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Search algorithms

Search algorithms. Outline. In this presentation, we will cover: Linear search Binary search Interpolation search, and A hybrid of these three. Linear search. Linearly searching an ordered list is straight-forward : We will always search on the interval [ a , b ]

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Search algorithms

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  1. Search algorithms

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

  3. Linear search Linearly searching an ordered list is straight-forward: • We will always search on the interval [a, b] template <typename Type> boollinear_search( Type const &obj, Type *array, int a, int b ) { for ( inti = a; i<= b; ++i ) { if ( array[i] == obj ) { return true; } } return false; }

  4. Binary search A binary search tests the middle entry and continues searching either the left or right halves, as appropriate: template <typename Type> boolbinary_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. Binary search Question: • Which of these should you choose? Does it matter, and if so, why? intb = a + (c - a)/2; intb = (a + b)/2; • Suppose both a, b < 231 but a + b ≥ 231

  6. 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. Binary search For very small lists, it would be better to use a linear search: template <typename Type> boolbinary_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. 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?

  9. 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

  10. Interpolation search Use linear interpolation to make a better guess as to where to look template <typename Type> boolinterpolation_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 ); }

  11. 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:

  12. Run times of searching algorithms The following table summarizes the run times:

  13. 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 <typename Type> boolharder_search( Type const &obj, Type *array, int a, int c ) { intuse_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 which alternates between interpolation andbinary searches

  14. Run Times of Searching Algorithms Now, the worst case is that of binary search while the best is that of interpolation search

  15. 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. 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.

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