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- W4287758060 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 $t$, $1le tle m$, comes with a hypergraph $mathcal{H}^{(t)}={B_1,B_2,ldots,B_{s_t}}$, and each edge $Binmathcal{H}^{(t)}$ contains at least a logarithmic number of bins. Given $dge 2$, our $d$-choice algorithm chooses an edge $Bin mathcal{H}^{(t)}$, 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 $log_dlog n+O(1)$, with high probability. The balancedness condition enforces that bins appear almost uniformly within the hyperedges of $mathcal{H}^{(t)}$, $1le tle m$, 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 $mathcal{H}^{(t)}$, $1le tle m$, 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." @default.
- W4287758060 created "2022-07-26" @default.
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- W4287758060 date "2020-06-13" @default.
- W4287758060 modified "2023-09-30" @default.
- W4287758060 title "Balanced Allocation on Hypergraphs" @default.
- W4287758060 doi "https://doi.org/10.48550/arxiv.2006.07588" @default.
- W4287758060 hasPublicationYear "2020" @default.
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