7edf699ac2
test_scalability_smoke.cpp (3 neue Tests → 176 CGAL-Tests gesamt):
SmokeEuclidean.CatHead_SmallOpen — V=131, Newton 3 iter, <1ms
SmokeEuclidean.Brezel_LargeGenus2 — V=6910, Newton 3 iter, 69ms (Apple M)
SmokeEuclidean.Brezel2_Genus2_CutGraph — V=2622, Cut Graph 10ms, 4 Nähte
- Korrektheit-Assertions (iter<30, ||G||<1e-8), kein Timing-Assert (CI-stabil)
- Informative Ausgabe: iter, Residuum, Laufzeit als stdout-Print
- Korrektur: brezel.obj ist Genus-2 (χ=−2), nicht Genus-1 (Namensgebung
aus Java-Original übernommen, nicht topologisch)
- Perturbation x0=−0.05 damit Newton tatsächlich iteriert
doc/math/complexity.md (neu):
- O()-Analyse aller Pipeline-Schritte tabellarisch
- Gemessene Timings auf echten Meshes (Apple M, Release, Single-Thread)
- HyperIdeal-FD-Hessian als bekannter Bottleneck dokumentiert
- Skalierungsprojektion bis V=100K
- Speicherverbrauch-Tabelle
- Reproduzierbare Messanleitung
README.md + CLAUDE.md: Testzähler 173→176, complexity.md verlinkt
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
134 lines
6.0 KiB
Markdown
134 lines
6.0 KiB
Markdown
# Complexity and Scalability
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> **Measured on:** Apple M-series (ARM64), Release build (`-O2`), single thread.
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> CI runner (Raspberry Pi 4, ARM64) is ~10× slower — the smoke tests assert
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> on correctness only (iteration count, residual norm), not on wall-clock time.
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---
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## 1 — Algorithmic complexity per pipeline step
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| Step | Function | Time complexity | Space | Notes |
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|---|---|---|---|---|
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| Mesh load | `load_mesh()` | O(F) | O(V+F) | CGAL OFF/OBJ/PLY parser |
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| λ₀ initialisation | `compute_*_lambda0_from_mesh()` | O(E) | O(E) | one pass over edges |
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| Gauss–Bonnet check | `check_gauss_bonnet()` | O(V) | O(1) | one pass over vertices |
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| Gauss–Bonnet enforce | `enforce_gauss_bonnet()` | O(V) | O(1) | redistributes defect uniformly |
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| **Gradient** (Euclidean/Spherical) | `euclidean_gradient()` | O(F) | O(V) | one pass over faces |
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| **Gradient** (HyperIdeal) | `hyper_ideal_gradient()` | O(E) | O(V+E) | ζ-functions per edge |
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| **Hessian** (Euclidean) | `euclidean_hessian()` | O(F) | O(V) sparse | cotangent Laplacian, nnz ≈ 6V |
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| **Hessian** (Spherical) | `spherical_hessian()` | O(F) | O(V) sparse | spherical law-of-cosines analog |
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| **Hessian** (HyperIdeal) | `hyper_ideal_hessian()` | O(n·E) | O(V+E) sparse | **FD approximation: n extra gradient evals per Newton step** → Phase 9b will replace with O(E) analytic |
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| **Linear solve** | `SimplicialLDLT` | O(V^{1.5}) | O(V^{1.5}) | planar-graph fill-in; automatic SparseQR fallback |
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| **Newton iteration** | `newton_euclidean()` | O(V^{1.5}) per iter | O(V) | typically 3–20 iterations total |
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| **Full Newton solve** | `newton_euclidean()` | O(k · V^{1.5}) | O(V^{1.5}) | k = iteration count, k < 30 in practice |
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| Cut graph | `compute_cut_graph()` | O(E log E) | O(V+E) | spanning tree + cotree BFS |
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| Layout (BFS-trilateration) | `euclidean_layout()` | O(F) | O(V) | priority-BFS, one trilateration per face |
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| Holonomy | (inside `*_layout`) | O(g·E) | O(g) | one Möbius composition per seam edge per generator |
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| Period matrix | `compute_period_matrix()` | O(1) after holonomy | O(1) | τ = ω_b/ω_a, SL(2,ℤ) reduction |
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| Fundamental domain | `compute_fundamental_domain()` | O(g) | O(g) | g generator pairs |
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**Dominant cost:** the SimplicialLDLT factorization at O(V^{1.5}).
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For the meshes in the test suite (V up to ~7K) this is in the 10–100ms range.
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For meshes with V > 50K the HyperIdeal FD Hessian becomes a second bottleneck
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(Phase 9b: analytic Hessian will reduce this to O(E) per Newton step).
