Choice-memory tradeoff in allocations

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 whp 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 krn/n. Roughly put, we show that for km >> n one can achieve a constant maximal load, while for km << nno 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 typically achieve a constant load 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 whp, whereas for km < en there are typically Ω(n) collisions. Furthermore, we show that the load is whp at least log(n/m)/log k+log log(n/m) In particular, whenever 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 non-adaptive allocation algorithms and give tight upper and lower bounds for their performance.