Practical Survey on Hash Tables. Aurelian Țuțuianu

Transcription

1 Practical Survey on Hash Tables Aurelian Țuțuianu

2 In memoriam Mihai Pătraşcu (17 July June 2012) I have no intention to ever teach computer science. I want to teach the love for computer science, and let the learning happen. Teaching Statement (

3 Abstract Hash table definition Collision resolving schemas: Chained hashing Linear and quadratic probing Cuckoo hashing Some hash function theory Simple tabulation hashing

4 Omnipresence of hash tables Symbol tables in compilers Cache implementations Database storages Manage memory pages in Linux Route tables Large number of documents

5 Hash Tables Considering a set of elements S from a finite and much larger universe U. A hash table consists of: hash function h: U {0,.., m 1} vector v of size m

6 Collisions same hash for two different keys What to do? Ignore them Chain colliding values Skip and try again Hash and displace Find a perfect hash function

7 War Story: cache with hash tables application Problem: An application which gets some data from an expensive repository. hash table Data source Solution: Hash table with collision replacement. Key point: a big chunk of users watched a lot of common data.

8 Collision Resolution Schemas Chained hashing Open hash: linear and quadratic probing Cuckoo hashing And many many others: perfect hashing, coalesced hashing, Robin Hood hashing, hopscotch hashing, etc.

9 Chained Hashing 0 Each slot contains a linked list. 1 O( n m ) = O(1) for all operations. 2 y Load factor: n m < x z w easy to implement works with weak hash functions consumes significant memory default implementation

10 Linear and quadratic probing All records are stored in the bucket array itself. h(x,i) = 4 + i w y z x Probe a try to find an empty place. Linear probing h x, i = h 0 (x) + i Quadratic probing i + i i h x, i = h 0 (x) + 2

11 War Story: Linear probing trick Min. 1st Qu. Median Mean 3rd Qu. Max linear probing chained hashing

12 War Story: Let it be quadratic! Replace library implementation with a home-made hash table 4 hours of work

13 Cuckoo hashing T 1 T 2 Two hash tables, T 1, T 2, of size m, and two hash functions h 1, h 2 : U -> {0,..., m 1}. h 1 (x) x z y Value x stored in cell h 1 (x) of T1 or in cell h 2 (x) of T2. Hash and displace. Lookup is constant in worst case! w h 2 (x) Updates in constant amortized time.

14 What about hash functions? Any hash function is good? What does a good hash function mean? Can I have my own?

15 The beginning of time Introduced by Alfred Dumey in 1956 for the symbol table in a compiler. He used a crazy, chaotic, random function h:u->{0..m-1}. h(x)=(x mod p) mod m, with p a big prime number. Is seems to work, but why?

16 First station: rigorous analysis Consider that h really is a random function! Knuth established a way to make a complete analysis, but based on a false assumption. No matter how long you stare at h(x)=(x mod p) mod m, it will not morph into a random function!

17 Next station: universality and k-independence Wegman and Carter (1978) A family of hash functions No need of perfect random hash function, but universal : x 1,x 2 S x 1 x 2, Pr[h(x 1 )=h(x 2 )] 1 N In generalized form the k-independence model uses statistics to measure how much random can a family of hash functions produce!

18 How it works? Random data x formula h(x) Universal multiplicative shift: h a x = a x l l out 2-independent multiplicative shift: h a,b x = a x + b 2l l out k-independent polynomial hashing: k 1 h x = i=0 a i x i mod p mod 2 l out

19 Facts on k-independence Chained hashing Wegman, Carter: requires only universal hashing Linear probing 1990 Siegel, Schmidt: O(logn)-independece is enough 2007 Pagh 5-independence suffices 2010 Patrascu,Thorup 4-independence is not enough Cuckoo hashing 2001 Pagh: O(logn)-independence is enough 2005 Cohen, Kane: 5-independence is not enough 2006 Cohen, Kane: 6-independence is enough

20 Simple tabulation hashing Simple tabulation is the fastest 3-independent family of hash functions known. Key x of length len (required bit width to store values) is divided into c chars x 1, x 2,.., x c We create c tables R 1, R 2,.., R c, filled with independent random values Hash value is created with function h x = R 1 x 1 R 2 x 2 R c x c x R 1 x 1 R 2 x 2 R 3 x 3 R 4 x 4 4 lookup tables with random 8-bit values h(x)

21 The power of simple tabulation! The power of simple tabulation hashing Mihai Pătrașcu, Mikkel Thorup December 6, 2011 According to this paper, even if is only 3-independent, we have: Constant time for linear probing Constant time for static cuckoo hashing => There are also other probabilistic properties which can be exploited, other than ones captured in k-independence theory

22 Summary Easy ways to implement optimal hash tables Simple scheme to generate a hash function family Theory produces practical results and is still alive! There are a lot of occasions to apply these ideas, so: Work hard, have fun and make history!

23 Questions?