w5ch11 hash index

7/31/2019 W5Ch11 Hash Index

1/63

CPSC 461Instructor: Marina Gavrilova


2/63

Goal Goal of todays lecture is to introduce concept of

hash-index. Hashing is an alternative to treestructure that allows near constant access time to ANYrecord in a very large database.


3/63

Presentation Outline Introduction

Did you know that?

Static hashing definition and methods

Good hash function Collision resolution (open addressing, chaining)

Extendible hashing

Linear hashing Useful links and current market trends

Summary


4/63

4

Approaches to Search

1. Sequential and list methods

(lists, tables, arrays).

2. Direct access by key value (hashing)

3. Tree indexing methods.

Introduction to Hashing


5/63

5

DefinitionHashing is the process of mapping a key value to a

position in a table.

Ahash function maps key values to positions.

Ahash table is an array that holds the records.

Searching in a hash table can be done in O(1) regardless of the

hash table size.



6/63

6



7/637

Example of Usefullness

10 stock details, 10 table positions

Stock numbers are between 0 and 1

1000.

Using the whole stock numbers may

require 1000 storage locations and

this is an obvious waste of memory.



8/638

Applications of Hashing

Compilers use hash tables to keep track of declared variables

A hash table can be used for on-line spelling checkersif

misspelling detection (rather than correction) is important, an entiredictionary can be hashed and words checked in constant time

Game playing programs use hash tables to store seen positions,

thereby saving computation time if the position is encountered

again

Hash functions can be used to quickly check for inequalityif

two elements hash to different values they must be different

Storing sparse data

Introduction of Hashing


9/639

Did you know that? Cryptography was once known only to the key people in the theNational Security Agency and a few academics.

Until 1996, it was illegal to export strong cryptography from theUnited States.

Fast forward to 2006, and the Payment Card Industry Data Security

Standard (PCI DSS) requires merchants to encrypt cardholderinformation. Visa and MasterCard can levy fines of up to $500,000for not complying!

Among methods recommended are:

Strong one-way hash functions (hashed indexes)

Truncation

Index tokens and pads (pads must be securely stored)

Strong cryptography[Hashing for fun and profit: Demystifying encryption for PCI DSS

Roger Nebel]
http://www.nsa.gov/http://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://www.nsa.gov/


10/6310

Did you know that? Transport Layer Security protocol on networks (TLS) uses the Rivest,

Shamir, and Adleman (RSA) public key algorithm for the TLS key

exchange and authentication, and the Secure Hashing Algorithm 1

(SHA-1) for the key exchange and hashing.

[System cryptography: Use FIPS compliant algorithms for encryption,

hashing, and signing, Microsoft TechNews, 2005]


11/6311

Did you know that? Spatial hashing studies performed at Microsoft Research, Redmond

combine hashing with computer graphics to create a new set of tools for

rendering, mesh reconstruction, and collision optimization (see publicposter by Hugues Hoppe on the next slide)


12/63

Perfect Spatial Hashing Sylvain Lefebvre Hugues Hoppe(Microsoft Research)

Hash table Offset table Hash table Offset table

Vector images Sprite maps

Alpha compression

3D textures 3D painting

Simulation Collision detection

2D 3D

1282 382 182 1283 353 193

ApplicationsHash function

p

p S

( )s h p

modq p r

[ ]q

Domain Hash

table H

Offset

table

( )h p p p

Perfect hash on multidimensional data

No collisions ideal for GPU

Single lookup into a small offset table

Offsets only ~4 bits per defined data

Access only ~4 instructions on GPU

Optimized spatial coherence

10243, 46MB, 530fps 20483, 56MB, 200fps

10243, 12MB, 140fps2563, 100fps

10242, 500KB, 700fps +900KB, 200fps

(modulo table sizes)

0.9bits/pixel, 800fps

1.8%

24372

83333

We design a perfect hash function to losslessly packsparse data while retaining efficient random access:

Simply:

453

nearest: 7.5MB, 370fps

11632


13/63

13

Did you know that? Combining hashing and encryption provides a much

stronger tool for database and password protection.

http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/

[Security Briefs, SMDN Magazine]
http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/


14/63

14

Hash Functions

Hashing is the process of chopping up the key andmixing it up in various ways in order to obtain an index

which will be uniformly distributed over the range of

indices - hence the hashing.

