Interpolation search

• Given: x, and a₁ < ... < a_n
• Return: i such that x = a_i
  0 if no such i exists
• Algorithm assumption: to have good (doubly logarithmic) time complexity, the values in {a_i} are distributed uniformly w.r.t. requests.

EXAMPLE: To look up cat in a dictionary, you expect to find cat near the beginning.

INTERPOLATON SEARCH
  i := 1
  j := n
  LO := ai
  HI := aj
  if x < LO then  return 0
  if x ≥ HI then i := j
                loop invariant:  x ≥ LO and x ≤ HI
  while (i<j) do
        m := [ i + (j-i)*(x-LO) / (HI-LO) ]
        MID := am
        if (x > MID) then
              i := m+1
              LO := MID
        else
              j := m
              HI := MID
  if (x ≠ ai) then i := 0
  return i

Worst case complexity

EXAMPLE: 1,2,3,4,5,...,999,1000,10⁹
        Look for 1000

Average case complexity