Controlled Perturbations

• Controlled Perturbations are algorithmic techniques that modify inputs slightly to prevent degenerate cases and maintain robust numerical and geometric computations.
• They integrate guard mechanisms and adaptive precision adjustments to detect and correct rounding errors near critical numerical boundaries.
• The framework employs direct, bottom-up, and top-down analysis methods to derive explicit error bounds and probabilistic performance guarantees.

Controlled perturbations are a class of algorithmic and analytical techniques designed to provide robustness, correctness, and efficiency in systems that would otherwise be vulnerable to degeneracies and rounding errors, particularly in numerical computation and computational geometry. The foundational principle is to introduce carefully bounded modifications—either to input data or to algorithmic conditions—such that the pathological behaviors caused by near-degenerate inputs or limited precision are avoided, while preserving the essential combinatorial or geometric structure of the problem.

1. Principles of Controlled Perturbation

Controlled perturbation addresses the inherent conflict in geometric and numerical computation between efficiency (favoring fast, floating-point arithmetic) and reliability (favoring slow, exact arithmetic). In practice, floating-point computation is susceptible to rounding errors that may cause sign misclassification or incorrect behavior near degenerate cases. Controlled perturbation algorithms introduce small, randomized or structured modifications to input values, drawing perturbed samples from an axis-parallel "perturbation area" around the original data. The purpose of these perturbations is to dislodge the computation away from unstable configurations (such as colinear points, nearly singular matrices, or points near a predicate's critical boundary).

Protection is enforced by embedding "guard mechanisms"—decision functions or checks—at the points where numerical uncertainty could cause a misclassification. If a guard detects that a computation is in an uncertainty region or too close to a critical value (e.g., the sign of a predicate cannot be deemed correct given the error bounds), then the algorithm either increases the floating-point precision or reselects a perturbed input. This combine fast arithmetic with an explicit error detection and correction strategy.

2. Analysis Tool Box and Formal Framework

The core contribution is the general analysis tool box that rigorously decomposes the analysis of controlled perturbation algorithms into modular components. The tool box is structured around two main stages:

Function Analysis

This stage focuses on the behavior of real-valued functions (predicates) over the perturbed input domain $U_\delta(A)$. Three quantitative properties must be derived:

• Region-suitability: For a function $f:U_\delta(A)\to\mathbb{R}$, the uncertainty region $R_{f, \gamma}$ is defined to enclose inputs where $f$ is near zero or near a pole. This is formally quantified by a volume upper bound $\nu_f(\gamma)$ or its complement $\chi_f(\gamma)$.
• Value-suitability: Outside $R_{f, \gamma}$, $f$ is bounded away from critical values; this is captured by a lower bound $\varphi_{\inf f}(\gamma)$ and, when rational functions are involved, an upper bound $\varphi_{\sup f}(\gamma)$ as well.
• Safety-suitability: An fp-safety bound defines the minimal magnitude of $f(x)$ needed to ensure that sign (or other qualitative properties) of its floating-point evaluation is correct, given floating point precision $L$.

Quantified Relations and Algorithm Analysis

After bounding uncertainty regions and predicate values, the "method of quantified relations" translates these bounds into precise requirements on floating-point precision (or exponent bit length) needed to guarantee correctness with a specified probability $p$. This involves calculating:

• The permissible uncertainty region volume $\varepsilon_\nu(p) = (1-p)\mu(U_\delta)$.
• The inverse volume bound $\gamma(p) = \nu_f^{-1}(\varepsilon_\nu(p))$.
• The minimal value bound $\varphi_f(\gamma(p))$, and from that, via the fp-safety bound, deduce the minimal precision $L_f(p)$ (and exponent bit-length $K_f(p)$ for range errors).

This process ensures probabilistic correctness in all predicate evaluations across the algorithm.

3. Derivation Techniques: Direct, Bottom-up, and Top-down

Three complementary approaches are provided to analyze the bounding functions needed for safe use of controlled perturbation:

Approach	Key Operation	Applicability
Direct	Derives bounds from geometric arguments	Efficient for low-dimensional and geometric predicates
Bottom-up	Combines rules for function composition	Suitable for polynomials, composite predicates
Top-down	Sequentially eliminates variables via infimum	Effective for recursive decomposition of predicates

• Direct Approach: For functions with simple geometric meaning, bounds are often available directly from geometric relationships (e.g., inside/outside tests, orientation predicates).
• Bottom-up Approach: Derivation uses rules for combining region- and value-suitable functions through operations such as product, min, or max, forming explicit bounds for multivariate polynomials, e.g., using Horner's scheme and reverse-lexicographic order to analyze polynomials efficiently. The product rule, min/max rule, and lower-bounding rule are formalized.
• Top-down Approach: In more general or higher-dimensional predicates, variables are recursively eliminated by "freezing" distances to the critical set. The final bounds are expressed as monotone functions of the distance parameters, facilitating inversion and composition.