There are several common ways of doing this:

Truncation Folding

Modular Arithmetic

Hash Functions


15/63

15

Hash FunctionsTruncation

Truncation is a method in which parts of the key are ignored andthe remaining portion becomes the index.

- For this, we take the given key and produce a hash location bytaking portions of the key (truncating the key).

ExampleIf a hash table can hold 1000 entries and an 8-digit

number is used as key, the 3rd, 5thand 7th digits starting

from the left of the key could be used to produce the index.

- e.g. .. Key is 62538194 and the hash location is 589.

-

Advantage: Simple and easy to implement.

Problems: Clustering and repetition.

Hash Functions


16/63

16

Hash FunctionsFolding

Folding breaks the key into several parts and combines the parts to forman index.

- The parts may be recombined by addition, subtraction, multiplications and may have

to be truncated as well.- Such a process is usually better than truncation by itself since it produces a better

distribution: all of the digits in the key are considered.

- Using a key 62538194 and breaking it into 3 numbers using the first 3 and the last 2

digits produced 625, 381 and 94. These could be added to get 1100 which could be truncated

to 100.

They could be also be multiplied together and then three digits chosen

from the middle of the number produced.

Hash Functions


17/63

17

Hash Functions(Modular Arithmetic)

Modular Arithmeticprocess essentially assures that the indexproduced is within a specified range. For this, the key is converted to

an integer which is divided by the range of the index with the resulting

function being the value of the remainder.

Uses: biometrics, encryption, compression

- If the value of the modulus is a prime number, the distribution ofindices

obtained is quite uniform.

- A table whose size is some number which has many factors provides thepossibility of many indices which are the same, so the size should be a prime

number.

Hash Functions


18/63

18

Good Hash Functions

Hash functions which use all of the key are almost always betterthan those which use only some of the key.

- When only portions are used, information is lost and therefore thenumber of possibilities for the final key are reduced.

- If we deal with the integer its binary form, then the number of

pieces that can be manipulated by the hash function is greatly

increased.

Hash Functions


19/63

19

Collision

It is obvious that no matter what function is used, the possibilityexists that the use of the function will produce an index which is a

duplicate of an index which already exists. This is a Collision.

Collision resolution strategy:

- Open addressing: store the key/entry in a different position

- Chaining: chain together several keys/entries in each position

Collision Resolution


20/63

20

Collision - Example- - Hash table size 11

- - Hash function: key mod hash size

So, the new positions in the hash table are:

Some collisions occur with this hash function.



21/63

21

Collision ResolutionOpen Addressing

Resolving collisions by open addressing is resolving the problem bytaking the next open space as determined by rehashing the key

according to some algorithm.

Two main open addressing collision resolution techniques:

- - Linear probing: increase by 1 each time [mod table size!]

- - Quadratic probing:to the original position, add 1, 4, 9, 16,

also in some cases key-dependent increment technique is used.


Probing

If the table position given by the hashed key is already

occupied, increase the position by some amount, until an empty

position is found


22/63


23/63

23


In order to try to avoid clustering, a method which does not look forthe first open space must be used.

Two common methods are used

- - Quadratic Probing

- - Key-dependent Increments



24/63

24


Quadratic Probingnew position = (collision position + j2) MOD hash size

{ j = 1, 2, 3, 4, }

Example

Before quadratic probing:

After quadratic probing:

ProblemOverflowmay occurs when there is still space in the

hash table.



25/63

25


Key-dependent Increments

This technique is used to solve the overflow problem of thequadratic probing method.

