A Framework for Algorithm Stability

TL;DR

This work introduces a formal stability framework for algorithms operating on time-varying data, specifically targeting combinatorial problems like kinetic EMSTs. It defines three stability notions—event stability, topological stability, and Lipschitz stability—and three algorithmic models—stateless, state-aware, and clairvoyant—then applies them to analyze stability-quality trade-offs. The paper provides both upper and lower bounds on stability-related metrics, including event counts and approximation factors, under various assumptions and data trajectories. The results yield practical insights into designing stable, high-quality algorithms for motion data visualization and dynamic networks, while highlighting fundamental limits and directions for future research. Overall, the framework unifies stability notions, clarifies their relationships, and demonstrates concrete bounds for kinetic EMST scenarios.

Abstract

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the trade-off between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. In addition, we need to refine the model of an algorithm based on how it interacts with the time-varying data, for which we consider several options. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.

Figures (17)

• Figure 1: Trade-offs between quality criteria for algorithms. Traditional criteria and their trade-off are shown in black; additional trade-offs for time-varying data are dashed blue.
• Figure 2: Algorithm $\mathcal{A}$ maps input instances from the input space $\mathcal{I}$ to the solution space $\mathcal{S}$. Metrics $d_\mathcal{I}$ and $d_\mathcal{S}$ allow us to reason about how different two instances are in respectively the input and solution space.
• Figure 3: Overview of the stability types and algorithmic models defined by our framework.
• Figure 4: The blue points are stationary, while the red point moves along a trajectory of degree $3$, triggering $\Omega(\frac{s}{k})$ events. The trajectory is shown as an arrow along the 1D space; the gray part has already been traversed.
• Figure 5: The blue points are stationary, while the red point moves along a trajectory of degree $8$ and is the second point to start moving through the blue points ($p_1(t) = \sum_{j=0}^{2} \frac{1}{(t-10\cdot j-20)^4+1}$). The trajectory is shown as an arrow along the 1D space; the gray part has already been traversed.

Theorems & Definitions (44)

• Lemma 1: markoff1916polynome
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• Lemma 4
• proof
• Lemma 5
• proof