This modular system allows detailed, function-specific and context-appropriate analysis for a wide class of predicates, including not just polynomial but also rational functions.

4. Extensions: Rational Functions and Object-Preserving Perturbations

The formalism is extended to rational function predicates, which introduces the necessity to control both underflow (region where the sign is almost zero) and overflow (near poles). The analysis admits upper fp-safety bounds so that overflow is properly detected and managed. The critical set $C_{f, \delta}(x)$ is extended to capture both points where $|f| \to 0$ and where $|f| \to \infty$.

A crucial development is the distinction between classical pointwise perturbations and object-preserving perturbations. In object-preserving perturbations, only the defining parameters of a geometric object (such as the center of a circle or anchor points of a segment) are perturbed, with the other measurements held fixed. This allows the combinatorial and geometric integrity of input objects (e.g., non-self-intersecting polygons) to be preserved through perturbation, which is essential for inputs like CAD data.

5. Practical Algorithm Design and Guarantees

The analysis provides a foundation for a guarded algorithm template, "cp", that repeatedly perturbs the input, performs the guarded computation, and conditionally increases precision or exponent bit-length in response to detected risk. The essential procedure is:

1. Randomly select perturbed input $y$ in the allowed region $_\delta(y)$.
2. Attempt execution with bounded-precision arithmetic.
3. On a guard failure (e.g., computed value insufficiently far from zero, or floating-point range overflow), either advance the precision parameter $L$/bit-length $K$ or redo the random perturbation.
4. Iterate until the guarded computation succeeds.

Probabilistic performance guarantees are computed precisely via the quantification machinery: with a bound on the number of predicate evaluations and a desired cumulative failure risk, appropriate pointwise risk is set for each predicate, and corresponding $L, K$ are assigned via the analytic bounds. This ensures that, over all computations, the failure probability remains below a defined threshold.

The modular composition of the method enables the reuse and adaptation of analysis results across different algorithms and predicates.

6. Key Mathematical Formulations

Critical analytic definitions provided include:

• Region of Uncertainty:

$$R_{f,\gamma}(x) := \overline{U_\delta(x)} \cap \Bigl( \bigcup_{c \in C_{f,\delta}(x)} U_\gamma(c) \Bigr)$$

• Value Suitability:

$$\varphi_{\inf f}(\gamma) \le |f(x)|, \quad x \notin R_{f,\gamma}$$

• FP-safety Bound for Univariate Polynomials (degree $d$):

$$(L) := (d+2) \cdot \max_{1\le i\le d}|a_i| \cdot 2^{e\,(d+1)+1-L}$$

If $|f(x)| > (L)$, floating-point sign is reliable.

• Precision Function via Quantified Relations:

$$\varepsilon_\nu(p) = (1-p)\,\mu(U_\delta) \ \gamma(p) = \nu_f^{-1}(\varepsilon_\nu(p)) \ L_f(p) = \text{{inverse fp-safety bound}}(\varphi_f(\gamma(p)))$$

For function combinations, bottom-up rules (e.g., for products $f(x) = g(x) h(x)$) specify

$$\varphi_f(\gamma) := \varphi_g(\gamma_1,\ldots,\gamma_\ell)\cdot \varphi_h(\gamma_{\ell+1},\ldots,\gamma_k)$$

and

$$\nu_f(\gamma) := \min\left\{\prod_{i=1}^k (2\delta_i),\, \nu_g(\gamma_1,\ldots,\gamma_\ell) \prod_{i=\ell+1}^k (2\delta_i) + \nu_h(\gamma_{\ell+1},\ldots,\gamma_k) \prod_{i=1}^{\ell} (2\delta_i)\right\}.$$

7. Significance and Impact on Computational Geometry

The controlled perturbation framework reconciles the need for exactness (reliability) with the efficiency of floating-point computation by:

• (i) providing probabilistically sound algorithmic templates that can adaptively increase precision or perturbation magnitude as required,
• (ii) unifying the analysis for a wide spectrum of predicates including polynomial and rational functions,
• (iii) extending to object-preserving perturbations, crucial for geometric object integrity,
• (iv) offering explicit formulas and code-level template guidance for implementers.

The separation of analysis into function-level and algorithm-level components, together with the three derivation approaches, establishes a reusable and extensible framework, bridging theoretical robustness and practical performance in robust geometric algorithm design.

In summary, controlled perturbations—systematically analyzed and implemented with this tool box—provide a methodological solution to the numerical and combinatorial fragility of geometric algorithms, ensuring correctness and efficiency across diverse input regimes.