gradient: For each vertex $v$, the steepest ascent (descent) neighbor is $n_{\max}(v) = \arg\max_{w\in N(v)} f_w$ and $n_{\min}(v) = \arg\min_{w\in N(v)} f_w$, respectively (Li et al., 5 Jan 2026, Will et al., 2024, Li et al., 2024).
• Integral lines: Discrete integral lines are polygonal chains traversed by following $n_{\max}$ (ascent) or $n_{\min}$ (descent); they terminate at extrema due to $f$ being injective (Li et al., 5 Jan 2026, Will et al., 2024, Li et al., 2024, Maack et al., 2023).
• Cells and labeling: Each vertex $v$ is associated with the pair $(m(v), M(v))$—the minimum and maximum it reaches by steepest descent/ascent—leading to a partition into Morse–Smale cells labeled by extremum pairs. The segmentation function $\ell(v) = (m(v), M(v))$ thus defines the PLMSS (Li et al., 5 Jan 2026, Li et al., 2024, Maack et al., 2023).

2. PLMSS Extraction Algorithms and Parallelization

Several high-performance algorithms exist for computing PLMSS, exploiting parallel hardware and distributed memory. Central to scalable workflows are pointer-doubling (path compression) and local ordering-preservation paradigms:

• Path compression: For descending (or ascending) segmentation, initialize $parent[v]=n_{\min}(v)$, then iteratively update $parent[v] \leftarrow parent[parent[v]]$. This union–find style pointer-doubling reduces the iteration count from the graph diameter $O(h)$ to $O(\log h)$, yielding $O(|V|\log h)$ runtime (Will et al., 2024, Maack et al., 2023).
• Parallelism: Algorithms are embarrassingly parallel over vertices in shared memory, and employ ghost layers and MPI communications in distributed memory. Critical performance factors include the cost of halo exchanges and global reductions in distributed settings (Li et al., 5 Jan 2026, Will et al., 2024, Maack et al., 2023).
• Boundary extraction: Multi-label marching tetrahedra or triangles (MLMT) are used to extract explicit region boundaries and separators within cells possessing non-trivial label diversity (Maack et al., 2023).

The pMSz algorithm (Li et al., 5 Jan 2026) innovates by replacing global integral-line tracing with strict local preservation of steepest edge directions, which enables near-ideal (>90%) parallel efficiency on GPU clusters and drastically reduces communication overhead. Only when a local block reaches local convergence is ghost-layer synchronization performed, further optimizing distributed scalability.

3. Correction and Preservation of PLMSS under Lossy Compression

Lossy compression can perturb scalar fields, distorting critical point structure and thereby segmentations. To guarantee the scientific validity of downstream analyses, edit-based correction frameworks such as MSz (single GPU) and pMSz (distributed) enforce PLMSS preservation within a prescribed $\ell_\infty$ error bound (Li et al., 2024, Li et al., 5 Jan 2026).

• MSz workflow: Iteratively alternates between C-loops (correcting false extrema) and R-loops (fixing misclassified regular points by following integral lines), applying local edits that monotonically reduce the value in a direction that restores original gradient orderings, strictly respecting the error bound. The workflow is proven to converge in finite steps (Li et al., 2024).
• pMSz simplification: Avoids global integral-line tracing. Instead, it ensures that each vertex's ordering relation with its steepest neighbors matches the original field, which is a sufficient condition for full PLMSS consistency (Li et al., 5 Jan 2026).
• Empirical results: Correction methods add minimal storage and runtime overhead (<5–10% bitrate increase, $<5\%$ additional edits compared to uncorrected decompression). pMSz achieves more than 14× speedup over MSz in distributed scenarios, with over 90% weak parallel efficiency on 128 A100 GPUs (Li et al., 5 Jan 2026).

