Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash

Transcription

1 Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash Martin Aumu ller, Martin Dietzfelbinger, Philipp Woelfel? Technische Universita t Ilmenau,? University of Calgary ESA Ljubljana, September 12, 2012 M. Aumu ller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 1/18

2 Cuckoo Hashing (Pagh/Rodler, 2001/2004) A hashing-based implementation of the dictionary data type. Setting: set S U of n keys two tables T 1 [0..m 1] and T 2 [0..m 1] two (hash) functions h 1, h 2 with h i : U [m] Rules: each table cell can hold exactly one key a key x must be stored either in T 1 [h 1 (x)] or T 2 [h 2 (x)] (fast lookup and deletions!) Definition If S can be stored according to these rules, we call (h 1, h 2 ) suitable for S. M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 2/18

3 Cuckoo Hashing (Pagh/Rodler, 2001/2004) A hashing-based implementation of the dictionary data type. Setting: set S U of n keys two tables T 1 [0..m 1] and T 2 [0..m 1] two (hash) functions h 1, h 2 with h i : U [m] Rules: each table cell can hold exactly one key a key x must be stored either in T 1 [h 1 (x)] or T 2 [h 2 (x)] (fast lookup and deletions!) Definition If S can be stored according to these rules, we call (h 1, h 2 ) suitable for S. M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 2/18

4 Cuckoo Hashing: Failure Probability Theorem (Pagh/Rodler, 2004) Let S U with S = n. If (h 1, h 2 ) are fully random (or Ω(log n)-wise independent), then In fact: Θ(1/n). Pr((h 1, h 2 ) unsuitable for S) = O(1/n). M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 3/18

5 Improving Standard Cuckoo Hashing: Stash (Kirsch/Mitzenmacher/Wieder, ESA 2008): Θ(1/n) is too large. Proposal: Can put up to s = O(1) keys into additional storage, called stash Theorem (K/M/W, 2008) Let S U with S = n. If (h 1, h 2 ) are fully random, then Pr((h 1, h 2 ) unsuitable for S with stash size s) = O(1/n s+1 ). Full Randomness Assumption (FRA) central in their analysis constructions for FRA known, but undesirable Open Question (Kirsch/Mitzenmacher/Wieder, SICOMP 09) A major open question is proving the above bounds for explicit hash families that can be represented, sampled, and evaluated efficiently. M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 4/18

6 Our contributions efficient hash functions that can realize cuckoo hashing with a stash variation of a known construction of a uniform hash function (not covered in this talk) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 5/18

7 Method extension of (Dietzfelbinger/Woelfel, 2003) they proposed efficient hash functions for standard cuckoo hashing we exhibit a new class of (even simpler) hash functions that allow running c.h. with a stash M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 6/18

8 Key ingredient: linear polynomials where p U is a prime, and h(x) = ((a x + b) mod p) mod m, a and b are chosen uniformly at random from {0,..., p 1}. very simple structure! (Remark: This function is 2-wise independent, i.e., for any pair x, y U, x y, h(x) and h(y) are fully random.) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 7/18

9 Our hash class (version for this talk) For given n 1, we combine linear polynomials with lookups in tables of size n filled with random values. h i (x) = f i (x) c j=1 z (i) j [ g j (x) ], i = 1, 2 Class of all these pairs (h 1, h 2 ) of hash functions: Z. M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 8/18

10 Main Objective For given S and stash size s, calculate Pr ((h 1, h 2 ) unsuitable for S with stash size s). (h 1,h 2 ) Z What is a criteria for (h 1, h 2 ) being unsuitable? Tool: Cuckoo graph M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 9/18

11 Suitability Criteria Fact (Kirsch/Mitzenmacher/Wieder, 2008) (h 1, h 2 ) suitable after removing s edges from G(S, h 1, h 2 ), all connected components are either trees or unicyclic. Consequence: unsuitable there exists a subgraph s.t. one cannot remove s edges to obtain only tree or unicyclic components. Minimal such bad subgraph : a MOS s. (Example: s = 2.) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 10/18

12 Suitability Criteria Fact (Kirsch/Mitzenmacher/Wieder, 2008) (h 1, h 2 ) suitable after removing s edges from G(S, h 1, h 2 ), all connected components are either trees or unicyclic. Consequence: unsuitable there exists a subgraph s.t. one cannot remove s edges to obtain only tree or unicyclic components. Minimal such bad subgraph : a MOS s. (Example: s = 2.) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 10/18

13 Suitability Criteria Fact (Kirsch/Mitzenmacher/Wieder, 2008) (h 1, h 2 ) suitable after removing s edges from G(S, h 1, h 2 ), all connected components are either trees or unicyclic. Consequence: unsuitable there exists a subgraph s.t. one cannot remove s edges to obtain only tree or unicyclic components. Minimal such bad subgraph : a MOS s. (Example: s = 2.) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 10/18

14 Suitability Criteria Fact (Kirsch/Mitzenmacher/Wieder, 2008) (h 1, h 2 ) suitable after removing s edges from G(S, h 1, h 2 ), all connected components are either trees or unicyclic. Consequence: unsuitable there exists a subgraph s.t. one cannot remove s edges to obtain only tree or unicyclic components. Minimal such bad subgraph : a MOS s. (Example: s = 2.) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 10/18