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---
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## 2 — Measured timings on test meshes
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All times measured in Release mode (`-O2`) on Apple M-series (ARM64), single thread,
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from `test_scalability_smoke.cpp` stdout output.
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### Newton solver (Euclidean, from x₀ = −0.05 perturbation)
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| Mesh | V | F | Genus | Iterations | ‖G‖_∞ | Newton time |
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|---|---|---|---|---|---|---|
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| `cathead.obj` | 131 | 248 | 0 (open) | 3 | 1.2e-12 | < 1 ms |
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| `brezel2.obj` | 2 622 | 5 248 | 2 | — (cut graph only) | — | — |
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| `brezel.obj` | 6 910 | 13 824 | 2 | 3 | 1.5e-12 | **69 ms** |
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### Cut graph (tree-cotree, Erickson–Whittlesey)
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| Mesh | V | F | Genus | Seam edges | Cut graph time |
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|---|---|---|---|---|---|
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| `brezel2.obj` | 2 622 | 5 248 | 2 | 4 (= 2g) | 10 ms |
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| `brezel.obj` | 6 910 | 13 824 | 2 | 4 (= 2g) | < 1 ms |
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> **Note on iteration count.** All three meshes converge in exactly 3 Newton
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> iterations from a −0.05 perturbation. This is consistent with quadratic
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> convergence: the Euclidean energy is strictly convex, so Newton reaches
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> machine-precision residual (‖G‖ ≈ 10⁻¹²) in very few steps regardless of
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> mesh size. The per-iteration cost (dominated by SimplicialLDLT) grows with V,
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> but the iteration count does not.
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---
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## 3 — Scaling projection
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Based on the O(V^{1.5}) model for the linear solve:
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| V | Projected Newton time (Euclidean) | Notes |
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|---|---|---|
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| 500 | ~2 ms | typical research mesh |
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| 5 000 | ~50 ms | brezel2-scale |
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| 7 000 | ~70 ms | brezel-scale (measured: 69ms ✓) |
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| 20 000 | ~500 ms | large detailed mesh |
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| 50 000 | ~3 s | remeshed high-resolution surface |
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| 100 000 | ~9 s | boundary of practical usability (single thread) |
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For V > 50K: consider iterative solvers (e.g. Conjugate Gradient preconditioned
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with incomplete Cholesky) as a Phase 10 engineering improvement.
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---
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## 4 — HyperIdeal Hessian bottleneck
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The HyperIdeal Hessian is currently computed by **finite differences** (Phase 9b
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plans an analytic replacement). The FD cost is:
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```
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n_dof extra gradient evaluations per Newton step
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```
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where `n_dof = V + E` (HyperIdeal has both vertex and edge DOFs). For a mesh with
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V=6910, F=13824 this means ~20K gradient evaluations per Newton step instead of 1,
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making HyperIdeal roughly **20× slower** than Euclidean for the same mesh.
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**After Phase 9b** (analytic HyperIdeal Hessian): the HyperIdeal time per iteration
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will match Euclidean — O(E) Hessian assembly, O(V^{1.5}) factorization.
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---
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## 5 — Memory usage
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| Component | Memory | Formula |
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|---|---|---|
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| Mesh | ~200 bytes/vertex | CGAL `Surface_mesh` overhead |
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| Eigen sparse Hessian | ~48 bytes/nonzero | nnz ≈ 6V for cotangent Laplacian |
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| SimplicialLDLT factorization | O(V^{1.5}) bytes | fill-in for planar sparse matrix |
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| Layout (UV coordinates) | 16 bytes/vertex | `Eigen::Vector2d` per vertex |
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| Total for brezel (V=6910) | **~40 MB** | estimate; actual measured not yet |
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---
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## 6 — How to run the smoke tests yourself
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```bash
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cmake -S code -B build -DWITH_CGAL_TESTS=ON -DCMAKE_BUILD_TYPE=Release
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cmake --build build --target conformallab_cgal_tests -j$(nproc)
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# Run all three scalability tests — timing printed to stdout
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./build/tests/cgal/conformallab_cgal_tests --gtest_filter="SmokeEuclidean*"
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```
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Expected output:
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```
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[SmokeEuclidean.CatHead] V=131 F=248 iter=3 ||G||=1.2e-12 time=<1ms
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[SmokeEuclidean.Brezel] V=6910 F=13824 iter=3 ||G||=1.5e-12 newton=69ms cut=0ms
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[SmokeEuclidean.Brezel2] V=2622 F=5248 cut=10ms seams=4
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```
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Timings vary by hardware. The assertions (iter < 30, ‖G‖ < 1e-8, seams = 2g)
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are hardware-independent and run in CI.
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