Fundamental Limits of Coded Polynomial Aggregation

2026-01-15

AI Summary

This work addresses the problem of efficiently recovering polynomial-weighted aggregation results in distributed computing systems with straggling nodes, without requiring per-term decoding. Focusing on a predetermined set of non-straggling (i.e., timely) workers, the paper proposes an aggregation framework that integrates coded computation with combinatorial design. It reveals that the intersection structure among subsets of non-straggling nodes fundamentally governs recoverability and establishes a precise threshold on the size of these intersections: when the number of such subsets is sufficiently large, this threshold becomes both necessary and sufficient for exact recovery. Moreover, the study provides an explicit coding construction that achieves recovery whenever the intersection size exceeds the threshold. Simulations demonstrate a sharp phase transition in feasibility at the threshold, confirming its tightness.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Coded polynomial aggregation (CPA) enables the master to directly recover a weighted aggregation of polynomial evaluations without individually decoding each term, thereby reducing the number of required worker responses. In this paper, we extend CPA to straggler-aware distributed computing systems and introduce a straggler-aware CPA framework with pre-specified non-straggler patterns, where exact recovery is required only for a given collection of admissible non-straggler sets. Our main result shows that exact recovery of the desired aggregation is achievable with fewer worker responses than required by polynomial coded computing based on individual decoding, and that feasibility is fundamentally characterized by the intersection structure of the non-straggler patterns. In particular, we establish necessary and sufficient conditions for exact recovery in straggler-aware CPA and identify an intersection-size threshold that is sufficient to guarantee exact recovery. We further prove that this threshold becomes both necessary and sufficient when the number of admissible non-straggler sets is sufficiently large. We also provide an explicit construction of feasible CPA schemes whenever the intersection size exceeds the derived threshold. Finally, simulations reveal a sharp feasibility transition at the predicted threshold, providing empirical evidence that the bound is tight in practice.

Problem

• coded polynomial aggregation
• straggler-aware computing
• exact recovery
• non-straggler patterns
• distributed computing

Innovation

• coded polynomial aggregation
• straggler-aware computing
• exact recovery
• intersection threshold
• distributed computing