Interpolation search

Perl, Itai, Avni CACM 21:7(1978), 550-553.

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 relatively uniformly.

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

INTERPOLATON SEARCH
  i := 1
  j := n
  LO := a_i
  HI := a_j
  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 := a_m
        if (x > MID) then
              i := m+1
              LO := MID
        else if (x < MID) then
              j := m-1
              HI := MID
        else
              return MID
  if (x ≠ a_i) then i := 0
  return i

The algorithm can be modified to use only one element comparison per loop iteration,
but care must be taken to avoid an infinite loop.

INTERPOLATON SEARCH
  i := 1
  j := n
  LO := a_i
  HI := a_j
  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) ]
                prevent probe at last index, thereby avoiding infinte loop
        if (m = j) then m := j-1
        MID := a_m
        if (x > MID) then
              i := m+1
              LO := MID
        else
              j := m
              HI := MID
  if (x ≠ a_i) then i := 0
  return i

Worst case complexity

O(n)

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

Average case complexity

O(lg lg n)

Dan Hirschberg
Last modified: Jul 2, 2013