ECE 250 Algorithms and Data Structures

Douglas Wilhelm Harder, M.Math. LELDepartment of Electrical and Computer EngineeringUniversity of WaterlooWaterloo, Ontario, Canada

ece.uwaterloo.cadwharder@alumni.uwaterloo.ca

© 2006-2013 by Douglas Wilhelm Harder. Some rights reserved.

Search algorithms

2Search algorithms

Outline

In this presentation, we will cover:– Linear search– Binary search– Interpolation search, and– A hybrid of these three

3Search algorithms

Linear search

Linearly searching an ordered list is straight-forward:– We will always search on the interval [a, b]

template <typename Type> bool linear_search( Type const &obj, Type *array, int a, int b ) { for ( int i = a; i <= b; ++i ) { if ( array[i] == obj ) { return true; } }

return false;}

4Search algorithms

Binary search

A binary search tests the middle entry and continues searching either the left or right halves, as appropriate:

template <typename Type>bool binary_search( Type const &obj, Type *array, int a, int c ) { while ( a <= c ) { int b = a + (c - a)/2;

if ( obj == array[b] ) { return true; } else if ( obj < array[b] ) { c = b – 1; } else { assert( obj > array[b] ); a = b + 1; } }

return false;}

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Binary search

Question:– Which of these should you choose? Does it matter, and if so, why?

int b = a + (c - a)/2;int b = (a + b)/2;

– Suppose both a, b < 231 but a + b ≥ 231

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Binary search

Question:– Should a binary search be called on a very small list?

• Hint: What is involved in the overhead of making a function call?

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Binary search

For very small lists, it would be better to use a linear search:

template <typename Type>bool binary_search( Type const &obj, Type *array, int a, int c ) { while ( c – a > 16 ) { int b = a + (c - a)/2;

if ( obj == array[b] ) { return true; } else if ( obj < array[b] ) { c = b – 1; } else { assert( obj > array[b] ); a = b + 1; } }

return linear_search( obj, array, a, c );}

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Binary search

Consider the following weakness with a binary search:– Who opens the telephone book at Larson—Law (the middle) when

searching for the name “Bhatti”?

Binary search, however, always searches the same sequence of entries– Consider searching for 5 in this list:

– Suggestions?

1 3 5 8 10 14 16 19 21 24 35 41 45 47 51 63

9Search algorithms

Binary search

We will assume that the object being searched for has properties similar to the real number where we can do linear interpolation

If we are dealing with a dictionary, we may need a refined definition of a linear interpolation based on the lexicographical ordering– Consider a string as the fractional part of a base 26 real number:

cat 0 . 2 0 1926

dog 0 . 3 14 6 26

Therefore, (cat + dog)/2 = 0 . 5 14 2526 / 2 ≈ 0.1072210

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Interpolation search

Use linear interpolation to make a better guess as to where to look

template <typename Type>bool interpolation_search( Type const &obj, Type *array, int a, int

c ) { while ( c – a > 16 ) { int b = a + static_cast<int>( ((c - a)*(obj – array[a])) / (array[c] – array[a]) );

if ( obj == array[b] ) { return true; } else if ( obj < array[b] ) { c = b – 1; } else { assert( obj > array[b] ); a = b + 1; } } return linear_search( obj, array, a, b );}

11Search algorithms

Interpolation search

Interpolation search is best if the list is:– Perfectly uniform: Q(1)– Uniformly distributed: O(ln(ln(n))

Unfortunately, interpolation search may fail dramatically:– Consider searching this array for 2:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 16

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Run times of searching algorithms

The following table summarizes the run times:

Algorithm Best Case Average Case Worst Case

Linear Search O(n) O(n) O(n)

Binary Search O(ln(n)) O(ln(n)) O(ln(n))

Interpolation Search Q(1) O(ln(ln(n))) O(n)

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Harder search

A hybrid of the two algorithm has the best of both worlds:– Start with interpolation search, use binary if interpolation doesn’t worktemplate <typename Type>bool harder_search( Type const &obj, Type *array, int a, int c ) { int use_binary_search = false;

while ( c – a > 16 ) { int midpoint = a + (c - a)/2; // point from binary search int b = use_binary_search ? midpoint : a + static_cast<int>( ((c - a)*(obj – array[a])) / (array[c] – array[a]) );

if ( obj == array[b] ) { return true; } else if ( obj < array[b] ) { c = b – 1; use_binary_search = ( midpoint < b ); } else { a = b + 1; use_binary_search = ( midpoint > b ); } }

return linear_search( obj, array, a, b );}

Based on introspective search whichalternates between interpolation andbinary searches

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Run Times of Searching Algorithms

Now, the worst case is that of binary search while the best is that of interpolation search

Algorithm Best Case Average Case Worst Case

Linear search O(n) O(n) O(n)

Binary search O(ln(n)) O(ln(n)) O(ln(n))

Interpolation search Q(1) O(ln(ln(n))) O(n)

Harder search Q(1) O(ln(ln(n))) O(ln(n))

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Summary

Searching a list is reasonably straight-forward, but there are some twists– In some cases, a linear search is simply quicker– Binary search has logarithmic run times– Interpolation search can be very good in specific cases– A hybrid prevents the worst-case scenario for an interpolation search

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References

Wikipedia, http://en.wikipedia.org/wiki/Search_algorithm

These slides are provided for the ECE 250 Algorithms and Data Structures course. The material in it reflects Douglas W. Harder’s best judgment in light of the information available to him at the time of preparation. Any reliance on these course slides by any party for any other purpose are the responsibility of such parties. Douglas W. Harder accepts no responsibility for damages, if any, suffered by any party as a result of decisions made or actions based on these course slides for any other purpose than that for which it was intended.