Summary: CHOICE-MEMORY TRADEOFF IN ALLOCATIONS
NOGA ALON, ORI GUREL-GUREVICH, AND EYAL LUBETZKY
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 (n) and the maximum number
of balls in a bin is ( 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 = (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 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 = (log n) and m = (log2
n), one can achieve a constant
load whp if km = (n), yet the load is unbounded if km = o(n).