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Dictionaries, Tables Hashing

• TCSS 342

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The Dictionary ADT

• a dictionary (table) is an abstract model of a
  database
• like a priority queue, a dictionary stores
  key-element pairs
• the main operation supported by a dictionary is
  searching by key

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Examples

• Telephone directory
• Library catalogue
• Books in print key ISBN
• FAT (File Allocation Table)

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Main Issues

• Size
• Operations search, insert, delete, ??? Create
  reports??? List?
• What will be stored in the dictionary?
• How will be items identified?

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The Dictionary ADT

• simple container methods
• size()
• isEmpty()
• elements()
• query methods
• findElement(k)
• findAllElements(k)

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The Dictionary ADT

• update methods
• insertItem(k, e)
• removeElement(k)
• removeAllElements(k)
• special element
• NO_SUCH_KEY, returned by an unsuccessful search

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Implementing a Dictionary with a Sequence

• unordered sequence
• searching and removing takes O(n) time
• inserting takes O(1) time
• applications to log files (frequent insertions,
  rare searches and removals) 34 14 12 22 18

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Implementing a Dictionary with a Sequence

• array-based ordered sequence (assumes keys can
  be ordered)- searching takes O(log n) time
  (binary search)- inserting and removing takes
  O(n) time- application to look-up tables
  (frequent searches, rare insertions and removals)

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Binary Search

• narrow down the search range in stages
• high-low game
• findElement(22)

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Binary Search
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Pseudocode for Binary SearchAlgorithm

• BinarySearch(S, k, low, high)if low high then
  return NO_SUCH_KEYelse mid (lowhigh) /
  2if k key(mid) then return key(mid)else
  if k k, low, mid-1)else return BinarySearch(S,
  k, mid1, high)

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Running Time of Binary Search

• The range of candidate items to be searched is
  halved after each comparison

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Running Time of Binary Search

• In the array-based implementation, access by rank
  takes O(1) time, thus binary search runs in O(log
  n) time
• Binary Search is applicable only to Random Access
  structures (Arrays, Vectors)

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Implementations

• Sorted? Non Sorted?
• Elementary Arrays, vectors linked lists
• Orgainization None (log file), Sorted, Hashed
• Advanced balanced trees

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Skip Lists

• Simulate Binary Search on a linked list.
• Linked list allows easy insertion and deletion.
• http//www.epaperpress.com/s_man.html

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Hashing

• Place item with key k in position h(k).
• Hope h(k) is 1-1.
• Requires unique key (unless multiple items
  allowed). Key must be protected from change (use
  abstract class that provides only a constructor).
• Keys must be comparable.

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Key class

• public abstract class KeyID
• Private Comparable searchKey
• Public KeyID(Comparable m)
• searchKey m
• //Only one constructor
• public Comparable getSearchKey()
• return searchKey

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Hash Tables

• RTT is a large phone company, and they want to
  provide enhanced caller ID capability
• given a phone number, return the callers name
• phone numbers are in the range 0 to R 10101
• n is the number of phone numbers used
• want to do this as efficiently as possible

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Alternatives

• There are a few ways to design this dictionary
• Balanced search tree (AVL, red-black, 2-4 trees,
  B-trees) or a skip-list with the phone number as
  the key has O(log n) query time and O(n) space
  --- good space usage and search time, but can we
  reduce the search time to constant?
• A bucket array indexed by the phone number has
  optimal O(1) query time, but there is a huge
  amount of wasted space O(n R)

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Bucket Array

• Each cell is thought of as a bucket or a
  container
• Holds key element pairs
• In array A of size N, an element e with key k is
  inserted in Ak.
• Table operations without searches!

(null)
(null)
Roberto
(null)


000-000-0000 000-000-0001
401-863-7639 ... 999-999-9999 Note we
need 10,000,000,000 buckets!
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Generalized indexing

• Hash table
• Data storage location associated with a key
• The key need not be an integer, but keys must be
  comparable.

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Hash Tables

• A data structure
• The location of an item is determined
• Directly as a function of the item itself
• Not by a sequence of trial and error comparisons
• Commonly used to provide faster searching.
• Comparisons of searching time
• O(n) for linear searches
• O (logn) for binary search
• O(1) for hash table

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Examples

• A symbol table constructed by a compiler.
• Stores identifiers and information about them in
  an array.
• File systems
• I-node location of a file in a file system.
• Personal records
• Personal information retrieval based on key

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Hashing Engine

• itemKey

Position Calculator
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Example

• Insert item (401-863-7639, Roberto) into a table
  of size 5
• calculate 4018637639 mod 5 4, insert item
  (401-863-7639, Roberto) in position 4 of the
  table (array, vector).
• A lookup uses the same process use the hash
  engine to map the key to a position, then check
  the array cell at that position.

401- 863-7639 Roberto
0 1 2 3
4
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Chaining

• The expected, search/insertion/removal time is
  O(n/N), provided the indices are uniformly
  distributed
• The performance of the data structure can be
  fine-tuned by changing the table size N

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From Keys to Indices

• The mapping of keys to indices of a hash table is
  called a hash function
• A hash function is usually the composition of two
  maps
• hash code map key ? integer
• compression map integer ? 0, N - 1
• An essential requirement of the hash function is
  tomap equal keys to equal indices.
• A good hash function is fast and minimizes the
  probability of collisions

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Perfect hash functions

• A perfect hash function maps each key to a unique
  position.
• A perfect hash function can be constructed if we
  know in advance all the keys to be stored in the
  table (almost never)

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A good hash function

• Be easy and fast to compute
• Distribute items evenly throughout the hash table
• Efficient collision resolution.

