Simple Tabulation, Fast Expanders, Double Tabulation, and High Independence

Abstract

Simple tabulation dates back to Zobrist in 1970 who used it for game playing programs. Keys are viewed as consisting of c characters from some alphabet Φ. We initialize c tables h 0 , . . . , h c−1 mapping characters to random hash values. A key x = (x 0 , . . . ,, where ⊕ denotes bit-wise exclusive-or. The scheme is extremely fast when the character hash tables h i are in cache. Simple tabulation hashing is not even 4-independent, but we show here that if we apply it twice, then we do get high independence. First we hash to some intermediate keys that are 6 times longer than the original keys, and then we hash the intermediate keys to the final hash values.The intermediate keys have d = 6c characters from Φ. We can then view the hash function as a highly unbalanced bipartite graph with keys on one side, each with edges to d output characters on the other side. We show that this graph has nice expansion properties, and from that it follows that if we perform another level of simple tabulation on the intermediate keys, then the composition is a highly independent hash function. More precisely, the independence we get is |Φ| Ω(1/c) . In our O-notation, we view both |Φ| and c is going to infinity, but with c much smaller than |Φ|.Our space is O(c|Φ|) and the hash function is evaluated in O(c) time. Siegel [FOCS'89, SICOMP'04] has proved that with this space, if the hash function is evaluated in o(c) time, then the independence can only be o(c), so our evaluation time is best possible for Ω(c) independence-our independence is much higher if c = |Φ| o(1/c) . Siegel used O(c)c evaluation time to get the same independence with similar space. Siegel's main focus was c = O(1), but we are exponentially faster when c = ω(1).Applying our scheme recursively, we can increase our independence to |Φ| Ω(1) with o(c log c ) evaluation time. Compared with Siegel's scheme this is both faster and higher independence.Siegel states about his scheme that it is "far too slow for any practical application". Our scheme is trivial to implement, and it does provide realistic implementations of 100-independent hashing for, say, 32-bit and 64-bit keys.

Non-empty Bins with Simple Tabulation Hashing

“…While this space and time matches that of simple tabulation within constant factors, it is slower by at least an order of magnitude. As mentioned in [16], double tabulation with 32-bit keys divided into 16-bit characters requires 11 times as many character table lookups as with simple tabulation and we lose the same factor in space. The larger space of double tabulation means that tables may expand into much slower memory, possibly costing us another order of magnitude in speed.…”

“…As in [16] we define a position character to be an element We start by describing an ordering of the position characters, introduced by Pǎtraşcu and Thorup [14] in order to prove that the number of balls hashing to a specific bin is Chernoff concentrated when using simple tabulation. If X ⊆ U is a set of keys and ≺ is any ordering of the position characters…”

Fast and powerful hashing using tabulation

“…More precisely, we consider having two independent simple tabulation functions h 0 : Σ c → Σ d and h 1 : Σ d → [2 r ], and then the claim is that h 1 • h 0 is likely to be highly independent. The main point from [50] is that the first simple tabulation h 0 is likely to have an expander-type property.…”

“…In this paper we will survey recent results from [12,13,14,15,41,42,50] showing how simple realistic hashing schemes based on tabulation provide unexpectedly strong guarantees for many popular randomized algorithms, e.g., linear probing, Cuckoo hashing, min-wise independence, treaps, planar partitions, powerof-two-choices, Chernoff-style concentration bounds, and even high independence. The survey is from a users perspective, explaining how these tabulation schemes can be applied.…”

From Independence to Expansion and Back Again

“…Cell probe upper bound [22] k(u/k) 1/t t O(t) probes space usage of [22,24] have likely not been optimized, their techniques do not seem to be able to yield space close to the cell probe lower bound.…”