These increments vary according to the key used for the hashfunction.

If the original hash function results in a good distribution, then key-

dependent functions work quite well for rehashing and all locations in the

table will eventually be probed for a free position.

Key dependent increments are determined by using the key to

calculate a new value and then using this as an increment to determine

successive probes.



26/63

26


Key-dependent IncrementsFor example, since the original hash function was key Mod 11,we might choose a function

of key MOD 7 to find the increment. Thus the hash function becomes - -

new position = current position + ( key DIV11) MOD 11

ExampleBefore key-dependent increments:

After key-dependent increments:



27/63

27


Key-dependent Increments In all of the closed hash functions it is important to ensure that an

increment of 0 does not arise.

If the increment is equal to hash size the same position will be probed all the

time, so this value cannot be used.

If we ensure that the hash size is prime and the divisors for the open and

closed hash are prime, the rehash function does not produce a 0 increment,

then this method will usually access all positions as does the linear

probe.

- Using a key-dependent method usually result reduces clustering and therefore

searches for an empty position should not be as long as for the linear method.



28/63

28

Collision ResolutionChaining Each table position is a linked list Add the keys and entries anywhere in the

list (front easiest)

Advantagesover open addressing:

- Simpler insertion and removal

- Array size is not a limitation (but should

still minimize collisions: make table size

roughly equal to expected number of keys

and entries)

Disadvantage- Memory overhead is large if entries are

small



29/63

29

Collision ResolutionChaining

Example:

Before chaining:

After chaining:



30/63

30

In analyzing search efficiency, the average is usually used. Searching withhash tables is highly dependent on how full the table is since as the tableapproaches a full state, more rehashes are necessary. The proportion of the

table which is full is called the Load Factor.

- When collisions are resolved using open addressing, the maximum load

factor is 1.

- Using chaining, however, the load factor can be greater than 1 when thetable is full and the linked list attached to each hash address has more than

one element.

- Chaining consistently requires fewer probes than open addressing.-Traversal of the linked list is slow and if the records are small, it may be just

as well to use open addressing.

- Chaining is the best under two conditions --- when the number of

unsuccessful searches is large or when the records are large.

- Open addressing would likely be a reasonable choice when most searches are

likely to be successful, the load factor is moderate and the records are

relatively small.

Analysis of Searching using Hash Tables


31/63

31

Average number of probes for different collision resolution

methods:[ The values are for large hash tables, in this case larger than

500]



32/63

32

When are other representations more suitable than hashing:

Hash tables are very good if there is a need for many searches in a

reasonably stable database

Hash tables are not so good if there are many insertions and

deletions, or if table traversals are neededin this case, trees arebetter for indexing

Also, hashing is very slow for any operations which require the

entries to be sorted (e.g. query to Find the minimum key)



33/63

33

Aperfect hashing functionmaps a key into a unique address. If the

range of potential addresses is the same as the number of keys, the

function is a minimal (in space) perfect hashing function.

What makes perfect hashing distinctive is that it is a process for

mapping a key space to a unique address in a smaller address space, that is

hash (key) unique address

Not only does a perfect hashing function improve retrieval performance,but a minimal perfect hashing function would provide 100 percent storage

utilization.

Perfect Hashing


34/63

34

Process of creating a perfect hash function

A general form of a perfect hashing function is:

p.hash (key) =(h0(key) + g[h

1(key)] + g[h

2(key)] mod N

Perfect Hashing


35/63

35

In Cichellis algorithm, the component functions are:

h0 = length (key)

h1 = first_character (key)

h2 = second_character (key)

and g = T (x)

where Tis the table of values associated with individual characters xwhich

may apply in a key.

The time consuming part of Cichellis algorithm is determining T.

Cichellis Algorithm


36/63

36

When we apply the Cichellis perfect hashing functionto the keyword

beginusing table 1, we can get

The keyword begin would be stored in location 33. Since the hash values

run from 2 through 37 for this set of data, the hash function is a minimal

perfect hashing function.

