1. ECE 250 Algorithms and Data Structures Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada ece.uwaterloo.ca dwharder@alumni.uwaterloo.ca © 2006-2013 by Douglas Wilhelm Harder. Some rights reserved. Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada ece.uwaterloo.ca dwharder@alumni.uwaterloo.ca © 2006-2013 by Douglas Wilhelm Harder. Some rights reserved. Search algorithms

2. 2 Outline In this presentation, we will cover: –Linear search –Binary search –Interpolation search, and –A hybrid of these three

3. 3 Search algorithms Linear search Linearly searching an ordered list is straight-forward: –We will always search on the interval [a, b] template 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; }

4. 4 Search algorithms Binary search A binary search tests the middle entry and continues searching either the left or right halves, as appropriate: template 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; }

5. 5 Search algorithms 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 < 2 31 but a + b ≥ 2 31

6. 6 Search algorithms 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?

7. 7 Search algorithms Binary search For very small lists, it would be better to use a linear search: template 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 ); }

8. 8 Search algorithms 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? 1358101416192124354145475163

9. 9 Search 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 19 26 dog 0. 3 14 6 26 Therefore, ( cat + dog )/2 = 0. 5 14 25 26 / 2 ≈ 0.10722 10

10. 10 Search algorithms Interpolation search Use linear interpolation to make a better guess as to where to look template bool interpolation_search( Type const &obj, Type *array, int a, int c ) { while ( c – a > 16 ) { int b = a + static_cast ( ((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 ); }

11. 11 Search algorithms Interpolation search Interpolation search is best if the list is: –Perfectly uniform:  (1) –Uniformly distributed: O(ln(ln(n)) Unfortunately, interpolation search may fail dramatically: –Consider searching this array for 2: 11111111111111216

12. 12 Search algorithms Run times of searching algorithms The following table summarizes the run times: AlgorithmBest Case Average Case Worst Case Linear Search O(n) Binary Search O(ln(n)) Interpolation Search  (1) O(ln(ln(n)))O(n)

13. 13 Search algorithms Harder search A hybrid of the two algorithm has the best of both worlds: –Start with interpolation search, use binary if interpolation doesn’t work template 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 ( ((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 which alternates between interpolation and binary searches

14. 14 Search algorithms Run Times of Searching Algorithms Now, the worst case is that of binary search while the best is that of interpolation search AlgorithmBest Case Average Case Worst Case Linear search O(n) Binary search O(ln(n)) Interpolation search  (1) O(ln(ln(n)))O(n) Harder search  (1) O(ln(ln(n)))O(ln(n))

15. 15 Search algorithms 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

16. 16 Search algorithms 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.