Balanced Allocations: Caching and Packing, Twinning and Thinning

Abstract

We consider the sequential allocation of m balls (jobs) into n bins (servers) by allowing each ball to choose from some bins sampled uniformly at random. The goal is to maintain a small gap between the maximum load and the average load.In this paper, we present a general framework that allows us to analyze various allocation processes that slightly prefer allocating into underloaded, as opposed to overloaded bins. Our analysis covers several natural instances of processes, including:• The Caching process (a.k.a. memory protocol) as studied by Mitzenmacher, Prabhakar and Shah (2002).• The Packing process: At each round we only take one bin sample. If the load is below some threshold (e.g., the average load), then we place as many balls until the threshold is reached; otherwise, we place only one ball.• The Twinning process: At each round, we only take one bin sample. If the load is below some threshold, then we place two balls; otherwise, we place only one ball.• The Thinning process as recently studied by Feldheim and Gurel-Gurevich (2021).As we demonstrate, using an interplay between several potential functions our general framework implies for all these processes a gap of O(log n) for any number of balls m ≥ n.

Tight Bounds for Repeated Balls-into-Bins

“…Next we prove an important relation between the quadratic potential Υ t and the number of empty bins F t . These relations are similar to the ones used in [21] to show that the absolute value potential is small in a constant fractions of the steps. The key insight is that the quadratic potentials drops in expectation as soon as the fraction of empty bins is of order Ω(n/m).…”

Balanced Allocations with the Choice of Noise

“…Lemma A.11 (cf. Lemma A.2 in [35]). Let (a k ) n k=1 , (b k ) n k=1 be non-negative and (c k ) n k=1 be non-negative and non-increasing.…”

“…In contrast to that, the two-Thinning process allocates in a two-stage procedure: Firstly, sample a random bin i. Secondly, based on the load of bin i (and additional information based on the history of the process), we can either place the ball into i, or place the ball into another randomly chosen bin j (without comparing its load with i). This process has received a lot of attention lately, and several variations were studied in [25] for m = n and [24,33,35] for m n. Finally, Czumaj and Stemann [20] investigated so-called adaptive allocation schemes for m = n. In contrast to Thinning, after having taken a certain number of bin samples, the ball is allocated into the least loaded bin among all samples. In another related model recently studied by the authors of this work, the load of a sampled bin can only be approximated through binary queries of the form "Is your load at least g?"…”

“…Roughly, this precondition says that a linear potential function is small (even though the exponential potential function may be very large). Extending [35], we establish this precondition by studying the interplay between three potential functions: the linear, quadratic and exponential potential functions.…”

Balanced Allocations in Batches: Simplified and Generalized

“…Related to (1 + β) process is the two-Thinning process [15,18,13] with some load threshold which works in a two-stage procedure: First, take a uniform bin sample i. Secondly, if the load of bin i is at most then allocate a ball into i, otherwise place a ball into another bin sample j (without comparing its load with i). This process has received some attention recently, and several variations were studied in [15] for m = n and [14,24,25] for m n. [24] investigated a variant of Thinning called Quantile, which uses relative instead of absolute loads. This means the ball is allocated in the first sample if its load is among the (1 − δ) • n lightest, for some quantile δ ∈ {1/n, 2/n, .…”