Hashing

1
Hashing

• ? ?? ????? ????????

2
??????????????????????????????? hash table

• Insert
• Delete
• find (constant time)
• sort ??????
• Findmin findmax ??????
• ????????????????????? binary search tree

3
Hash table

• ????? key ??? value ??????????? Map ??
• ?????? key ?????? hash function ??????????? index
  ????????? value
• ????????????????????????
• ?????????
• ?????????????????????????????????? index ???????
  (????????????????????????????????????????)

4
Hash function

• ??????????????????????????? ?????????
• ????????? mod tableSize
• ?????????????? 10, 20, 30, ?????????????????????
  ?
• ??????????????? String ??? ??????????????????
  hash function ???????????

5
Hash function (1st example)

• ?????? ASCII ????????????????
• public static int hash(String key, int
  tableSize)
• int hashVal 0
• for(int i 0 iltkey.length() i)
• hashVal key.charAt(i)
• return hashValtableSize

6

• ??????????????????????????
• ?????????????????????? ?????????????????
• ASCII ???????????????????? 127
• ????????????????????????????????????? 1278
• ???????????? ???????????

???????????

Index ????????????????????
7
Hash function (2nd example)

• ??????????????????????????????????????????????????
  ?????????????
• ???????????????????????????
• public static int hash(String key, int
  tableSize)
• return (key.charAt(0) 27key.charAt(1) 729
  key.charAT(2))tableSize

2727
????????????? ??????????
???????????????????????????????? (10007
????????????????????????????????? prime
?????????????????????????)
8

• ??????????????????????????????
• ???????????????????????
• ???????????????????????????

9
Hash function (3rd example)

• ????? polynomial function ??? 37 ?????? Horners
  Rule
• ?????????????? k0 37k1 3737k2 ?????????
• (k237)k137 k0
• Horner rule ???????????????? n ???
  ????????????????????????

10

• public static int hash(String key, int
  tableSize)
• int hashVal 0
• for(int i 0 iltkey.length() i)
• hashVal 37hashValkey.charAt(i)
• hashVal tableSize
• if(hashVallt0)
• hashVal tableSize
• return hashVal

????? overflow
11

• ??????????????????? ????????????????????
• ???????????????? ????????????
• ?????????????????????????? ???????????????????????
  ????????????????????????????????
• ???????? hash function ?????????????????????100
• ?????????????????????????????????? ???????????
  collision
• ????????????????????????????????? collision

12
??????? separate chaining

• ???????????????? linked list
• ?????????? ????? hash function ???????????????????
  ????
• ??????????????? ?? ??? hash function
  ??????????????????? ??????????????????????????????
  ????????????????? ?????????????????????????
  (?????????????????????????????????????????????????
  ?)

13
??????????????????????

1. public interface Hashable
3. /
4. Compute a hash function for this
   object.
5. _at_param tableSize the hash table size.
6. _at_return (deterministically) a number
   between
7. 0 and tableSize-1, distributed
   equitably.
8. /
9. int hash( int tableSize )

14
?????????????????

• Public class Student implements Hashable
• private String name
• private double number
• private int year
• public int hash(int tableSize)
• return SeparateChainingHashTable.hash(name,
  tableSize)
• public boolean equals(Object rhs)
• return name.equals(((Student)rhs).name)

???????????? static method ???????
15

1. public class SeparateChainingHashTable
3. /
4. Construct the hash table.
5. /
6. public SeparateChainingHashTable( )
8. this( DEFAULT_TABLE_SIZE )
10. /
11. Construct the hash table.
12. _at_param size approximate table size.
13. /
14. public SeparateChainingHashTable( int
    size )
16. theLists new LinkedList nextPrime(
    size )
17. for( int i 0 i lt theLists.length
    i )
18. theLists i new LinkedList(
    )

16

1. /
2. Insert into the hash table. If the
   item is
3. already present, then do nothing.
4. _at_param x the item to insert.
5. /
6. public void insert( Hashable x )
8. LinkedList whichList theLists
   x.hash( theLists.length )
9. LinkedListItr itr whichList.find( x
   )
10. if( itr.isPastEnd( ) )
11. whichList.insert( x,
    whichList.zeroth( ) )
13. /
14. Remove from the hash table.
15. _at_param x the item to remove.
16. /
17. public void remove( Hashable x )

??? Student ????????
17

1. /
2. Find an item in the hash table.
3. _at_param x the item to search for.
4. _at_return the matching item, or null if
   not found.
5. /
6. public Hashable find( Hashable x )
8. return (Hashable)theLists x.hash(
   theLists.length ) .find( x ).retrieve( )
10. /
11. Make the hash table logically empty.
12. /
13. public void makeEmpty( )
15. for( int i 0 i lt theLists.length
    i )
16. theLists i .makeEmpty( )

