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Tight Bounds for Online Balanced Partitioning in the Generalized Learning Model

Harald Räcke, Stefan Schmid, Ruslan Zabrodin

TL;DR

Two deterministic online algorithms are presented: an online algorithm with competitive ratio O(max(√kℓ log k, ℓ log k)) and augmentation 1 + ϵ; and an online algorithm with competitive ratio O(√k) and augmentation 2 + ϵ.

Abstract

Resource allocation in distributed and networked systems such as the Cloud is becoming increasingly flexible, allowing these systems to dynamically adjust toward the workloads they serve, in a demand-aware manner. Online balanced partitioning is a fundamental optimization problem underlying such self-adjusting systems. We are given a set of $\ell$ servers. On each server we can schedule up to $k$ processes simultaneously. The demand is described as a sequence of requests $σ_t=\{p_i, p_{j}\}$, which means that the two processes $p_i,p_{j}$ communicate. A process can be migrated from one server to another which costs 1 unit per process move. If the communicating processes are on different servers, it further incurs a communication cost of 1 unit for this request. The objective is to minimize the competitive ratio: the cost of serving such a request sequence compared to the cost incurred by an optimal offline algorithm. Henzinger et al. (at SIGMETRICS 2019) introduced a learning variant of this problem where the cost of an online algorithm is compared to the cost of a static offline algorithm that does not perform any communication, but which simply learns the communication graph and keeps the discovered connected components together. This problem variant was recently also studied at SODA 2021. In this paper, we consider a more general learning model (i.e., stronger adversary), where the offline algorithm is not restricted to keep connected components together. Our main contribution are tight bounds for this problem. In particular, we present two deterministic online algorithms: (1) an online algorithm with competitive ratio $O(\max(\sqrt{k\ell \log k}, \ell \log k))$ and augmentation $1+ε$; (2) an online algorithm with competitive ratio $O(\sqrt{k})$ and augmentation $2+ε$. We further present lower bounds showing optimality of these bounds.

Tight Bounds for Online Balanced Partitioning in the Generalized Learning Model

TL;DR

Two deterministic online algorithms are presented: an online algorithm with competitive ratio O(max(√kℓ log k, ℓ log k)) and augmentation 1 + ϵ; and an online algorithm with competitive ratio O(√k) and augmentation 2 + ϵ.

Abstract

Resource allocation in distributed and networked systems such as the Cloud is becoming increasingly flexible, allowing these systems to dynamically adjust toward the workloads they serve, in a demand-aware manner. Online balanced partitioning is a fundamental optimization problem underlying such self-adjusting systems. We are given a set of servers. On each server we can schedule up to processes simultaneously. The demand is described as a sequence of requests , which means that the two processes communicate. A process can be migrated from one server to another which costs 1 unit per process move. If the communicating processes are on different servers, it further incurs a communication cost of 1 unit for this request. The objective is to minimize the competitive ratio: the cost of serving such a request sequence compared to the cost incurred by an optimal offline algorithm. Henzinger et al. (at SIGMETRICS 2019) introduced a learning variant of this problem where the cost of an online algorithm is compared to the cost of a static offline algorithm that does not perform any communication, but which simply learns the communication graph and keeps the discovered connected components together. This problem variant was recently also studied at SODA 2021. In this paper, we consider a more general learning model (i.e., stronger adversary), where the offline algorithm is not restricted to keep connected components together. Our main contribution are tight bounds for this problem. In particular, we present two deterministic online algorithms: (1) an online algorithm with competitive ratio and augmentation ; (2) an online algorithm with competitive ratio and augmentation . We further present lower bounds showing optimality of these bounds.

Paper Structure

This paper contains 40 sections, 40 theorems, 25 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions and Novelty
  3. Further Related Work
  4. Future Work
  5. Model and Preliminaries
  6. Lower Bounds
  7. Lower Bounds for Algorithms with Unlimited Augmentation
  8. Lower Bounds for Algorithms with Augmentation $1+\epsilon$
  9. Notation
  10. Algorithmic Concepts
  11. Time-Expanded Graph
  12. Linear Program Formulation
  13. Clustering
  14. Flow Clustering Procedure
  15. Request Processing
  16. ...and 25 more sections

Key Result

Theorem 1.1

For any deterministic online algorithm $\operatorname{ALG}$ with unlimited resource augmentation, there exists a request sequence $\sigma$, such that $\operatorname{ALG}(\sigma)\ge\Omega(\sqrt{k})\cdot\operatorname{OPT_S}(\sigma)$, where $\operatorname{OPT_S}$ is the optimum static offline algorithm

Figures (2)

  • Figure 1: An illustration of the time expanded graph $\bar{G}_\sigma$ for $\ell=3$ servers, server capacity $k=4$, and $|\sigma|=4$ requests. The bottom row are the server nodes, the remaining nodes are process time nodes. The request sequence is $\sigma = (\{p_2, p_3\}, \{p_4, p_6\}, \{p_6, p_9\}, \{p_1,p_9\})$. The blue/bend/thick/horizontal edges are communication edges. All other edges are migration edges.
  • Figure 2: Linear program relaxation of the generalized learning problem

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.1
  • Lemma 3.1
  • ...and 64 more