Balanced Allocation on Hypergraphs

TL;DR

This paper extends the balls-into-bins model by using hypergraphs and a d-choice algorithm, showing that under certain conditions, the maximum load remains very low after allocating a linear number of balls.

Contribution

It introduces a hypergraph-based allocation model with balancedness and pair visibility conditions, proving low maximum load with the power of multiple choices.

Findings

Maximum load is at most log_d log n + O(1) with high probability.

The balancedness and pair visibility conditions are crucial for achieving low maximum load.

Lower bounds relate maximum load to pair visibility, emphasizing its importance.

Abstract

We consider a variation of balls-into-bins which randomly allocates $m$ balls into $n$ bins. Following Godfrey's model (SODA, 2008), we assume that each ball $t$, $1\le t\le m$, comes with a hypergraph $\mathcal{H}^{(t)}=\{B_1,B_2,\ldots,B_{s_t}\}$, and each edge $B\in\mathcal{H}^{(t)}$ contains at least a logarithmic number of bins. Given $d\ge 2$, our $d$-choice algorithm chooses an edge $B\in \mathcal{H}^{(t)}$, uniformly at random, and then chooses a set $D$ of $d$ random bins from the selected edge $B$. The ball is allocated to a least-loaded bin from $D$, with ties are broken randomly. We prove that if the hypergraphs $\mathcal{H}^{(1)},\ldots, \mathcal{H}^{(m)}$ satisfy a \emph{balancedness} condition and have low \emph{pair visibility}, then after allocating $m=\Theta(n)$ balls, the maximum number of balls at any bin, called the \emph{maximum load}, is at most $\log_d\log n+O(1)$, with high probability. The balancedness condition enforces that bins appear almost uniformly within the hyperedges of $\mathcal{H}^{(t)}$, $1\le t\le m$, while the pair visibility condition measures how frequently a pair of bins is chosen during the allocation of balls. Moreover, we establish a lower bound for the maximum load attained by the balanced allocation for a sequence of hypergraphs in terms of pair visibility, showing the relevance of the visibility parameter to the maximum load. In Godfrey's model, each ball is forced to probe all bins in a randomly selected hyperedge and the ball is then allocated in a least-loaded bin. Godfrey showed that if each $\mathcal{H}^{(t)}$, $1\le t\le m$, is balanced and $m=O(n)$, then the maximum load is at most one, with high probability. However, we apply the power of $d$ choices paradigm, and only query the load information of $d$ random bins per ball, while achieving very slow growth in the maximum load.