The Power of Two Choices with Simple Tabulation

TL;DR

This paper proves that simple tabulation hashing, which is efficient and constant-time, achieves high-probability load balancing guarantees similar to fully random hash functions in the two choices paradigm, with some limitations.

Contribution

It establishes that simple tabulation hashing provides high-probability bounds on maximum load in the two choices paradigm, matching fully random hashing in expectation.

Findings

Maximum load is O(log log n) with high probability using simple tabulation.

There exists a case where simple tabulation does not bound maximum load by log log n.

Expected maximum load remains lg log n + O(1) with simple tabulation.

Abstract

The power of two choices is a classic paradigm for load balancing when assigning $m$ balls to $n$ bins. When placing a ball, we pick two bins according to two hash functions $h_0$ and $h_1$, and place the ball in the least loaded bin. Assuming fully random hash functions, when $m=O(n)$, Azar et al.~[STOC'94] proved that the maximum load is $\lg \lg n + O(1)$ with high probability. In this paper, we investigate the power of two choices when the hash functions $h_0$ and $h_1$ are implemented with simple tabulation, which is a very efficient hash function evaluated in constant time. Following their analysis of Cuckoo hashing [J.ACM'12], P\v{a}tra\c{s}cu and Thorup claimed that the expected maximum load with simple tabulation is $O(\lg\lg n)$. This did not include any high probability guarantee, so the load balancing was not yet to be trusted. Here, we show that with simple tabulation, the maximum load is $O(\lg\lg n)$ with high probability, giving the first constant time hash function with this guarantee. We also give a concrete example where, unlike with fully random hashing, the maximum load is not bounded by $\lg \lg n + O(1)$, or even $(1+o(1))\lg \lg n$ with high probability. Finally, we show that the expected maximum load is $\lg \lg n + O(1)$, just like with fully random hashing.