Optimal Fast Hashing

Yossi Kanizo (Technion, Israel)

Joint work with Isaac Keslassy (Technion, Israel) and David Hay (Hebrew Univ., Israel)

Hash Tables for Networking Devices

Hash tables and hash-based structures are often used in high-speed devices Heavy-hitter flow identification Flow state keeping Flow counter management Virus signature scanning IP address lookup algorithms

Hash tables

In theory, hash tables are particularly suitable: O(1) memory accesses per operation (element insertion/query/deletion) for reasonable load

But in practice, there is a big difference between an average of 1.1 memory accesses per operation, and an average of 4

Why not only 1 memory access? Collisions

Hash Tables for Networking Devices Collisions are unavoidable wasted memory

accesses For load≤1, let a and d be the average and worst-

case time (number of memory accesses) per element insertion

Objective: Minimize a and d

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Memory

Why We Care

On-chip memory: memory accesses power consumption

Off-chip memory: memory accesses lost on/off-chip pin capacity

Datacenters: memory accesses network & server load

Parallelism does not help reduce these costs d serial or parallel memory accesses have same cost

Traditional Hash Table Schemes Example 1: linked lists (chaining)

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Traditional Hash Table Schemes Example 1: linked lists (chaining) Example 2: linear probing (open addressing)

Problem: the worst-case time cannot be bounded by a constant d

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High-Speed Hardware Enable overflows: if time exceeds d → overflow list

Can be stored in expensive CAM Otherwise, overflow elements = lost elements

Bucket contains h elements E.g.: 128-bit memory word h=4 elements of 32 bits Assumption: Access cost (read & write word) = 1 cycle

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Possible Settings

Static setting - Insertions and queries only

Dynamic setting – Insertions, deletions, and queries.

Generalized setting – Balancing between the buckets’ load.

Problem Formulation

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Given average a and worst-case d of memory accesses per operation,

Minimize overflow rate

Example: Power of d-Random Choices

d hash functions: pick least loaded bucket. Break ties u.a.r. [Azar et al.]

Intuition: can reach low … but average time a = worst-case time d wasted memory accesses

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Other Examples

d-left [Vöcking] Same as d-random, but break ties to the left.

Cuckoo [Pagh et al.] Whenever collision occurs, moves stored

elements to their other choices. Typically, uses much more than d memory

accesses on average.

Outline

Static Case Overflow Lower Bound Optimal Schemes: SIMPLE, GREEDY, MHT.

Dynamic Case Comparison with Static Case. Overflow Lower Bound

Overflow Fraction Depending on d.

Overflow Lower Bound

Objective: given any online scheme with average a and worst-case d, find lower-bound on overflow .

[h=4, load=n/(mh)=0.95, fixed d]

No scheme can achieve (capacity region)

Overflow Lower Bound

Result: closed-form lower-bound formula Given n elements in m buckets of height h:

Valid also for non-uniform hashes For n=m and h=1, we get simply

Defines a capacity region for high-throughput hashing

Lower-Bound Example

[h=4, load=n/(mh)=0.95]

For 3% overflow rate, throughput can be at most

1/a = 2/3 of memory rate

Overflow Lower Bound

Example: d-left scheme: low overflow , but high average memory access rate a

[h=4, load=n/(mh)=0.95, m=5,000]

The SIMPLE Scheme

SIMPLE scheme: single hash function Looks like truncated linked list

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Performance of SIMPLE Scheme

[h=4, load=0.95, m=5,000]

The lower bound can actually be achieved

for a=1

The GREEDY Scheme

Using uniform hashes, try to insert each element greedily until either inserted or d

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d=2

Performance of GREEDY Scheme

[d=4, h=4, load=0.95, m=5,000]

The GREEDY scheme is always optimal until aco

Performance of GREEDY Scheme

[d=4, h=4, load=0.95, m=5,000]

Overflow rate worse than 4-left, but better throughput (1/a)

The MHT Scheme

MHT (Multi-Level Hash Table) [Broder&Karlin]: d successive subtables with their d hash functions

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1st Subtable 2nd Subtable 3rd Subtable

Performance of MHT Scheme Optimality of MHT until cut-off point aco(MHT)

Proof that subtable sizes fall geometrically Confirmed in simulations

[d=4, h=4, load=0.95, m=5,000]

Overflow rate close to 4-left, with much better throughput (1/a)

Outline

Static Case Overflow Lower Bound Optimal Schemes: SIMPLE, GREEDY, MHT.

Dynamic Case Comparison with Static Case. Overflow Lower Bound

Overflow Fraction Depending on d.

Dynamic vs. Static

Dynamic hash tables are harder to model than the static ones [Kirsch et al.]

But past studies show same asymptotic behavior with infinite buckets (insertions only vs. alternations) traditional hashing using linked lists – maximum bucket

size of approx. log n / log log n [Gonnet, 1981] d-random, d-left schemes – maximum bucket size of log

log n / log 2 + O(1) [Azar et al.,1994; Vöcking, 1999] As a designer, using the static model seems natural.

Even if real-life devices have finite buckets

Degradation with Finite Buckets

Finite buckets are used. Surprising result: degradation in performance

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H(1) = 3H(2) = 3

Element “2” is lost although its corresponding bucket is empty

Comparing Static and Dynamic

Static setting: insertions only n = number of elements m = number of buckets

Dynamic setting: alternations between element insertions and deletions of randomly chosen elements. fixed load of c = n / (mh)

Fair comparison Given an average number of memory accesses a,

minimize overflow fraction .

Overflow Lower Bound

Overflow lower bound of

where r = ach. Also holds for non-uniformly distributed

hash functions (under some constraints). The lower bound is tight (Simple, Greedy)

Numerical Example

For h=1 and c=1 (100% load) we get a lower bound of 1/(1+a).

To get an overflow fraction of 1%, one needs at least 99 memory accesses per element. Infeasible for high-speed networking devices

Compared to a tight upper bound of e-a in the static case. [Kanizo et al., INFOCOM 2009] need ~4.6 memory accesses.

Outline

Static Case Overflow Lower Bound Optimal Schemes: SIMPLE, GREEDY, MHT.

Dynamic Case Comparison with Static Case. Overflow Lower Bound

Overflow Fraction Depending on d.

Overflow Fraction Depending on d

So far, we relaxed the constraint on d. We considered n elements with an average of

a memory accesses, as na distinct elements.

To take into account d, we must consider each element along with its own hash values.

Graph Theory Approach

Consider a bipartite graph. Left vertices = Elements Right vertices = Buckets

(assume h=1). Edge = The bucket is one

of the element’s d choices

Graph Theory Approach

We get a random bipartite graph where each left vertex has degree d.

Expected maximum size matching = Expected number of elements that can be inserted to the table, that is, a lower bound.

We derived an explicit expression for d=2.

Upper bound can be achieved by Cuckoo hashing (equivalent to finding maximum size matching).

Summary

We found lower and upper bounds on the achievable overflow fraction both for the static and dynamic cases.

Static models are not necessarily exact with dynamic hash tables.

Improved lower bound for d=2 and a characterization of the performance of Cuckoo hashing.

Thank you.