18

1. /
2. A hash routine for String objects.
3. _at_param key the String to hash.
4. _at_param tableSize the size of the hash
   table.
5. _at_return the hash value.
6. /
7. public static int hash( String key, int
   tableSize )
9. int hashVal 0
10. for( int i 0 i lt key.length( )
    i )
11. hashVal 37 hashVal
    key.charAt( i )
12. hashVal tableSize
13. if( hashVal lt 0 )
14. hashVal tableSize
15. return hashVal

19

1. private static final int
   DEFAULT_TABLE_SIZE 101
2. / The array of Lists. /
3. private LinkedList theLists
4. /
5. Internal method to find a prime number
   at least as large as n.
6. _at_param n the starting number (must be
   positive).
7. _at_return a prime number larger than or
   equal to n.
8. /
9. private static int nextPrime( int n )
11. if( n 2 0 )
12. n
13. for( !isPrime( n ) n 2 )
15. return n

20

1. /
2. Internal method to test if a number is
   prime.
3. Not an efficient algorithm.
4. _at_param n the number to test.
5. _at_return the result of the test.
6. /
7. private static boolean isPrime( int n )
9. if( n 2 n 3 )
10. return true
11. if( n 1 n 2 0 )
12. return false
13. for( int i 3 i i lt n i 2 )
14. if( n i 0 )
15. return false
16. return true

21

• // Simple main
• public static void main( String args
  )
• SeparateChainingHashTable H new
  SeparateChainingHashTable( )
• final int NUMS 4000
• final int GAP 37
• System.out.println( "Checking... (no
  more output means success)" )
• for( int i GAP i ! 0 i ( i
  GAP ) NUMS )
• H.insert( new MyInteger( i ) )
• for( int i 1 i lt NUMS i 2 )
• H.remove( new MyInteger( i ) )
• for( int i 2 i lt NUMS i2 )
• if( ((MyInteger)(H.find( new
  MyInteger( i ) ))).intValue( ) ! i )
• System.out.println( "Find
  fails " i )

22
?????

• Load factor
• ???????????????????????????????????
• Search time time to do hashing time to search
  list
• constant time to search list
• Unsuccessful search
• Search time ???????????? ??? load factor

23

• Successful search
• ???????????????? ??????????????????
  ?????????????????????????? 0 ?????????
• ????????????????????????????????? N
  ?????????????????????? M ?????
• ???????? ???????????????????????????????????? N-1
  ???
• ????????????????????????????????????????????????
  (N-1)/M ???
• lambda- (1/M)
• lambda ??????? ???????? M ????????
• ????????????? ??????????????????????????????
  ??????? lambda/2
• ?????? ??????????????????????????????????? 1
  (lambda/2)
• ?????????????? ????????????????? ??????????? load
  factor

24
?????????? Open addressing

• ?????????????????? ???????????????
• ????? collision ??????????????????????????????????
  ????
• ????????????? h0(x), h1(x),
• hi(x)hash(x)f(i)tableSize, f(0)0
• ??????????????????????? ????????????????????
  ????????????????????
• Load factor lt0.5

25
Open addressing ??? linear probing

• F ?????????????????????????? i
• ????????????? f(i)i
• ??????????????????????????????????????????????????
  ????
• ?????????? ???????????????????????????

?????????????????????? (primary clustering)
????????????? collide ?????????
??????????????????????????????????
26
Open addressing ??? quadratic probing

• ????????? primary clustering
• ????????????????????????? f(i)i2
• ?????????? hi(x)hash(x)f(i)tableSize

a
??? b ???????????????? a ????????? 12
??????????????
??? c ???????????????? a ????????? 12 ??????? b
????????????? 22 ??????????????
27

• ????????????????????????????????
  ?????????????????????????????????
  ?????????????????????????????????
• ??????????????????????????????????????????????????
  ????????????????? ????????????????????????????????
  ???????

28
???????

• ????????????????????????????????????????
• ??? (h(x)i2) mod tableSize
• (h(x)j2) mod tableSize
• Prove by contradiction
• ???????????????????????????????????????????? i
  ?????????? j

???????????????????????????
29
???????(???)

• i-j 0 ???????????????? ??????????????????????????
  ?????????????
• ij0 ?????????????? ?????????????
• ?????????????????????? ???????????????
  ??????????????????? ??????
• ??????? ??????????????? ???????????????????
• ????????????????????????????????????????????????
  ?????????????????????

30
???????????????????????????

• ?????????? ????????????????????????????????????
• ???????? ????????? 16 ????????????????????????????
  ????
• ??? quadratic probing

32
12
22
42
52
62
72
??????????????????????
31
??????????????????????

• ????????????????????? ????????????????(??????
  insert ?????????????????????????????)
  ?????????????????????????????? ???????????????????
  ??????????????

