Perfectly Balanced Allocation With Estimated Average Using Expected Constant Retries

Abstract

Balanced allocation of online balls-into-bins has long been an active area of research for efficient load balancing and hashing applications.There exists a large number of results in this domain for different settings, such as parallel allocations~\cite{parallel}, multi-dimensional allocations~\cite{multi}, weighted balls~\cite{weight} etc. For sequential multi-choice allocation, where $m$ balls are thrown into $n$ bins with each ball choosing $d$ (constant) bins independently uniformly at random, the maximum load of a bin is $O(\log \log n) + m/n$ with high probability~\cite{heavily_load}. This offers the current best known allocation scheme. However, for $d = \Theta(\log n)$, the gap reduces to $O(1)$~\cite{soda08}.A similar constant gap bound has been established for parallel allocations with $O(\log ^*n)$ communication rounds~\cite{lenzen}. In this paper we propose a novel multi-choice allocation algorithm, \emph{Improved D-choice with Estimated Average} ($IDEA$) achieving a constant gap with a high probability for the sequential single-dimensional online allocation problem with constant $d$. We achieve a maximum load of $\lceil m/n \rceil$ with high probability for constant $d$ choice scheme with \emph{expected} constant number of retries or rounds per ball. We also show that the bound holds even for an arbitrary large number of balls, $m>>n$. Further, we generalize this result to (i)~the weighted case, where balls have weights drawn from an arbitrary weight distribution with finite variance, (ii)~multi-dimensional setting, where balls have $D$ dimensions with $f$ randomly and uniformly chosen filled dimension for $m=n$, and (iii)~the parallel case, where $n$ balls arrive and are placed parallely in the bins. We show that the gap in these case is also a constant w.h.p. (independent of $m$) for constant value of $d$ with expected constant number of retries per ball.