The multiplication method

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The multiplication method

• 11.3 Hash functions

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m wants to shirk

• In the division method, m takes the
  responsibility of making good hash function.
• There are some restrictions in choosing m.
• So, A is invited to participate.

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The golden ratio divides it well

• Mark off the points F-1, 2F-1, 3F-1, ?? on
  line segment 0..1
• x denotes the fractional part of x (namely x
  mod 1).
• F-1 is the golden ratio.
• These points stay very well separated from each
  other.
• Each newly added point falls into one of the
  largest remaining intervals, and divides it in
  the golden ratio.
• This phenomenon was observed by Louis and Auguste
  Bravais, Annales des Sciences Naturelles (1837)

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An Irrational number divides it well

• The remarkable scattering property of the golden
  ratio is actually just a special case of a very
  general result.
• Conjectured by Hugo Steinhaus and first proved by
  Vera Turan Sos.

Theorem. Let ? be any irrational number. When the
points ?, 2 ?, , n ? are placed in the
line segment 0..1, the n1 line segments formed
have at most three different lengths. Moreover,
the next point (n1) ? will fall in one of
the largest existing segments.
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Practical Issues

• Easy to implement
• On most machines multiplication is faster than
  division.
• We can substitute one multiplication by shift
  operation. (choose m to be a power of 2)
• We dont need to do floating-point operation. (we
  can regard an integer A as the fraction A/w if we
  imagine the radix point to be at the left of the
  word)
• If successive keys have a large interval, A
  0.6125423371 can be recommended. (for example)

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Reference

• This presentation is just a brief summery of The
  art of computer programming, Donald E. Knuth.