Online Multi-Level Aggregation with Per-Batch Maximum Delay

2026-08-07

AI Summary

This work addresses the online multilevel aggregation problem on finite rooted trees, aiming to minimize the sum of the maximum waiting time per batch and the service cost over subtrees. By characterizing the dynamic programming structure of the offline optimum, the authors propose a class of deadline-based DP-Envelope online algorithms. On any non-degenerate rooted tree, this framework achieves—for the first time—a deterministic competitive ratio of 2 and a randomized competitive ratio of $e/(e-1)$, both of which are proven to be tight lower bounds, thus attaining theoretical optimality. Key technical contributions include polynomial-time dynamic programming, geometric nested partitioning (for line metrics), parameterized random sampling, and submodular analysis of service costs.

Abstract

We study online multi-level aggregation on finite rooted trees with a per-batch maximum-delay objective. A service pays for a rooted subtree and for the maximum waiting time among the requests cleared by that service. We show that the offline optimum admits a consecutive-arrival-block normal form and can be computed by a polynomial-time dynamic program. The same dynamic program defines the deadlines of a family of online algorithms, which we call DP-Envelope. Its deterministic endpoint is $2$-competitive. Sampling one global parameter with density $e^θ/(e-1)$ leads to an $e/(e-1)$-competitive randomized algorithm against an oblivious adversary. The deterministic guarantee matches the known fixed-node lower bound, and we prove a matching randomized lower bound. Thus, both guarantees are optimal on every nondegenerate rooted tree. We first develop the line metric as a warm-up, where the algorithm and its nested block partitions have a direct geometric interpretation. Finally, we show that the upper bounds extend to every realizable static service system with a normalized, nondecreasing, submodular joint service cost.

Problem

• online multi-level aggregation
• maximum delay
• rooted trees
• competitive analysis
• service cost

Innovation

• online multi-level aggregation
• maximum-delay objective
• dynamic programming
• competitive analysis
• submodular cost