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Popular Hash-Code Maps

• Integer cast for numeric types with 32 bits or
  less, we can reinterpret the bits of the number
  as an int
• Component sum for numeric types with more than
  32 bits (e.g., long and double), we can add the
  32-bit components.

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Sample of hash functions

• Digit selection
• h(2536924520) 590
• (select 2-nd, 5-th and last digits).
• This is usually not a good hash function. It will
  not distribute keys evenly.
• A hash function should use every part of the key.

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Sample (continued)

• Folding add all digits
• Modulo arithmetic
• h(key) h(x) x mod table_size.
• The modulo arithmetic is a very popular basis for
  hash functions. To better the chance of even
  distribution table_size should be a prime number.
  If n is the number of items there is always a
  prime p, n

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Popular Hash-Code Maps

• Polynomial accumulation for strings of a natural
  language, combine the character values (ASCII or
  Unicode) a 0 a 1 ... a n-1 by viewing them as the
  coefficients of a polynomial a 0 a 1 x ...
  a n-1 x n-1
• For instance, choosing x 33, 37, 39, or 41
  gives at most 6 collisions on a vocabulary of
  50,000 English words.

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Popular Hash-Code Maps

• Why is the component-sum hash code bad for
  strings?

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Popular Compression Maps

• Division h(k) k mod N
• the choice N 2 k is bad because not all the bits
  aretaken into account
• the table size N is usually chosen as a
  primenumber
• certain patterns in the hash codes are propagated
• Multiply, Add, and Divide (MAD)
• h(k) ak b mod N
• eliminates patterns provided a mod N ¹ 0
• same formula used in linear congruential
  (pseudo)random number generators

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Java Hash

• Java provides a hashCode() method for the Object
  class, which typically returns the 32-bit memory
  address of the object.
• This default hash code would work poorly for
  Integer and String objects
• The hashCode() method should be suitably
  redefined by classes.

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Collision

• A collision occurs when two distinct items are
  mapped to the same position.
• Insert (401-863-9350, Andy) ? 0
• And insert (401-863-2234, Devin). 4018632234 ? 4.
  We have a collision!

401- 863-9350 Andy
401- 863-7639 Roberto
0 1 2 3
4
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Collision Resolution

• How to deal with two keys which map to the same
  cell of the array?
• Need policies, design good Hashing engines that
  will minimize collisions.

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Chaining I

• Use chaining
• Each position is viewed as a container of a list
  of items, not a single item. All items in this
  list share the same hash value.

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Chaining II
0 1 2 3 4
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Collisions resolution policies

• A key is mapped to an already occupied table
  location
• what to do?!?
• Use a collision handling technique
• Chaining (may have less buckets than items)
• Open Addressing (load factor
• Linear Probing
• Quadratic Probing
• Double Hashing

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Linear Probing

• If the current location is used, try the next
  table location
• linear_probing_insert(K)if (table is full)
  errorprobe h(K)while (tableprobe
  occupied)probe (probe 1) mod Mtableprobe
  K

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Linear Probing

• Lookups walk along table until the key or an
  empty slot is found
• Uses less memory than chaining
• dont have to store all those links
• Slower than chaining
• may have to walk along table for a long way
• Deletion is more complex
• either mark the deleted slot
• or fill in the slot by shifting some elements down

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Linear Probing Example

• h(k) k mod 13
• Insert keys
• 18 41 22 44 59 32 31 73

0 1 2 3 4 5 6 7
8 9 10 11 12
41
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32
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72
0 1 2 3 4 5 6 7
8 9 10 11 12
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Double Hashing

• Use two hash functions
• If M is prime, eventually will examine every
  position in the table
• double_hash_insert(K)if(table is full)
  errorprobe h1(K)offset h2(K)while
  (tableprobe occupied) probe (probe
  offset) mod Mtableprobe K

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Double Hashing

• Many of same (dis)advantages as linear probing
• Distributes keys more uniformly than linear
  probing does

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Double Hashing Example

• h1(K) K mod 13
• h2(K) 8 - K mod 8
• we want h2 to be an offset to add
• 18 41 22 44 59 32 31 73
• h1(44) 5 (occupied) h2(0) 8 44 ? 58 Mod 13

0 1 2 3 4 5 6 7
8 9 10 11 12
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73
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Why so many Hash functions?

• Its different strokes for different folks.
• We seldom know the nature of the object that will
  be stored in our dictionary.

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A FAT Example

• Directory Key file name. Data (time, date,
  size ) location of first block in the FAT table.
• If first block is in physical location 23 (Disk
  block number) look up position 23 in the FAT.
  Either shows end of file or has the block number
  on disk.
• Example Directory entry block 4
• FAT x x x F 5 6 10 x 23 25
• 3
• The file occupies blocks 4,5,6,10, 3.