Bounding Distance Between Outputs in Distributed Lattice Agreement
Abdullah Rasheed, Nidhi Dubagunta
TL;DR
The paper studies lattice agreement with an added $\varepsilon$-tightness constraint by introducing a lattice quasi-metric space with distance function $\delta_X$ and formalizing $\varepsilon$-bounded lattice agreement. It establishes equivalences with standard lattice agreement under certain bounds, and analyzes solvability in synchronous versus asynchronous systems, showing straightforward solvability in the synchronous setting but strong impossibility results asynchronously. To cope with asynchronous limits, it presents a heuristic DR$(k)$ reconciliation protocol and a simplified Markov-model abstraction that quantify tightness improvement probabilities, reporting that a small constant number of rounds (around $k=5$) suffices to achieve high-probability improvement in simulations. The work provides a discrete tightness taxonomy, highlights fundamental asynchronous limitations, and offers a practical reconciliation approach to accelerate convergence in distributed systems.
Abstract
This paper studies the lattice agreement problem and proposes a stronger form, $\varepsilon$-bounded lattice agreement, that enforces an additional tightness constraint on the outputs. To formalize the concept, we define a quasi-metric on the structure of the lattice, which captures a natural notion of distance between lattice elements. We consider the bounded lattice agreement problem in both synchronous and asynchronous systems, and provide algorithms that aim to minimize the distance between the output values, while satisfying the requirements of the classic lattice agreement problem. We show strong impossibility results for the asynchronous case, and a heuristic algorithm that achieves improved tightness with high probability, and we test an approximation of this algorithm to show that only a very small number of rounds are necessary.