Presentation is loading. Please wait.

Presentation is loading. Please wait.

Optimal Fast Hashing Yossi Kanizo (Technion, Israel) Joint work with Isaac Keslassy (Technion, Israel) and David Hay (Hebrew Univ., Israel)

Similar presentations


Presentation on theme: "Optimal Fast Hashing Yossi Kanizo (Technion, Israel) Joint work with Isaac Keslassy (Technion, Israel) and David Hay (Hebrew Univ., Israel)"— Presentation transcript:

1 Optimal Fast Hashing Yossi Kanizo (Technion, Israel) Joint work with Isaac Keslassy (Technion, Israel) and David Hay (Hebrew Univ., Israel)

2 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

3 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

4 Hash Tables for Networking Devices 123  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 Memory

5 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

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

7 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 Memory

8 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 Memory h CAM 9

9 Possible Settings  Static setting - Insertions and queries only  Dynamic setting – Insertions, deletions, and queries.  Generalized setting – Balancing between the buckets’ load.

10 Problem Formulation Memory h CAM 9 Given average a and worst-case d of memory accesses per operation, Minimize overflow rate  Given average a and worst-case d of memory accesses per operation, Minimize overflow rate 

11 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 Memory h CAM

12 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.

13 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.

14 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)

15 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

16 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

17 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]

18 The SIMPLE Scheme  SIMPLE scheme: single hash function  Looks like truncated linked list Memory h CAM

19 Performance of SIMPLE Scheme [h=4, load=0.95, m=5,000] The lower bound can actually be achieved for a=1

20 The GREEDY Scheme  Using uniform hashes, try to insert each element greedily until either inserted or d Memory h CAM d=2

21 Performance of GREEDY Scheme [d=4, h=4, load=0.95, m=5,000] The GREEDY scheme is always optimal until a co

22 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)

23 The MHT Scheme  MHT (Multi-Level Hash Table) [Broder&Karlin]: d successive subtables with their d hash functions Memory h CAM st Subtable2 nd Subtable3 rd Subtable

24 Performance of MHT Scheme  Optimality of MHT until cut-off point a co (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)

25 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.

26 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

27 Degradation with Finite Buckets  Finite buckets are used.  Surprising result: degradation in performance FiniteInfinite H(1) = 3H(2) = 3 Remove 1 Element “2” is lost although its corresponding bucket is empty

28 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 .

29 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)

30 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.

31 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.

32 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 n  a distinct elements.  To take into account d, we must consider each element along with its own hash values.

33 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

34 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).

35 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.

36 Thank you.


Download ppt "Optimal Fast Hashing Yossi Kanizo (Technion, Israel) Joint work with Isaac Keslassy (Technion, Israel) and David Hay (Hebrew Univ., Israel)"

Similar presentations


Ads by Google