Choice-memory tradeoff in allocations

Abstract

In the classical balls-and-bins paradigm, where n balls are placed independently and uniformly in n bins, typically the number of bins with at least two balls in them is Θ(n) and the maximum number of balls in a bin is Θ(log n / log log n). It is well known that when each round offers k independent uniform options for bins, it is possible to typically achieve a constant maximal load if and only if k = Ω(log n). Moreover, it is possible w.h.p. to avoid any collisions between n / 2 balls if k > log₂ n.

In this work, we extend this into the setting where only m bits of memory are available. We establish a tradeoff between the number of choices k and the memory m, dictated by the quantity km / n. Roughly put, we show that for km ≫ n one can achieve a constant maximal load, while for km ≪ n no substantial improvement can be gained over the case k = 1 (i.e., a random allocation).

For any k = Ω(log n) and m = Ω(log² n), one can achieve a constant load w.h.p. if km = Ω(n), yet the load is unbounded if km = o(n). Similarly, if km > Cn then n / 2 balls can be allocated without any collisions w.h.p., whereas for km < ɛn there are typically Ω(n) collisions. Furthermore, we show that the load is w.h.p. at least log(n/m) / (log k+log log(n / m)). In particular, for k ≤ polylog (n), if m = n^1−δ the optimal maximal load is Θ(log n / log log n) (the same as in the case k = 1), while m = 2n suffices to ensure a constant load. Finally, we analyze nonadaptive allocation algorithms and give tight upper and lower bounds for their performance.