hash tablesds202/wiki.files/08-hash.pdf · 2020-04-29 · open addressing hash tables. hash table...

Hash tables

Dictionary

Definition

A dictionary is a data structure that stores a set of elementswhere each element has a unique key, and supports thefollowing operations:

Search(S , k) Return the element whose key is k .

Insert(S , x) Add x to S .

Delete(S , x) Remove x from S

Assume that the keys are from U = 0, . . . , u − 1.

Hash tables

Direct addressing

S is stored in an array T [0..u − 1]. The entry T [k]contains a pointer to the element with key k , if suchelement exists, and NULL otherwise.

Search(T , k): return T [k].

Insert(T , x): T [x .key]← x .

Delete(T , x): T [x .key]← NULL.

What is the problem with this structure?0123456789

keysatellitedata

Hash tables

Hashing

In order to reduce the space complexity of directaddressing, map the keys to a smaller range0, . . .m − 1 using a hash function.

There is a problem of collisions (two or more keys thatare mapped to the same value).

There are several methods to handle collisions.

ChainingOpen addressing

Hash tables

Hash table with chaining

Let h : U → 0, . . . ,m − 1 be a hash function (m < u).

S is stored in a table T [0..m − 1] of linked lists. Theelement x ∈ S is stored in the list T [h(x .key)].

Search(T , k): Search the list T [h(k)].

Insert(T , x): Insert x at the head of T [h(x .key)].

Delete(T , x): Delete x from the list containing x .

S=30,9,26,19,1,6

m=5, h(x)=x mod 5

Hash tables

S=30,9,26,19,1,6

m=5, h(x)=x mod 5

Hash tables

S=30,9,26,19,1,6

m=5, h(x)=x mod 5

Hash tables

S=30,9,26,19,1,6

m=5, h(x)=x mod 5

Hash tables

S=30,9,26,19,1,6

m=5, h(x)=x mod 5

Hash tables

S=30,9,26,19,1,6

m=5, h(x)=x mod 5

Hash tables

S=30,9,26,19,1,6

m=5, h(x)=x mod 5

Hash tables

Analysis — Worst case

The worst case running times of the operations are:

Search: Θ(n).

Insert: Θ(1).

Delete: Θ(n) (can be reduced to Θ(1) using doublylinked list).

Hash tables

Analysis

Assumption of simple uniform hashing: any element isequally likely to hash into any of the m slots,independently of where other elements have hashed into.

The above assumption is true when the keys are chosenuniformly and independently at random (withrepetitions), h(x) = x mod m, and u is divisible by m.

We want to analyze the performance of hashing under theassumption of simple uniform hashing. This is the ballsinto bins problem.

Suppose we randomly place n balls into m bins. Let X bethe number of balls in bin 0.

The expected time of an unsuccessful search in a hashtable is Θ(1 + E [X ]).

Hash tables

The expectation of X

Each ball has probability 1/m to be in bin 0.

The random variable X has binomial distribution withparameters n and 1/m.

Therefore, E [X ] = n/m.

The expected time of an unsuccessful search in a hashtable is Θ(1 + α), where α = n/m.

The value α is called the load factor.

The expected time of searching a random key from S isalso Θ(1 + α).

Hash tables

Example

If α = 1 and n is large:

Pr[X = 0] ≈ 0.368 Pr[X = 4] ≈ 0.015

Pr[X = 1] ≈ 0.368 Pr[X = 5] ≈ 0.003

Pr[X = 2] ≈ 0.184 Pr[X = 6] ≈ 0.0005

Pr[X = 3] ≈ 0.061 Pr[X = 7] ≈ 0.00007

Hash tables

The maximum number of balls in a bin

Suppose that n = m = 106. Most bins contain 0–3 balls.

The probability that a specific bin contains at least 8 ballsis ≈ 0.000009.

However, it is very likely that some bin will contain 8balls.

Theorem

If m ∈ Θ(n) then the size of the largest bin isΘ(log n/ log log n) with probability at least 1− 1/nΘ(1).

Hash tables

Universal hash functions

Random keys + fixed hash function (e.g. mod)⇒ The hashed keys are random numbers.

A fixed set of keys + random hash function (selectedfrom a universal set)⇒ The hashed keys are semi-random numbers.

Hash tables

Definition

A collection H of hash functions is a universal if for every pairof distinct keys x , y ∈ U , Prh∈H[h(x) = h(y)] ≤ 1

Example

Let p be a prime number larger than u.fa,b(x) = ((ax + b) mod p) mod mHp,m = fa,b|a ∈ 1, 2, . . . , p − 1, b ∈ 0, 1, . . . , p − 1

Theorem

Suppose that H is a universal collection of hash functions.If a hash table for S is built using a randomly chosen h ∈ H,then for every k ∈ U, the expected length of T [h(k)] is atmost 1 + n/m.

