Complexity

Computational Complexity Analysis

Formal time and space complexity analysis for all LIRiAP algorithms.

Symbols

Symbol	Meaning
n	Total polygon vertices (exterior + holes)
g_coarse	Coarse grid size (GRID_COARSE)
g_fine	Fine grid size (GRID_FINE)
k	Candidate count kept for refinement (TOP_K)
m	Edge-guided initial angle candidates (<=12 in Contained/BCRS, <=10 in Approximation)
s90	Fallback sweep size: ceil(90 / a), where a = ANGLE_STEP
s180	Approximation fallback sweep size: ceil(180 / a)
p	Brent objective evaluations (maxiter=60 where explicitly set)
t	Stage 4-5 angle trials per candidate (<=4 in BCRS, <=2 in BCRS Fast)
X, Y	Unique boundary x/y coordinates after rotation (BCRS grid lines)
nu	BCRS cell count: (X-1) * (Y-1), max 89401 (300x300)

Primitive Solver Costs

Uniform Grid Solver

T_grid(g) = O(g^2), M_grid(g) = O(g^2)

BCRS Variable-Pitch Solver

T_bcrs = O(n log n + nu), M_bcrs = O(n + nu)

Implementation guard: if X > 300 or Y > 300, BCRS is skipped (seed fallback).

CABF Expansion

T_cabf = O(n)

Bounded iteration counts; worst-case geometric predicate cost.

Certification + Best-Effort Shrink

T_cert = O(n), T_shrink = O(n)

Axis-Aligned Solvers (Alt/Amenta, Daniels et al.)

T_axis = O(n^2), M_axis = O(n^2)

Per-Feature Complexity

Generic Pipeline

T_feature = T_angles + T_stage1 + T_refine + T_cert/fallback

BCRS-Family

T_feature,BCRS = T_angles + T_stage1 + T_refine + T_expand + T_cert/fallback

Algorithm Worst-Case Complexity

Algorithm	Time (single feature)	Memory
Approximation Standard	O(n + (m+s180)*g_coarse^2 + (p+1)*g_fine^2)	O(max(g_coarse^2, g_fine^2))
Approximation Fast	Same as Standard (batch changes constants)	Same
Contained Standard	O(n + (m+s90)g_coarse^2 + k((p+2)*g_fine^2 + n))	O(max(g_coarse^2, g_fine^2))
Contained Fast	O(n + (m+s90)g_coarse^2 + k(p*g_coarse^2 + g_fine^2 + n))	O(max(g_coarse^2, g_fine^2))
BCRS	O(n + (m+s90)g_coarse^2 + k(pg_coarse^2 + t(g_fine^2 + n log n + nu + n) + n))	O(max(g_fine^2, nu))
BCRS Fast	O(n + (m+s90)g_coarse^2 + k((p+4)g_coarse^2 + t(n log n + nu + n) + n)), t<=2	O(max(g_coarse^2, nu))
Axis-Aligned LIR	O(n^2)	O(n^2)

Fast Variant Speedup

Contained

Delta approx: p * (g_fine^2 - g_coarse^2) saved per candidate

BCRS

Delta approx: (4-2)(n log n + nu) + 4g_fine^2 - 4*g_coarse^2

Default Parameter Operation Model

Defaults used for calculation:

Family	g_coarse	g_fine	k	ANGLE_STEP
Approximation	40	100	1 effective	5
Contained/BCRS	40	120	3	5

Derived values: s180 = 36, s90 = 18, 40^2 = 1600, 100^2 = 10000, 120^2 = 14400

Dominant Term Estimates

Algorithm	Estimated Grid Units
Approximation Standard/Fast	(10+36)*1600 + (60+1)*10000 = 683,600
Contained Standard	(12+18)1600 + 3(60+2)*14400 = 2,726,400
Contained Fast	(12+18)1600 + 3(60*1600 + 14400) = 379,200
BCRS	(12+18)1600 + 3(601600 + 4(14400 + 89401)) = 1,581,612
BCRS Fast	(12+18)1600 + 3((60+4)1600 + 289401) = 891,606
Axis-Aligned LIR	O(n^2) - depends on vertex count, not grid resolution

Note: These are complexity-weighted operation counts, not wall-clock predictions.

Verification Against Wall-Clock Times

Using baseline 5406-feature wall times (N_WORKERS=1, no chunking):

Relation Check	Model Expectation	Measured	Result
Approximation Fast vs Standard	Nearly equal	125.93s vs 127.25s	Consistent
Contained Fast vs Standard	Fast should be lower	226.05s < 574.13s	Consistent
BCRS Fast vs BCRS	Fast should be lower	445.01s < 772.05s	Consistent
Contained Standard vs BCRS	Model estimate favors BCRS	574.13s < 772.05s	Mismatch

The model captures intra-family speed relations well; cross-family ordering can vary due to solver semantics, settings, and constant factors not captured by asymptotic terms.

Scaling Behavior

Observed Scaling Exponents (5406 features / 290 features = 18.65x)

Algorithm	1 Worker	12 Workers
Approximation Standard	0.9851	1.0031
Approximation Fast	0.9888	0.9951
Contained Standard (strict)	1.0039	0.9923
Contained Standard (fallback)	1.0002	0.9993
Contained Fast (fallback)	0.9965	1.0009
BCRS (strict)	0.9879	0.9851
BCRS (fallback)	0.9994	0.9975
BCRS Fast (fallback)	1.0038	1.0005
Axis-Aligned LIR	1.0058	1.0067

All exponents near 1.0 indicate linear scaling with feature count.

Parallel Efficiency

Algorithm	Speedup @290	Efficiency @290	Speedup @5406	Efficiency @5406
Approx Standard	1.19	9.95%	1.13	9.44%
Approx Fast	1.18	9.86%	1.16	9.68%
Contained Standard	1.37	11.39%	1.41	11.79%
Contained Fast	1.02	8.49%	1.01	8.38%
BCRS	0.83	6.90%	0.83	6.96%
BCRS Fast	0.79	6.59%	0.80	6.66%
Axis-Aligned LIR	0.80	6.63%	0.76	6.32%

BCRS and Axis-Aligned families show negative speedup (slowdown) with multithreading due to synchronization overhead exceeding per-feature computation time.