1. Hash Tables With Finite Buckets Are Less Resistant to Deletions Yossi Kanizo (Technion, Israel) Joint work with David Hay (Columbia U. and Hebrew U.) and Isaac Keslassy ( Technion )

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  In many applications elements are also deleted (a.k.a. dynamic hash tables)

3. Dynamic vs. Static  Dynamic hash tables are harder to model than the static ones, that is, insertions only [Kirsch et al.]  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]  Using the static model seems natural.

4. High-Speed Hardware  Bucket is a memory word that contains h elements  E.g.: 128-bit memory word  h=4 elements of 32 bits  Assumption: Access cost (read & write word) = 1 cycle  Enable overflows: after d memory accesses → overflow list  Can be stored in expensive CAM  Otherwise, overflow elements = lost elements  Overflow fraction =  123456789 Memory 4 7 15 3 6 28 h CAM 9

5. Degradation with Finite Buckets  Finite buckets are used.  Degradation in performance 1234 12 FiniteInfinite 1 1 2 1234 H(1) = 3H(2) = 3 Remove 1 Element “2” is not stored although its corresponding bucket is empty

6. Degradation with Finite Buckets  What we had is  Insert element “1”  Insert element “2”  Remove element “1”  Equivalent to only inserting element “2” in the static case 1234 FiniteInfinite 2 1234 2

7. Simulations [h=1, load=n/(mh)=1, d = 2]

8. 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 .

9. Why We Care about Average Number of Memory Accesses?  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

10. From Discrete to Fluid Model  Discrete model  Models the system accurately but induces complex interactions between the elements  Approximation using a fluid model  Based on differential equations with an infinite number of elements and buckets.  Elements stay in the system for exponentially- distributed duration of average 1. Bucket departure rate is proportional to its occupancy.  Upon departure, a new element arrives. arrival rate is constant (fixed load in the system). Assuming uniformly distributed hash functions, bucket arrival rate is n / m = ch

11. Main Results  Case Study: Single choice hashing scheme  Lower bound on overflow fraction  Mitigating the degradation in performance.

12. Case Study: Analysis of Single Choice Hashing Scheme  Departure rate is proportional to bucket occupancy; arrival rate is constant  We show that (limit of) discrete Markov chain  fluid model  Intuition: No dependency between the buckets because of the single choice. No “complex interaction”  Bucket occupancy distribution is  The Overflow fraction is (Erlang-B formula) 12h0 1/m·(1-1/n) (1-1/m) ·1/n … 1/m·(1-2/n)1/m·(1-3/n)1/m·(1-h/n) (1-1/m) ·2/n(1-1/m) ·3/n h/n

13. Case Study: Numerical Example  For bucket size h=1, we get:  =c/(1 + c).  In case of 100% load (c=1):  dynamic: 50%.  static: 36.79%. [Kanizo et al., INFOCOM 2009]  In case of 10% load (c=0.1):  dynamic: 9.1%.  static: 4.84%.  As load  0, dynamic systems has twice the overflow fraction of static systems.

14. Main Results  Case Study: Single choice hashing scheme  Lower bound on overflow fraction  Mitigating the degradation in performance.

15. Overflow Lower Bound  Objective: given any online scheme with average a, find lower-bound on the overflow fraction .  We use the fluid model  Elements arrival rate is ch = n / m.  Hashing rate per element is a.  In the best case, all memory accesses are used to store elements.

16. Overflow Lower Bound  Overflow lower bound of where r = ach.  Also holds for non-uniformly distributed hash functions (under some constraints).

17. Numerical Example  For bucket size h=1, lower bound of 1-a/(1+ac).  100% load (c=1) implies 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.

18. The Lower Bound is Tight  Single choice hashing scheme  Optimal for a = 1  Multiple choice: Try to insert each element greedily until either inserted or d trials.  Optimal for larger number of memory accesses, depending on system parameters.  Example:  h = 4, c = 1, d = 4  Multiple choice is optimal for a  2.19.

19. Main Results  Case Study: Single choice hashing scheme  Lower bound on overflow fraction  Mitigating the degradation in performance.

20. Moving Back Elements  Recall the example from the beginning 1234 FiniteInfinite 2 1234 Element “2” is not stored although its corresponding bucket is empty

21. Moving Back Elements  Overflow elements are stored in CAM.  Moving back elements from the CAM to the buckets.  We cannot check upon a deletion every element in the CAM.  Store the hash values along with the elements in the CAM.  Upon departure check if an element can be moved back.  Can be combined with any hashing insertion scheme.

22. Evaluation  Single choice hashing scheme  Performance is exactly as in the static case.  Multiple choice hashing scheme  Performance is better than the static case, albeit with more memory accesses. [h=4, d=1]

23. Wrap-up  Initial simulation results show degradation in performance.  We found lower and upper bounds on the achievable overflow fraction.  We compared it with upper bounds of the static case.  Mitigating the degradation in performance.  Also in the paper  Simulations with synthetic data  Other dynamic models  Trace-driven simulations

24. Thank you.