Balls into Bins: strict Capacities and Edge Weights

Ankur Sahai

Home/Archive/arXiv

arXiv · 1206.3634

Balls into Bins: strict Capacities and Edge Weights

Ankur Sahai

View original source ↗Read source document ↗Export metadata ↓

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)).

Explore related subjects

cs.DS cs.DC

Keep this discovery

BibTeX RIS

Ankur Sahai. 2012-06-16. Balls into Bins: strict Capacities and Edge Weights. https://arxiv.org/abs/1206.3634

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