The Role Of Condition Numbers And Numerical Stability In Robust Computational Geometry Algorithms

Numerical stability refers to how sensitive an algorithm’s output is to slight changes or errors in the input data. Condition numbers quantify this sensitivity – high condition numbers imply greater numerical instability. Unstable algorithms can produce wildly inaccurate outputs even for reasonable inputs. This is problematic in computational geometry where robustness and reliability are critical.

Definition of Condition Number and Numerical Stability

The condition number provides a bound on how much the output value of a function can change for a small change in the input argument. A problem with a low condition number is considered well-conditioned – small errors in inputs produce small errors in outputs. As the condition number grows, the problem becomes ill-conditioned – output errors become disproportionately large.

For example, consider the function f(x) = 1/x. When x = 1, f(x) = 1. A tiny change in x near 1 causes a small change in f(x). But when x approaches 0, f(x) approaches infinity. Now a tiny change in x causes an enormous change in f(x). The problem is ill-conditioned when x is near 0.

More formally, for a function f(x) with input x and output y = f(x), the condition number is defined as:

Condition Number = lim|_{delta x->0} (Fractional change in y) / (Fractional change in x)

The higher this limit, the more ill-conditioned f(x) is, and the more unstable its output. Well-conditioned functions have condition numbers near 1. Geometry problems often have high condition numbers – a small error in a coordinate makes a large difference in intersection outcomes.

Role of Condition Number in Robustness of Algorithms

The condition number directly impacts an algorithm’s numerical robustness – its capacity to handle imperfect calculations and still produce valid outputs. Numerical robustness requires computational stability – small changes in input must produce proportionally small changes in output.

In computational geometry, condition numbers arise when solving geometric predicates – tests to determine spatial relations like “do two line segments intersect?” These decisions depend on calculating parameters like slopes, distances, intersection points etc. High condition numbers here can completely flip an algorithm’s logic with tiny input errors.

For example, when finding the convex hull of a point set, an unstable predicate for checking left/right turns could incorrectly classify a point. This single mistake then propagates, producing a wildly incorrect hull. Such fragility is clearly unacceptable in practice.

Numerical stability powered by low condition numbers is thus crucial for computational geometry. Knowing the prevailing condition numbers then helps guide algorithm design towards robustness.

Techniques to Improve Numerical Stability

Various techniques can enhance numerical stability and control condition numbers:

Exact Arithmetic

Instead of fixed precision floats, exact rational arithmetic avoids representation errors. But operations on expensive rational fractions slow algorithms down considerably. Hybrid approaches use exact arithmetic only for critical predicates.

Controlled Perturbation

Deliberately perturbing input data can move predictors away from unstable ill-conditioned regimes. For example, slightly shifting a point sideways can dramatically improve convex hull stability. The challenge is perturbing just enough to enable stability but not affect algorithm correctness.

Algorithm Redesign

Selecting numerically stable algorithms, or redesigning unstable algorithms for stability can fundamentally improve robustness. This includes using stable geometric predicates, avoiding cancellation errors, improved convergence criteria and other problem-specific techniques.

Examples of Numerically Unstable Algorithms

Many textbook computational geometry algorithms have poor numerical properties. Two representative examples are:

Intersection of Lines

Consider finding whether two lines intersect. The standard algebraic approach solves a system of linear equations. But this breaks down if the lines are nearly parallel – the high condition number flips the result with tiny input errors. Such “almost parallel” cases are common in practice.

Convex Hull

The textbook Gift Wrapping algorithm for computing convex hulls has terrible numerical fragility. Decision predicates for left/right turns have high condition numbers. Tiny input uncertainty leads to wrong turn classifications and a completely incorrect hull.

Modifying Unstable Algorithms for Improved Robustness

Techniques like controlled perturbation, enhanced precision and topological methods can modify numerically unstable computational geometry algorithms.

Using Symbolic Perturbation

Instead of numeric perturbation, symbolic techniques inject uncertainty intrinsically in the computation flow. For convex hull, this symbolically perturbs the input to mitigate conditioning issues. The output remains fully deterministic.

Switching to Higher Precision

Using long double variables and data types in place of float/double can minimize representation errors. Precision is increased only in critical fragile sections like predicate evaluation. This balances stability with computational cost.

Topological Methods

Topological techniques replace floating point computations with robust integer decisions based on geometric topology and connectivity. This avoids numeric conditioning entirely. For convex hull, a topologically stable pivot algorithm provably handles degeneracies.

Testing and Validation of Robustness

Testing computational geometry code for numerical reliability involves intentionally inducing instability:

Random Input Testing

Monte Carlo methods using randomly generated geometric inputs check for sporadic unstable outputs across large samples. But they often miss corner cases.

Adversarial Input Generation

Unlike random testing, adversarial techniques surgically insert ill-conditioned inputs to trigger instability. This includes numerically challenging scenarios like nearly identical points, multiple intersections, infinitely steep lines etc. Code that survives adversarial attacks inspires confidence.

Conclusion and Future Directions

Numerical stability enforced by low condition numbers is critical for writing robust computational geometry implementations. The path ahead includes hybrid techniques combining topological and numerical methods for efficiency and reliability along with formal algorithmic analysis to bound condition numbers across problem domains.