Cichellis Algorithm

Table 1: Values associated with the characters of the Pascal

reserved words


37/63

37

Links for interactive hashing examples:

http://www.engin.umd.umich.edu/CIS/course.des/cis350/hashing/WEB/HashApplet.htm

http://www.cs.auckland.ac.nz/software/AlgAnim/hash_tables.html

http://www.cse.yorku.ca/~aaw/Hang/hash/Hash.html

http://www.cs.pitt.edu/~kirk/cs1501/animations/Hashing.html

Some Links to Hashing Animation


38/63

Hashing as Database Index

The basic idea is to use hashing function, which maps asearch key value(of a field) into a record or bucket ofrecords.

As for any index, 3 alternatives for data entries k*: Data record with key value k

Hash-basedindexes are best for equalityselections.Cannot support range searches.

Static and dynamic hashing techniques exist; trade-offssimilar to ISAM vs. B+ trees.


39/63

Static Hashing # primary pages fixed, allocated sequentially, never

de-allocated; overflow pages allowed if needed.

h(k) mod M = bucket to which data entry withkeykbelongs. (M = # of buckets)

h(key) mod N

hkey

Primary bucket pages Overflow pages

2

0

N-1


40/63

Static Hashing (Contd.) Buckets contain data entries.

Hash function depends on search key field of record r.Must distribute values over range 0 ... M-1 (table size).

h(key) = (a * key + b) mod M usually works well. a and b are constants; lots known about how to tune h.

Long overflow chains can develop and degradeperformance.

Extendible and LinearHashing: Dynamic techniques to fixthis problem.


41/63

Rule of thumb Try to keep space utilization between 50% and 80%

If 80%, overflow significantDepends on how good hashing function is

&

On # keys/bucket


42/63

42

Extendible Hashing (Fagin et. al. 1979)

Expandable hashing (Knott 1971)

Dynamic Hashing (Larson 1978)

Hash Functions for Extendible Hashing


43/63

43

Assume that a hashing technique is applied to a dynamicallychanging file composed of buckets, and each bucket can holdonly a fixed number of items.

Extendible hashing accesses the data stored in bucketsindirectly through an index that is dynamically adjusted toreflect changes in the file.

The characteristic feature of extendible hashing is theorganization of the index, which is an expandable table.

Extendible Hashing


44/63

44

A hash function applied to a certain key indicates a position in the indexand not in the file (or table or keys). Values returned by such a hash

function are calledpseudokeys.

The database/file requires no reorganization when data are added to or

deleted from it, since these changes are indicated in the index.

Only one hash function hcan be used, but depending on the size of the

index, only a portion of the added h(K)is utilized.

A simple way to achieve this effect is by looking at the address into the

string of bits from which only the ileftmost bits can be used.

The number i is the depth of the directory.

In figure 1(a) (in the next slide), the depth is equal to two.

Extendible Hashing


45/63

45

Figure 1. An example of extendible

hashing (Drozdek Textbook)

Example


46/63

Extendible Hashing as Index

Situation: Bucket (primary page) becomes full. Why notre-organize file bydoubling # of buckets? Reading and writing all pages is expensive!

Idea: Use directory of pointers to buckets, double # ofbuckets bydoubling the directory, splitting just the bucketthat overflowed!

Directory much smaller than file, so doubling it is muchcheaper. Only one page of data entries is split. No

overflowpage! Trick lies in how hash function is adjusted!

2LOCAL DEPTH


47/63

Example Directory is array of size 4.

To find bucket for r, take lastglobal depth # bits ofh(r); wedenote rbyh(r).

Ifh(r) = 5 = binary 101, it isin bucket pointed to by 01.

Insert

: If bucket is full, splitit (allocate new page, re-distribute).If necessary, double the directory. (As we will see, splitting a

bucket does not always require doubling; we can tell bycomparingglobal depth with local depth for the split bucket.)