Proof.

Let X = length of T [h(k)].X =

∑y∈S Iy where Iy = 1 if h(y .key) = h(k) and Iy = 0

otherwise.

E [X ] = E

[∑y∈S

∑y∈S

E [Iy ] =∑y∈S

Prh∈H

[h(y .key) = h(k)]

If k /∈ S , E [X ] ≤ n · 1m

. Otherwise, E [X ] ≤ 1 + (n − 1) 1m

Hash tables

Rehashing

We wish to maintain n ∈ O(m) in order to have Θ(1)expected search time.

This can be achieved by rehashing. Suppose we wantn ≤ m. When the table has m elements and a newelement is inserted, create a new table of size 2m andcopy all elements into the new table.

h(x)=x mod 5

h'(x)=x mod 10

Hash tables

Rehashing

We wish to maintain n ∈ O(m) in order to have Θ(1)expected search time.

This can be achieved by rehashing. Suppose we wantn ≤ m. When the table has m elements and a newelement is inserted, create a new table of size 2m andcopy all elements into the new table.The cost of rehashing is Θ(n).

Hash tables

Rehashing

In order to keep the space usage Θ(n), we needn ∈ Ω(m).

We can keep α ∈ [ 14, 1]. If an insert would make α > 1,

create a new table of size 2m. If a delete would makeα < 1

4, create a new table of size m/2.

Hash tables

Time complexity of chaining

Under the assumption of simple uniform hashing, theexpected time of a search is Θ(1 + α) time.

If α = Θ(1), and under the assumption of simple uniformhashing, the worst case time of a search isΘ(log n/ log log n), with probability at least 1− 1/nΘ(1).

If the hash function is chosen from a universal collectionat random, the expected time of a search is Θ(1 + α).

The worst case time of insert is Θ(1) if there is norehashing.

Using doubly linked lists, the worst case time of delete isΘ(1).

Hash tables

Linear probing

In the following, we assume that the elementsin the hash table are keys with no satelliteinformation.

To insert element k , try inserting to T [h(k)].If T [h(k)] is not empty, try T [h(k) + 1 mod m],then try T [h(k) + 2 mod m] etc.

Insert(T , k)

(1) j ← h(k)(2) for i = 0 to m − 1(3) if T [j ] = NULL(4) T [j ]← k(5) return(6) j ← j + 1 mod m(7) error “hash table overflow”

0123456789

h(k)=k mod 10

Hash tables

Linear probing

Insert(T , k)

0123456789

insert(T,14)

Hash tables

Linear probing

Insert(T , k)

0123456789

insert(T,20)

Hash tables

Linear probing

Insert(T , k)

0123456789

insert(T,34)

Hash tables

Linear probing

Insert(T , k)

0123456789

19insert(T,19)

Hash tables

Linear probing

Insert(T , k)

0123456789

insert(T,55)

Hash tables

Linear probing

Insert(T , k)

0123456789

insert(T,49)

Hash tables

Searching

Search(T , k)

(1) j ← h(k)(2) for i = 0 to m − 1(3) if T [j ] = k(4) return j(5) if T [j ] = NULL(6) return NULL(7) j ← j + 1 mod m(8) return NULL

0123456789

search(T,55)Hash tables

Searching

Search(T , k)

(1) j ← h(k)(2) for i = 0 to m − 1(3) if T [j ] = k(4) return j(5) if T [j ] = NULL(6) return NULL(7) j ← j + 1 mod m(8) return NULL

0123456789

search(T,25)Hash tables

Deletion

If we want to delete the element T [i ], we cannotwrite NULL into T [i ].

0123456789

delete 14

Hash tables

Deletion

If we want to delete the element T [i ], we cannotwrite NULL into T [i ].

0123456789

delete 14

Hash tables

Deletion

Delete method 1: To delete element T [i ], storein T [i ] a special value DELETED.

Change line (3) in Insert to:

if T [j ] = NULL OR T [j ] = DELETED

0123456789

delete 14

Hash tables

Deletion

Delete method 2: Erase T [i ] from the table (re-place it by NULL) and also erase the consecutiveblock of elements following T [i ]. Then, reinsertthe latter elements to the table.

0123456789

delete 14

Hash tables

Deletion

0123456789

delete 14

Hash tables

Deletion

0123456789

delete 14

Hash tables

Deletion

0123456789

delete 14

Hash tables

Time complexity of linear probing

Assuming simple uniform hashing and that α ≤ 1− ε for somefixed constant ε > 0:

The expected time of search is Θ(1).