15 Suitability Criteria Fact (Kirsch/Mitzenmacher/Wieder, 2008) (h 1, h 2 ) suitable after removing s edges from G(S, h 1, h 2 ), all connected components are either trees or unicyclic. Consequence: unsuitable there exists a subgraph s.t. one cannot remove s edges to obtain only tree or unicyclic components. Minimal such bad subgraph : a MOS s. (Example: s = 2.) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 10/18

16 Suitability Criteria Fact (Kirsch/Mitzenmacher/Wieder, 2008) (h 1, h 2 ) suitable after removing s edges from G(S, h 1, h 2 ), all connected components are either trees or unicyclic. Consequence: unsuitable there exists a subgraph s.t. one cannot remove s edges to obtain only tree or unicyclic components. Minimal such bad subgraph : a MOS s. (Example: s = 2.) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 10/18

17 Thus, we have Pr ((h 1, h 2 ) unsuitable for S with stash size s) (h 1,h 2 ) Z = Pr (h 1,h 2 ) Z ( T S : G(T, h 1, h 2 ) forms a MOS s ) T S Pr (G(T, h 1, h 2 ) forms a MOS s ) (h 1,h 2 ) Z if (h 1, h 2 ) are fully random, we provide a direct counting argument that this is O(1/n s+1 ) giving an alternative proof to the orginal analysis by Kirsch, Mitzenmacher and Wieder (who used machinery like Markov chain coupling) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 11/18

18 Behavior of our hash class on fixed T Recall: h i (x) = f i (x) c j=1 z (i) j [g j (x)] Central Observation Let T U. If there is a g j such that at most one pair of keys in T collides under g j (i.e., g j (x) = g j (y)), then h 1, h 2 are fully random on T. if this is the case: (h 1, h 2 ) T -good. otherwise (each g j has more than one colliding pair of keys): (h 1, h 2 ) is T -bad. M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 12/18

19 Collecting bad hash functions We split our set of hash functions into harmful and harmless ones. (h 1, h 2 ) are harmful, if there exists T S s.t. G(T, h 1, h 2 ) forms a MOS s, and (h 1, h 2 ) is T -bad. B MOSs := the set of all the harmful pairs (h 1, h 2 ). (An event in our probability space!) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 13/18

20 Splitting the calculation Let A be the event T S : G(T, h 1, h 2 ) forms a MOS s. We calculate: Pr (h1,h 2 ) Z(A) Pr (h1,h 2 ) Z(A B MOSs ) + Pr(B MOSs ) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 14/18

21 Splitting the calculation Let A be the event T S : G(T, h 1, h 2 ) forms a MOS s. We calculate: Pr (h1,h 2 ) Z(A) Pr (h1,h 2 ) Z(A B MOSs ) + Pr(B MOSs ) for this summand, we can use full randomness of (h 1, h 2 ) as before: this summand is O(1/n s+1 ) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 14/18

22 Splitting the calculation Let A be the event T S : G(T, h 1, h 2 ) forms a MOS s. We calculate: Pr (h1,h 2 ) Z(A) O(1/n s+1 ) + Pr(B MOSs ) for this summand, we can use full randomness of (h 1, h 2 ) as before: this summand is O(1/n s+1 ) M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 14/18

23 Splitting the calculation Let A be the event T S : G(T, h 1, h 2 ) forms a MOS s. We calculate: Pr (h1,h 2 ) Z(A) O(1/n s+1 ) + Pr(B MOSs ) for this summand, we can use full randomness of (h 1, h 2 ) as before: this summand is O(1/n s+1 ) The hard part: Calculating/bounding Pr(B MOSs ) = Pr( T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad ) Wish: Use full randomness nonetheless Idea: Find a suitable event that contains B MOSs M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 14/18

24 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad. #collisions g 1 5 g 2 7 g 3 4 M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

25 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad. Recall: h i (x) = f i (x) c j=1 z(i) j [g j (x)] T -bad each g j has more than one pair of colliding keys #collisions g 1 5 g 2 7 g 3 4 M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

26 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 5 g 2 7 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

27 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 4 g 2 5 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

28 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 4 g 2 5 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

29 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 4 g 2 5 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

30 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 4 g 2 4 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

31 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 3 g 2 4 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

32 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 3 g 2 4 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

33 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 2 g 2 3 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

34 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 2 g 2 3 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

35 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 2 g 2 3 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

36 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 2 g 2 3 g M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

37 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 1 g 2 3 g 3 3 M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

38 Peeling of bad graphs (simplified) Assume T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad 4 8 T -good exists g j with at most one pair 5 of colliding keys #collisions g 1 1 g 2 3 g 3 3 M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

39 Peeling of bad graphs (simplified) If T S : G(T, h 1, h 2 ) forms a MOS s (h 1, h 2 ) are T -bad #collisions g 1 1 g 2 3 g 3 3 then T S : G(T, h 1, h 2 ) forms peeled graph (h 1, h 2 ) are T -good M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 15/18

40 Using a direct counting argument, we show Pr(B MOSs ) = O(n/n c/2 ). We thus have Pr((h 1, h 2 ) unsuitable) = O(1/n s+1 ) + O(n/n c/2 ) Setting c = 2(s + 2) gives our result. M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 16/18

41 Conclusion we proposed efficient hash functions that can run c. h. with a stash approach has more potential: applicable to balanced allocation (forthcoming paper) Open question: Explicit hash functions for generalized cuckoo hashing with more than 2 hash functions. M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 17/18

42 Thanks! Any questions? M. Aumüller Explicit and Efficient Hash Functions Suffice for Cuckoo Hashing with a Stash 18/18