Balls into Bins: strict Capacities and Edge Weights

TL;DR

This paper introduces a new theoretical model for distributed storage management that considers consumer capacities and edge weights, analyzing optimal offline algorithms and competitive online strategies with provable performance bounds.

Contribution

It extends the classic Balls-into-bins model to include capacities and weights, providing algorithms and competitive analysis for both simplified and general cases.

Findings

LP and Primal-Dual algorithms compute optimal offline solutions in O(r*n) time.

A simple randomized online algorithm achieves a competitive ratio of AVG(d)/MIN(d) in the equal capacity case.

The model's extension to arbitrary capacities yields an average-case competitive ratio of AVG(d*c)/(AVG(c)*MIN(d)).

Abstract

We explore a novel theoretical model for studying the performance of distributed storage management systems where the data-centers have limited capacities (as compared to storage space requested by the users). Prior schemes such as Balls-into-bins (used for load balancing) neither consider bin (consumer) capacities (multiple balls into a bin) nor the future performance of the system after, balls (producer requests) are allocated to bins and restrict number of balls as a function of the number of bins. Our problem consists of finding an optimal assignment of the online producer requests to consumers (via weighted edges) in a complete bipartite graph while ensuring that the total size of request assigned on a consumer is limited by its capacity. The metric used to measure the performance in this model is the (minimization of) weighted sum of the requests assigned on the edges (loads) and their corresponding weights. We first explore the optimal offline algorithms followed by competitive analysis of different online techniques. Using oblivious adversary. LP and Primal-Dual algorithms are used for calculating the optimal offline solution in O(r*n) time (where r and n are the number of requests and consumers respectively) while randomized algorithms are used for the online case. For the simplified model with equal consumer capacities an average-case competitive ratio of AVG(d) / MIN(d) (where d is the edge weight / distance) is achieved using an algorithm that has equal probability for selecting any of the available edges with a running time of $O(r)$. In the extending the model to arbitrary consumer capacities we show an average case competitive ratio of AVG(d*c) / (AVG(c) *MIN(d)).