Fundamental Limits of Coded Polynomial Aggregation

TL;DR

This paper extends coded polynomial aggregation to account for stragglers in distributed systems, establishing conditions for exact recovery with fewer responses and demonstrating a sharp transition at a derived intersection-size threshold.

Contribution

It introduces a straggler-aware CPA framework with necessary and sufficient conditions for exact recovery based on non-straggler pattern intersections.

Findings

Exact recovery is achievable with fewer responses than traditional methods.

Feasibility depends on the intersection structure of non-straggler patterns.

Simulation results confirm a sharp transition at the theoretical threshold.

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.