[Paper Review] Computing the Skorokhod Distance between Polygonal Traces (Full Paper)

Rupak Majumdar, Vinayak S. Prabhu|arXiv (Cornell University)|Oct 22, 2014

Formal Methods in Verification11 references3 citations

TL;DR

This paper presents the first fully polynomial-time algorithm for computing the Skorokhod distance between polygonal traces in $\mathbb{R}^n$ under the $L_1$, $L_2$, and $L_\infty$ norms. By reducing the Skorokhod distance computation to a Fréchet distance problem and introducing novel geometric primitives, the method enables exact, efficient computation of timing and value mismatches in hybrid system traces, with a time complexity of $O(m^3 \cdot \log m \cdot \text{poly}(n))$ for $m$ affine segments.

ABSTRACT

The \emph{Skorokhod distance} is a natural metric on traces of continuous and hybrid systems. For two traces, from $[0,T]$ to values in a metric space $O$, it measures the best match between the traces when allowed continuous bijective timing distortions. Formally, it computes the infimum, over all timing distortions, of the maximum of two components: the first component quantifies the {\em timing discrepancy} of the timing distortion, and the second quantifies the mismatch (in the metric space $O$) of the values under the timing distortion. Skorokhod distances appear in various fundamental hybrid systems analysis concerns: from definitions of hybrid systems semantics and notions of equivalence, to practical problems such as checking the closeness of models or the quality of simulations. Despite its popularity and extensive theoretical use, the \emph{computation} problem for the Skorokhod distance between two finite sampled-time hybrid traces has remained open. We address in this work the problem of computing the Skorokhod distance between two polygonal traces (these traces arise when sampled-time traces are completed by linear interpolation between sample points). We provide the first algorithm to compute the exact Skorokhod distance when trace values are in $ eals^n$ for the $L_1$, $L_2$, and $L_{\infty}$ norms. Our algorithm, based on a reduction to Frechet distances, is fully polynomial-time, and incorporates novel polynomial-time procedures for a set of geometric primitives in $ eals^n$ over the three norms.

Motivation & Objective

• To address the long-open problem of computing the Skorokhod distance between finite, sampled-time hybrid traces that are completed via linear interpolation (i.e., polygonal traces).
• To provide a fully polynomial-time algorithm for computing the Skorokhod distance in $\mathbb{R}^n$ under the $L_1$, $L_2$, and $L_\infty$ norms.
• To generalize and formalize the sketch of a Fréchet-based approach from Alt and Godau (1995) to higher dimensions and multiple norms, filling in missing details and correcting inaccuracies.
• To develop efficient, polynomial-time procedures for key geometric primitives required for Skorokhod distance computation, including ball-line intersections and distance to line segments under various norms.

Proposed method

• Reduces the Skorokhod distance computation to computing a related Fréchet distance between curves in a transformed normed space.
• Introduces a novel reduction that maps the Skorokhod problem into a geometric optimization problem involving timing distortions and value mismatches.
• Develops polynomial-time algorithms for geometric primitives in $\mathbb{R}^n$ under $L_1$, $L_2$, $L_\infty$, $L_S^1$, and $L_S^2$ norms using linear programming and convex geometry.
• Uses parametric search and parallel sorting techniques to improve asymptotic complexity, though with large constants.
• Applies windowing optimizations to restrict retiming to constant-size sliding windows, reducing complexity to $O(m \cdot \log m)$ for value computation and $O(m)$ for decision problems.
• Provides a self-contained treatment, deriving and proving all necessary geometric results from first principles without relying on prior computational geometry results.

Experimental results

Research questions

• RQ1 Can the Skorokhod distance between two polygonal traces in $\mathbb{R}^n$ be computed exactly in polynomial time for the $L_1$, $L_2$, and $L_\infty$ norms?
• RQ2 Is it possible to reduce the Skorokhod distance computation to a Fréchet distance problem in a higher-dimensional or transformed space?
• RQ3 What are the computational complexities of the geometric primitives (e.g., distance from point to line, intersection of balls and lines) required for Skorokhod distance computation under different norms in $\mathbb{R}^n$?
• RQ4 Can the algorithm be optimized for practical settings where retiming is constrained to a sliding window of constant size?
• RQ5 Can the approach be extended to norms that involve summing discrepancies rather than taking the maximum?

Key findings

• The Skorokhod distance between two polygonal traces in $\mathbb{R}^n$ can be computed in $O\left((m_x m_y^2 + m_y m_x^2) \cdot (P(\chi) + \log(m_x m_y)) + m_x m_y \cdot H(\chi)\right)$ time, where $m_x$ and $m_y$ are the number of affine segments in the traces.
• For the $L_1$ and $L_\infty$ norms, the geometric primitives require $O(n^2)$ time, with $P(L_1) = P(L_\infty) = \text{LP}(n)$, where $\text{LP}(n)$ is the time bound for linear programming.
• For the $L_2$ norm, the primitives are computed in $O(n)$ time, with $P(L_2) = n$ and $H(L_2) = n$, using vector algebra and convex geometry.
• When retiming is restricted to a constant-size sliding window $W$, the value computation runs in $O(M \cdot P(\chi) + M \cdot \log M \cdot H(\chi))$ time, with $M = \max(m_x, m_y)$, and the decision problem in $O(M \cdot H(\chi))$ time.
• The algorithm also yields the first fully polynomial-time method for computing the Fréchet distance between polygonal curves in $\mathbb{R}^n$ for the $L_1$, $L_2$, $L_\infty$, $L_S^1$, and $L_S^2$ norms.
• The method is self-contained and formally proves all geometric lemmas and generalizations of prior work, correcting inaccuracies in earlier sketches, particularly those from Alt and Godau (1995).