The expected time of insert/delete without rehashing isΘ(1).

Hash tables

Open addressing

Open addressing is a generalization oflinear probing.

Leth : U × 0, . . .m − 1 → 0, . . .m − 1be a hash function such thath(k , 0), h(k , 1), . . . h(k ,m − 1) is apermutation of 0, . . .m − 1 for everyk ∈ U .

The slots examined during search/insertare h(k , 0), then h(k , 1), h(k , 2) etc.

In the example on the right,h(k , i) = (h1(k) + ih2(k)) mod 13 whereh1(k) = k mod 13h2(k) = 1 + (k mod 11)

0123456789

101112

insert(T,14)

h(14,0)

h(14,1)

h(14,2)

Hash tables

Open addressing

Insertion is the same as in linear probing:Insert(T , k)

(1) for i = 0 to m − 1(2) j ← h(k , i)(3) if T [j ] = NULL OR T [j ] = DELETED(4) T [j ]← k(5) return(6) error “hash table overflow”

Deletion is done using delete method 1 defined above(using special value DELETED).

Hash tables

Double hashing

In the double hashing method,

h(k , i) = (h1(k) + ih2(k)) mod m

for some hash functions h1 and h2.The value h2(k) must be relatively prime to m for theentire hash table to be searched. This can be ensured byeither

Taking m to be a power of 2, and the image of h2

contains only odd numbers.Taking m to be a prime number, and the image of h2

contains integers from 1, . . . ,m − 1.For example,

h1(k) = k mod m

h2(k) = 1 + (k mod m′)

where m is prime and m′ < m.Hash tables

Comparison of hash table methods – Search

8 10 12 14 16 18 20 220

Successful Lookup

CuckooTwo−Way ChainingChained HashingLinear Probing

Hash tables

Comparison of hash table methods – Search

8 10 12 14 16 18 20 220

Unsuccessful Lookup

Hash tables

Comparison of hash table methods – Insert

8 10 12 14 16 18 20 220

Insert

Hash tables

Comparison of hash table methods – Delete

8 10 12 14 16 18 20 220

Delete

Hash tables

Hash function for strings

A string can be viewed as a sequence of integer, whereeach character is replaced by its ASCII value.

For example, the string ABXY is the sequence65,66,88,89.

To hash a sequence of integers x0, . . . , xr−1 from1, . . . , u, we use a prime number p > u. We select arandom number z ∈ 0, . . . , p − 1 and use the functionhz(x0, . . . , xr−1) = (x0z0 + x1z1 + · · ·+ xr−1z r−1) mod p.

For example h300(65, 66, 88, 89) =(65 + 66 · 300 + 88 · 3002 + 89 · 3003) mod p.

Hash tables

Hash function for strings

We definedhz(x0, . . . , xr−1) = (x0z0 + x1z1 + · · ·+ xr−1z r−1) mod p

Theorem

Given two different sequences x0, . . . , xr−1 and y0, . . . , yr−1

over 1, . . . , u,Pr[hz(x0, . . . , xr−1) = hz(y0, . . . , yr−1)] ≤ (r − 1)/p when z ischosen randomly from 0, . . . , p − 1.

Proof.

A non-trivial polynomial of degree d over the field Zp has atmost d roots. Therefore, the polynomialQ(z) = hz(x0, . . . , xr−1)− hz(y0, . . . , yr−1) =(x0 − y0)z0 + · · ·+ (xr−1 − yr−1)z r−1

has at most r − 1 roots over the field Zp.The prob. of choosing z to be a root of Q is ≤ (r − 1)/p.

Hash tables

Applications of hash functions

Data deduplication: Suppose that we have many files, andsome files have duplicates. In order to save storage space,we want to store only one instance of each distinct file.

Rabin-Karp pattern matching algorithm To find theoccurrences of a string P of length m in a string T ,compute a hash function on every substring of T oflength m and compare with h(P).

Distributed storage: Storing files on several servers.

Rabin-Karp pattern matching algorithm

Suppose we are given strings P and T , and we want tofind all occurrences of P in T .

The Rabin-Karp algorithm is as follows:Compute h(P)for every substring T ′ of T of length |P |

if h(T ′) = h(P) check whether T ′ = P .

The values h(T ′) for all T ′ can be computed in Θ(|T |)time using rolling hash.

Example

Let T = CABXYZ, P = ABXY.

h300(CABX ) = (67 + 65 · 300 + 66 · 3002 + 88 · 3003) mod p

h300(ABXY ) = (65 + 66 · 300 + 88 · 3002 + 89 · 3003) mod p

= ((h300(CABX )− 67)/300 + 89 · 3003) mod p

Distributed storage

Suppose we have many files, and we want to store themon several servers.

