Interpolation Search
Douglas Wilhelm Harder, M.Math. LEL
Department of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario, Canada
ece.uwaterloo.ca
[email protected]
© 2012 by Douglas Wilhelm Harder. Some rights reserved.
Interpolation Search
Outline
In this laboratory, we will
– Determine how to make very fast searches on appropriately
ordered data
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Interpolation Search
Outcomes Based Learning Objectives
By the end of this laboratory, you will:
– Understand how to quickly find a point in a sorted list with certain
properties
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Interpolation Search
The Problem
Given a solution to an IVP in the form of two vectors tout
and yout, given a point t where
t0 < t < tfinal
how do we find the index k such that
tout,k < t < tout,k + 1
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Interpolation Search
Finding the Point
Possible solutions:
– Linear search—just go through the vector t_out
• Problem: very slow: O(n)
– Binary search?
• Faster—O( ln(n) )—but still slower than necessary...
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Interpolation Search
Finding the Point
Assume equal spacing—the same h is used
– A binary search would start us at k = 9
– Suppose t is very close to t5—should we not start close to 5 and
not 9?
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Interpolation Search
Finding the Point
Question, proportionally, how far is t into the interval
[t0, tfinal]?
t  t0
tfinal  t0
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Interpolation Search
Finding the Point
Question, proportionally, how far is t into the interval [t0,
tfinal]?
t  t0
tfinal  t0
Why not calculate

t  t0 
k  1  17  1

t

t
final
0

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Interpolation Search
Finding the Point
Consider binary search:
– A binary search starts with [0, 17]
– Based on where t falls relative to t9, we continue with [0, 9] or
[9, 17]
With the interpolation search, use this interpolated point
– Again, we start with [0, 17]
– If we calculate

t  t0 
k  1  17  1

t

t
final
0

and get the value 5, we continue with [0, 5] or [5, 17]
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Interpolation Search
Interpolation Search
Being careful about how we update allows us to
implement:
function [k] = interpolation_search( t, t_vec )
lower = 1;
upper = length( t_vec );
while true
k = lower + floor( (upper - lower)*(t - t_vec(k1))/(t_vec(k2) - t_vec(k1)) );
if t == t_vec(k)
return
elseif t > t_vec(k) && t < t_vec(k + 1)
return;
elseif t < t_vec(k)
upper = k;
else % t > t_vec(k + 1)
lower = k + 1;
end
end
end
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Interpolation Search
Interpolation Search
Why use an interpolation search?
If the points within t_vec are distributed uniformly, the
run time will be Q( ln(ln(n)) )
– The distribution of widths of n intervals in a uniform distribution
on [a, b] is given by the exponential distribution
f :t
Distribution function:
n  b n a t
e
ba
t
F :t
n  b n a
0 b  a e d
Cumulative distribution:
1 e

n
t
ba
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Interpolation Search
Interpolation Search
You can try this:
N = 1000;
t = sort( rand( N + 1, 1 ) ); % 1001 uniformly random entries
dt = sort( diff( t ) );
% a sorted list of the interval widths
plot( dt, linspace( 0, 1, N ), '.' );
% the cumulative distribution
lambda = N/1; % the width is 1
ts = linspace( 0, 10*1/lambda, 10000 );
hold on
plot( ts, 1 - exp(-lambda*ts), 'r' );
% the actual cumulative distribution
% of the exponential function
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Interpolation Search
Interpolation Search
Our data is even more well behaved than a uniform
distribution
If the entries of t_vec are equally spaced, an
interpolation search will run in Q(1) time
For many of the returned t_out vectors, the width sizes
have been reasonably consistent except when we have
discontinuities
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Interpolation Search
Interpolation Search
Therefore, given t_out and a point t, we can find which
points
surround it
[t8b, y8b] = dp45( @f8b, [1, 2],
t8b =
1.0000
1.1000
1.3000
y8b =
1.5000
1.6508
1.9047
1.7000
1.3609
1.2663
[1.5, 1.7]', 0.1, 1e-1 )
1.5000
1.7000
1.9000
2.0000
2.1816
1.5376
2.5316
1.9875
2.9883
2.6124
3.2688
3.0092
interpolation_search( 1.253, t6c )
ans =
4
interpolation_search( 1.793, t6c )
ans =
8
interpolation_search( 1.813, t6c )
ans =
9
Searching for 10000 numbers on [1, 2]:
1834 intervals were found in one step
5932 intervals were found in two steps
2234 intervals were found in three steps
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Summary
In this quick topic, we’ve looked at interpolation searches
– Finding points in o( ln(n) ) time
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Interpolation Search
References
[1]
Glyn James, Modern Engineering Mathematics, 4th Ed., Prentice Hall,
2007, p.778.
[2]
Glyn James, Advanced Modern Engineering Mathematics, 4th Ed.,
Prentice Hall, 2011, p.164.
[3]
J.R. Dormand and P. J. Prince, "A family of embedded Runge-Kutta
formulae," J. Comp. Appl. Math., Vol. 6, 1980, pp. 19-26.
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