32
12
22
42
52
62
72
?????????????? lazy deletion ?????????????????????
???????????
32
Open addressing implementation

• class HashEntry
• Hashable element // the element
• boolean isActive // false is deleted
• public HashEntry( Hashable e )
• this( e, true )
• public HashEntry( Hashable e, boolean i
  )
• element e
• isActive i

33

• public class QuadraticProbingHashTable
• private static final int DEFAULT_TABLE_SIZE
  11
• / The array of elements. /
• private HashEntry array // The
  array of elements
• private int currentSize // The
  number of occupied cells
• public QuadraticProbingHashTable( )
• this( DEFAULT_TABLE_SIZE )
• /
• Construct the hash table.
• _at_param size the approximate initial
  size.
• /
• public QuadraticProbingHashTable( int
  size )
• allocateArray( size )
• makeEmpty( )

nonactive
null
active
34

• /
• Internal method to allocate array.
• _at_param arraySize the size of the
  array.
• /
• private void allocateArray( int arraySize
  )
• array new HashEntry arraySize
• /
• Make the hash table logically empty.
• /
• public void makeEmpty( )
• currentSize 0
• for( int i 0 i lt array.length i
  )
• array i null

35

• /
• Return true if currentPos exists and
  is active.
• _at_param currentPos the result of a call
  to findPos.
• _at_return true if currentPos is active.
• /
• private boolean isActive( int currentPos
  )
• return array currentPos ! null
  array currentPos .isActive

36

• /
• Method that performs quadratic probing
  resolution.
• _at_param x the item to search for.
• _at_return the position where the search
  terminates.
• /
• private int findPos( Hashable x )
• / 1/ int collisionNum 0
• / 2/ int currentPos x.hash( array.length
  )
• / 3/ while( array currentPos ! null
• !array currentPos
  .element.equals( x ) )
• / 4/ currentPos 2 collisionNum
  - 1 // Compute ith probe
• / 5/ if( currentPos gt array.length )
  // Implement the mod
• / 6/ currentPos - array.length
• / 7/ return currentPos

f(i)i2f(i-1)2i-1
37

• /
• Find an item in the hash table.
• _at_param x the item to search for.
• _at_return the matching item.
• /
• public Hashable find( Hashable x )
• int currentPos findPos( x )
• return isActive( currentPos ) ? array
  currentPos .element null

38

• /
• Insert into the hash table. If the
  item is
• already present, do nothing.
• _at_param x the item to insert.
• /
• public void insert( Hashable x )
• // Insert x as active
• int currentPos findPos( x )
• if( isActive( currentPos ) )
• return //x is already
  inside, so do nothing
• array currentPos new HashEntry(
  x, true )
• // Rehash see Section 5.5
• if( currentSize gt array.length / 2
  )
• rehash( )

39
???? O(N) ??????? N ????????? rehash
???????????????????? 2N ???????????? ???????
rehash ?????????????????????????????

• /
• Expand the hash table.
• /
• private void rehash( )
• HashEntry oldArray array
• // Create a new double-sized,
  empty table
• allocateArray( nextPrime( 2
  oldArray.length ) )
• currentSize 0
• // Copy table over
• for( int i 0 i lt oldArray.length
  i )
• if( oldArray i ! null
  oldArray i .isActive )
• insert( oldArray i .element
  )
• return

40
????????? rehash

• Rehash ?????????????????
• ?????????????????????????????
• ??????? insert ?????????????????? ????
• ??????????????????????????????
• ????????????????????????????? ???? insert
  ????????
• ??????????????????? ???????????????????????

41
hash, nextPrime, isPrime ??????????

• /
• Remove from the hash table.
• _at_param x the item to remove.
• /
• public void remove( Hashable x )
• int currentPos findPos( x )
• if( isActive( currentPos ) )
• array currentPos .isActive
  false

42
?????????? quadratic probing

• Secondary clustering
• ?????? double hashing ???????
• f(i) ihash2(x) ?????????????????????????
  hash2(x), 2 hash2(x), ?????????
• ????????????????????????????????????????
• ??????????????????? hash2(x) ???? x9
  ???????????????????????? 99 ????????? 0 ????
• ????????????????????????? 0
• ???????????????????????????????????????

43
???????? hash2

• hash2(x)R-(xR) ??? R ?????????????????????????
  tableSize
• ??????????????? 16 ?????? 9, 25, 26, 41, 42, 58

???????? hash ?????????? xtableSize
26 9 25
25 ???????????????? 13-(2513)1
26 ???????????????? 13-(2613)13
44
41 26 9 25 42
41 ???????????????? 13-(4113)11
42 ???????????????? 13-(4213)10
??? 42 ???????????? ???????????????????? 210
45
58 41 26 9 25 42
58 ????? ?????????????? 13-(5813)7