Let m denote the number of servers.

The simple solution is to use a hash functionh : U → 1, . . . ,m, and assign file x to server h(x).

The problem with this solution is that if we add orremove a server, we need to do rehashing which will movemost files between servers.

Distributed storage: Consistent hashing

Suppose that the server have identifiers s1, . . . , sm.

Let h : U → [0, 1] be a hash function.

For each server i associate a point h(si) on the unit circle.

For each file f , assign f to the server whose point is thefirst point encountered when traversing the unit cycleanti-clockwise starting from h(f ).

server 1

server 3

server 2

Suppose that the server have identifiers s1, . . . , sm.

Let h : U → [0, 1] be a hash function.

For each server i associate a point h(si) on the unit circle.

For each file f , assign f to the server whose point is thefirst point encountered when traversing the unit cycleanti-clockwise starting from h(f ).

server 1

server 3

server 2

When a new server m + 1 is added, let i be the serverwhose point is the first server point after h(sm+1).

We only need to reassign some of the files that wereassigned to server i .

The expected number of files reassignments is n/(m + 1).

server 1

server 3

server 2

server 4

Bloom Filter

Blocking malicious web pages

Suppose we want to implement blocking of malicious webpages in a web browser.

Assume we have a list of 10,000,000 malicious pages.

We can store all pages in a hash table, but this requires alarge amount of memory.

Bloom filter is a solution for the problem that uses smallamount of memory.

Bloom filter stores a set of elements and supportMembership and Insert operations.

For an element that is in the set, the Membershipoperation always returns a positive answer.

For an element that is not in the set, the Membershipoperation can return either a negative answer, or apositive answer (false positive).

Bloom Filter

Blocking malicious web pages

Suppose that the list of malicious pages is stored in aBloom filter.

For a query URL, if the answer is negative, we know thatthe URL is not a malicious page.

If the answer is positive, we do not know the correctanswer. In this case, we get the answer from a server.

We want a small probability for a false positive.

Bloom Filter

Bloom filter — simple version

Suppose we want to store a set S .

We use an array A of size m initialized to 0.

We then choose a hash function h, and set A[h(x)]← 1for every x ∈ S .

For a query y , if A[h(y)] = 0, we know that y /∈ S .

If A[h(y)] = 1, either y is in S or not.

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

Probability of false positive

Assume simple uniform hashing: any element is equallylikely to hash into any of the m slots, independently ofwhere other elements have hashed into.

For fixed i , the probability that A[i ] = 0 isp = (1− 1/m)n.

For y /∈ S , the probability that A[h(y)] = 1 is 1− p.

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

Reducing the false positive probability

We choose t hash functions h1, h2, . . . , ht , and setA[hj(x)]← 1 for j = 1, 2, . . . t, for every x ∈ S .

For a query y , if A[hj(y)] = 0 for some j , we know thaty /∈ S .

If A[hj(y)] = 1 for all j , either y is in S or not.

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

01 1111

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

01 1111

Bloom Filter

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

01 1111

Bloom Filter

Probability of false positive

For fixed i , the probability that A[i ] = 0 isp = (1− 1/m)tn.

For y /∈ S , the probability that A[h(y)] = 1 is (1− p)t .

000 0 0 0 0 0 0 00 1 2 3 4 5 6 7 8 9

01 1111

Bloom Filter

Optimizing the probability of false positive

For the previous example (n = 2, m = 10):

For t = 1, p = 0.81, (1− p)1 = 0.19.

For t = 2, p ≈ 0.656, (1− p)2 ≈ 0.118.

For t = 3, p ≈ 0.531, (1− p)3 ≈ 0.103.

For t = 4, p ≈ 0.430, (1− p)4 ≈ 0.105.

For t = 5, p ≈ 0.349, (1− p)5 ≈ 0.117.

Bloom Filter

Optimizing the probability of false positive

Let f (t) be the probability of false positive when using thash functions.

p = (1− 1/m)tn ≈ e−tn/m

f (t) = (1− p)t ≈ (1− e−tn/m)t = et·ln(1−e−tn/m)

Minimizing f (t) is equivalent to minimizingg(t) = t · ln(1− e−tn/m).dgdt

= ln(1− e−tn/m) + tnm

e−tn/m

1−e−tn/m .

The minimum of g(t) is when dg/dt = 0 =⇒ t = mn

The minimum of f (t) is ≈ 0.6185m/n.

Example: For m = 8n, mn

ln 2 ≈ 5.55. f (5) ≈ 0.0217,f (6) ≈ 0.0216.

Bloom Filter

hash tablesds202/wiki.files/08-hash.pdf · 2020-04-29 · open addressing hash tables. hash table...

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