13*00

01

10

11

2

2

2

2

2

LOCAL DEPTH

GLOBAL DEPTH

DIRECTORY

Bucket A

Bucket B

Bucket C

Bucket D

DATA PAGES

10*

1* 21*

4* 12* 32* 16*

15* 7* 19*

5*


48/63

Insert h(r)=20 (Causes Doubling)

20*

00

01

10

11

2 2

2

2

LOCAL DEPTH 2

2

DIRECTORY

GLOBAL DEPTHBucket A

Bucket B

Bucket C

Bucket D

Bucket A2(`split image'of Bucket A)

1* 5* 21*13*

32*16*

10*

15* 7* 19*

4* 12*

19*

2

2

2

000001

010

011

100

101

110

111

3

3

3

DIRECTORY

Bucket A

Bucket B

Bucket C

Bucket D

Bucket A2

(`split image'of Bucket A)

32*

1* 5* 21* 13*

16*

10*

15* 7*

4* 20*12*

LOCAL DEPTH

GLOBAL DEPTH


49/63

Points to Note 20 = binary 10100. Last 2 bits (00) tell us rbelongs in A or

A2. Last3 bits needed to tell which.

Global depth of directory: Max # of bits needed to tell whichbucket an entry belongs to.

Local depth of a bucket: # of bits used to determine if anentry belongs to this bucket.

When does bucket split cause directory doubling?

Before insert, local depth of bucket =global depth. Insertcauses local depth to become >global depth; directory isdoubled bycopying it overand `fixing pointer to split imagepage. (Use of least significant bits enables efficient doubling

via copying of directory!)


50/63

Directory Doubling

00

01

10

11

2

Why use least significant bits in directory? Allows for doubling via copying!

000

001

010

011

3

100

101110

111

vs.

0

1

1

6*6*

6*

6 = 110

00

10

01

11

2

3

0

1

1

6* 6*6*

6 = 110000

100

010

110

001

101

011

111

Least Significant Most Significant


51/63

Comments on Extendible Hashing

If directory fits in memory, equality search answered withone disk access; else two.

100MB file, 100 bytes/rec, 4K pages contains 1,000,000records (as data entries) and 25,000 directory elements;

chances are high that directory will fit in memory. Directory grows in spurts, and, if the distribution of hash

values is skewed, directory can grow large.

Multiple entries with same hash value cause problems!

Delete: If removal of data entry makes bucket empty,can be merged with `split image. If each directoryelement points to same bucket as its split image, canhalve directory.


52/63

52

Expandable Hashing

Similar idea to an extendible hashing.But binary tree is used to store an index on the buckets.

Dynamic Hashing

multiple binary trees are used.

Outcome:- To shorten the search.- Based on the key --- select what tree to search.

Hybrid methods


53/63

Linear Hashing

This is another dynamic hashing scheme, an alternative toExtendible Hashing.

LH handles the problem of long overflow chains without

using a directory, and handles duplicates. Idea: Use a family of hash functions h0, h1, h2, ...

hi(key) = h(key) mod(2iN); N = initial # buckets

h is some hash function (range is not 0 to N-1)

If N = 2d0, for some d0, hi consists of applying h and looking atthe last di bits, where di = d0 + i.

hi+1 doubles the range ofhi (similar to directory doubling)


54/63

Linear Hashing (Contd.) Directory avoided in LH by using overflow pages, and

choosing bucket to split round-robin.

Splitting proceeds in `rounds. Round ends when allNRinitial (for round R) buckets are split. Buckets 0 toNext-1have been split; Next toNR yet to be split.

Current round number is Level.

Search: To find bucket for data entryr, findhLevel(r):

IfhLevel

(r) in range Next toNR

, rbelongs here.

Else, r could belong to bucket hLevel(r) or bucket hLevel(r)+NR; must applyhLevel+1(r) to find out.


55/63

Overview of LH File

In the middle of a round.

