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Delayed Assignments in Online Non-Centroid Clustering with Stochastic Arrivals

Saar Cohen

TL;DR

This work addresses online non-centroid clustering with delays under unknown i.i.d. arrivals in a finite metric space. It introduces the OCD problem and a ratio-of-expectations performance measure, proposing the Delayed Greedy (DGreedy) algorithm with a geometric ball-intersection interpretation to balance throughput and waiting costs. The authors prove a tight constant RoE bound for DGreedy by deriving upper bounds on its cost and a matching lower bound on the offline optimum, and they extend the framework to non-metric spaces and fixed-size cluster settings, including hedonic games. The results offer a practical, theoretically solid approach to batching in streaming clustering and coalition formation, with potential applications in gaming, ride-sharing, and distributed learning, where delayed decisions can improve batching without sacrificing long-run efficiency.

Abstract

Clustering is a fundamental problem, aiming to partition a set of elements, like agents or data points, into clusters such that elements in the same cluster are closer to each other than to those in other clusters. In this paper, we present a new framework for studying online non-centroid clustering with delays, where elements, that arrive one at a time as points in a finite metric space, should be assigned to clusters, but assignments need not be immediate. Specifically, upon arrival, each point's location is revealed, and an online algorithm has to irrevocably assign it to an existing cluster or create a new one containing, at this moment, only this point. However, we allow decisions to be postponed at a delay cost, instead of following the more common assumption of immediate decisions upon arrival. This poses a critical challenge: the goal is to minimize both the total distance costs between points in each cluster and the overall delay costs incurred by postponing assignments. In the classic worst-case arrival model, where points arrive in an arbitrary order, no algorithm has a competitive ratio better than sublogarithmic in the number of points. To overcome this strong impossibility, we focus on a stochastic arrival model, where points' locations are drawn independently across time from an unknown and fixed probability distribution over the finite metric space. We offer hope for beyond worst-case adversaries: we devise an algorithm that is constant competitive in the sense that, as the number of points grows, the ratio between the expected overall costs of the output clustering and an optimal offline clustering is bounded by a constant.

Delayed Assignments in Online Non-Centroid Clustering with Stochastic Arrivals

TL;DR

This work addresses online non-centroid clustering with delays under unknown i.i.d. arrivals in a finite metric space. It introduces the OCD problem and a ratio-of-expectations performance measure, proposing the Delayed Greedy (DGreedy) algorithm with a geometric ball-intersection interpretation to balance throughput and waiting costs. The authors prove a tight constant RoE bound for DGreedy by deriving upper bounds on its cost and a matching lower bound on the offline optimum, and they extend the framework to non-metric spaces and fixed-size cluster settings, including hedonic games. The results offer a practical, theoretically solid approach to batching in streaming clustering and coalition formation, with potential applications in gaming, ride-sharing, and distributed learning, where delayed decisions can improve batching without sacrificing long-run efficiency.

Abstract

Clustering is a fundamental problem, aiming to partition a set of elements, like agents or data points, into clusters such that elements in the same cluster are closer to each other than to those in other clusters. In this paper, we present a new framework for studying online non-centroid clustering with delays, where elements, that arrive one at a time as points in a finite metric space, should be assigned to clusters, but assignments need not be immediate. Specifically, upon arrival, each point's location is revealed, and an online algorithm has to irrevocably assign it to an existing cluster or create a new one containing, at this moment, only this point. However, we allow decisions to be postponed at a delay cost, instead of following the more common assumption of immediate decisions upon arrival. This poses a critical challenge: the goal is to minimize both the total distance costs between points in each cluster and the overall delay costs incurred by postponing assignments. In the classic worst-case arrival model, where points arrive in an arbitrary order, no algorithm has a competitive ratio better than sublogarithmic in the number of points. To overcome this strong impossibility, we focus on a stochastic arrival model, where points' locations are drawn independently across time from an unknown and fixed probability distribution over the finite metric space. We offer hope for beyond worst-case adversaries: we devise an algorithm that is constant competitive in the sense that, as the number of points grows, the ratio between the expected overall costs of the output clustering and an optimal offline clustering is bounded by a constant.
Paper Structure (14 sections, 10 theorems, 32 equations, 1 figure, 1 algorithm)

This paper contains 14 sections, 10 theorems, 32 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Preliminaries
  5. A Constant Competitive Algorithm
  6. Geometric Interpretation.
  7. Upper Bounding DGreedy's total cost
  8. Lower Bounding the Optimal total cost
  9. DGreedy's Constant Ratio-of-Expectations
  10. Extensions
  11. Non-Metric Spaces and Hedonic Games
  12. Online Additively Separable Hedonic Games with Agent Types
  13. Lower and Upper Bounds on Cluster Sizes
  14. Conclusions and Future Work

Key Result

Lemma 1

Given a finite metric space $\mathcal{M}=(\mathcal{X},d)$, for any sequence of $n$ points $\sigma$, the total cost of the final clustering $\mathcal{C}$ generated by DGreedy satisfies:

Figures (1)

  • Figure 1: An example of how DGreedy works for clusterings with a single cluster of size $3$ on a sequence of three points $1, 2, 3$ arriving at times $1, 3, 4$ in a finite metric space consisting of $3$ locations, where the distance between each pair of locations is $2$.

Theorems & Definitions (23)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Lemma 4
  • proof
  • ...and 13 more