Insight on how to improve the search performance of arrays

  • Fact on arrays:

    • Array access is (very) fast if access uses an array index

  • Fact on dictionaries:

    • Entries in a dictionary are looked up using its key

  • The problem with the ArrayMap implementation of the dictionary:

    • Entries of the dictionary are stored using an index that is unrelated to the key

Insight on how to improve the search performance of arrays

  • How to improve the search operation for a dictionary stored in an array:

    • Find a way to relate (= map) the key k to an index h of the array: 

          h = hashFunction( k )

    • Store the entry (k, v) at index h in the array

    Example: how to store a map (dictionary) using hashing

  • This way of storing data into a array is called hashing

Hash functions

  • Hash function H( ):

    • Hash function is a function that maps a key k to a number h in the
      range [0, M-1] where M = length of the array

            I.e.:   h = H(k)   where   h ∈ [0..(M-1)]

    • is consistent (= always gives the same answer for a given key)

    • is uniform (= function values are distributed "evenly" across [0..(M-1)])

  • A hash function is usually specified as the composition of 2 functions: 

           H(k) = H2( H1(k) )

    where:

    • H1(k) = the hash code function that returns the integer value of the key k

    • H2(x) = a compression function that maps a value x uniformly to [0..(M-1)]

The hash code of a key

  • Fact:

    • All data inside a computer is stored as a binary number

  • The Object class in Java contains a hashCode() method that returns the data stored in the Object as an integer

  • Examples: 

      Integers (byte, short, int, long):           

            stored as binary numbers

  Floating point numbers (float, double):      

            stored as 2 binary numbers 

 Characters:                                   

            stored as binary numbers in Unicode

  • We can use the hashCode() method as our h1(k) function

DEMO: 15-hashing/03-hashcode/HashCode.java

The compression function h2(x)

  • Notice from the previous discussion on the hash code H1(k):

    • H1(k) uses the data stored in the key k to compute (deterministically) a hash code value

  • The compression function H2(x) has 2 purpose:

    1. Make sure that the return value is in the range [0..(M-1)] (where M = size of array)

    2. Scatter/randomize the input value x = H1(k), so that the value H2(x) is evenly/uniformly distributed over the range [0..(M-1)]

  • Why use uniform randomization ?

    • The array element used to store the dictionary entry (k, v) is: 

        array index = H(k) = H2( H1( k ) )

    • Uniform randomization will minimize the likelihood/chance that 2 different keys being hashed to the same value (= array index) (a.k.a. collision)

Commonly used compression function

  • A commonly used compression function is the Multiply Add Divide (MAD) function: 

       H2(x) = ( ( ax + b ) % p ) % M    where p = a prime number
             ^^^^^^^^^^^^^^
                randomizes

  • In my examples, I will use: 

       p = 109345121
   a = 123
   b = 456

    Note:

    • p must be greater than M (i.e.: p > M) -- otherwise, you will not use the full capacity of the array

DEMO: 15-hashing/03-hashcode/HashValue.java

Summary on the hashing technique

  • How to improve the search operation for a dictionary stored in an array:

    • Compute the hash value for a given key k: 

          h = H2( H1( k ) )

    H1(k) = k.hashCode()
    H2(x) = Math.abs( a*x + b ) ) % p % M

    • Store the entry (k, v) at index h in the array

    Example: how to store a map (dictionary) using hashing