Levelh

Buckets that existed at the

beginning of this round:

this is the range of

Next

Bucket to be split

of other buckets) in this round

Levelh search key value )(

search key value )(

Buckets split in this round:

If

is in this range, must use

h Level+1

`split image' bucket.

to decide if entry is in

created (through splitting

`split image' buckets:


56/63

Linear Hashing (Contd.) Insert: Find bucket by applying hLevel / hLevel+1:

If bucket to insert into is full:

Add overflow page and insert data entry.

(Maybe) SplitNext bucket and incrementNext.

Can choose any criterion to `trigger split. Since buckets are split round-robin, long overflow chains

dont develop!

Doubling of directory in Extendible Hashing is similar;switching of hash functions is implicit in how the # of bitsexamined is increased.


57/63

Example of Linear Hashing On split, hLevel+1 is used to

re-distribute entries.

0hh1

(This info

is for illustration

only!)

Level=0, N=4

00

01

10

11

000

001

010

011

(The actual contents

of the linear hashed

file)

Next=0PRIMARYPAGES

Data entry rwith h(r)=5

Primarybucket page

44* 36*32*

25*9* 5*

14* 18*10*30*

31* 35* 11*7*

0hh1

Level=0

00

01

10

11

000

001

010

011

Next=1

PRIMARYPAGES

44* 36*

32*

25*9* 5*

14* 18*10*30*

31* 35* 11*7*

OVERFLOWPAGES

43*

00100


58/63

Example: End of a Round

0hh1

22*

00

01

10

11

000

001

010

011

00100

Next=3

01

10

101

110

Level=0

PRIMARYPAGES

OVERFLOW

PAGES

32*

9*

5*

14*

25*

66* 10*18* 34*

35*31* 7* 11* 43*

44*36*

37*29*

30*

0hh1

37*

00

01

10

11

000

001

010

011

00100

10

101

110

Next=0

Level=1

111

11

PRIMARY

PAGES OVERFLOWPAGES

11

32*

9* 25*

66* 18* 10* 34*

35* 11*

44* 36*

5* 29*

43*

14* 30* 22*

31* 7*

50*


59/63

LH Described as a Variant of EH

The two schemes are actually quite similar: Begin with an EH index where directory hasNelements.

Use overflow pages, split buckets round-robin.

First split is at bucket 0. (Imagine directory being doubled at

this point.) But elements , , ... are the same.So, need only create directory elementN, which differs from 0,now.

When bucket 1 splits, create directory elementN+1, etc.

So, directory can double gradually. Also, primary bucketpages are created in order. If they are allocatedin sequencetoo (so that finding ith is easy), we actually dont need adirectory!


60/63

Useful Links http://www.cs.ucla.edu/classes/winter03/cs143/l1/han

douts/hash.pdf

http://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htm

http://www.smckearney.com/adb/notes/lecture.exten

dible.hashing.pdf
http://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htm


61/63

Summary Hash-based indexes: best for equality searches, cannot

support range searches.

Static Hashing can lead to long overflow chains.

Extendible Hashing avoids overflow pages by splitting afull bucket when a new data entry is to be added to it.(Duplicates may require overflow pages.)

Directory to keep track of buckets, doubles periodically.

Directoryless schemes (linear dynamic hashing) available Can get large with skewed data; additional I/O if this does

not fit in main memory.


62/63

Summary Linear Hashing avoids directory by splitting buckets

round-robin, and using overflow pages.

Overflow pages not likely to be long.

Duplicates handled easily.

Space utilization could be lower than Extendible Hashing,since splits not concentrated on `dense data areas.


63/63

Check ListWhat is the intuition behind hash-structured indexes?

Why are they especially good for equality searches butuseless for range selections?

What is Extendible Hashing? How does it handle

search, insert, and delete?

What is Linear Hashing?

What are the similarities and differences betweenExtendible and Linear Hashing?

How does perfect hash function works?

w5ch11 hash index

Documents