4. Implementation Considerations and Scalability

PLMSS computation has been implemented in both open-source pipelines (e.g., the Topology ToolKit/TTK) and custom parallel algorithms (Will et al., 2024, Maack et al., 2023). Key implementation strategies include:

• Efficient storage: Only $O(n)$ per-vertex storage is required for parent pointers, label arrays, and possibly ghost/halo buffers in distributed memory (Maack et al., 2023).
• Value disambiguation: Global tie-breaking ensures injectivity needed for unique gradient directions (Maack et al., 2023, Will et al., 2024).
• Scaling metrics: Weak and strong scaling performance is quantified via parallel efficiency $E(P)$; for connected component labeling, communication costs depend on the size of the ghost set $G$ and benefit from neighbor-only communication (Will et al., 2024, Maack et al., 2023).
• Strong empirical scaling: PLMSS algorithms achieve near-linear speedups on both multicore CPUs and large GPU clusters, outperforming discrete-Morse-based MS complex codes by up to an order of magnitude (Maack et al., 2023, Will et al., 2024).

5. Comparison to Classical and Alternative MS Complexes

PLMSS occupies a specific role distinct from both smooth Morse–Smale complexes and discrete Morse-theoretic complexes:

• PL vs. smooth: In PLMSS, all critical points are vertices; integral lines are edge paths. The segmentation does not explicitly represent saddle–saddle connections or separatrix topology, often simplifying downstream geometric output and acceleration (Maack et al., 2023, Ludmány et al., 2021).
• Combinatorial gradient schemes: The standard steepest-descent PL approach may suffer geometric inaccuracy on axis-aligned grids. Mesh-augmentation and triangulation methods overcome this, yielding both topologically correct and geometrically accurate PLMSS (Thanh et al., 2024).
• Saddle treatment: PLMSS in its standard form uses only minima and maxima for segmentation labels. Saddles are necessary only for the explicit construction of the full Morse–Smale complex (for separatrix extraction, feature simplification, etc.) (Maack et al., 2023, Ludmány et al., 2021, Subhash et al., 2020).

6. Applications, Limitations, and Directions

PLMSS underpins numerous topological workflows in scientific computing, visualization, and data analysis:

• Large-scale simulations: Correction techniques (MSz, pMSz) enable safe use of aggressive lossy compressors in cosmology, combustion, and materials simulation, with fully guaranteed topological validity for segmentation-based analysis (Li et al., 5 Jan 2026, Li et al., 2024).
• Feature extraction and tracking: PLMSS provides robust region labeling for quantitative analysis, segmentation-driven visualization (e.g., isolating crystal grains, flamelet zones) (Li et al., 5 Jan 2026, Will et al., 2024, Maack et al., 2023).
• Machine learning integration: Morse–Smale regression combines PLMSS with local model fitting in each segment, affording both interpretability and potential for hybridization with nonparametric learners. The segmentation offers regimes for model partitioning by data topology (Farrelly, 2017).
• Limitations: Algorithms require minor extension for unstructured grids with variable valence. Communication bottlenecks and load balance in the distributed context may be further improved with asynchronous/block-relaxed schemes or neighbor-only exchanges (Li et al., 5 Jan 2026, Will et al., 2024).
• Future avenues: Extension to vector field critical-point matching, higher-order descriptors, and fully topology-and-geometry-accurate separatrix extraction are active research directions (Li et al., 5 Jan 2026, Thanh et al., 2024).

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References

• (Li et al., 5 Jan 2026) "pMSz: A Distributed Parallel Algorithm for Correcting Extrema and Morse Smale Segmentations in Lossy Compression"
• (Will et al., 2024) "Distributed Path Compression for Piecewise Linear Morse-Smale Segmentations and Connected Components"
• (Li et al., 2024) "MSz: An Efficient Parallel Algorithm for Correcting Morse-Smale Segmentations in Error-Bounded Lossy Compressors"
• (Maack et al., 2023) "Parallel Computation of Piecewise Linear Morse-Smale Segmentations"
• (Subhash et al., 2020) "A GPU Parallel Algorithm for Computing Morse-Smale Complexes"
• (Ludmány et al., 2021) "Morse-Smale complexes on convex polyhedra"
• (Thanh et al., 2024) "Revisiting Accurate Geometry for Morse-Smale Complexes"
• (Farrelly, 2017) "Extensions of Morse-Smale Regression with Application to Actuarial Science"